THE SCIENCE AND APPLICATIONS OF ACOUSTICS - H. H. Arnold ...

THE SCIENCE ANDAPPLICATIONS OFACOUSTICSSECOND EDITIONDaniel R. RaichelCUNY Graduate CenterandSchool of Architecture, Urban Design and Landscape DesignThe City College of the City University of New YorkWith 253 Illustrations

PrefaceThe science of acoustics deals with the creation of sound, sound transmissionthrough solids, and the effects of sound on both inert and living materials. As amechanical effect, sound is essentially the passage of pressure fluctuations throughmatter as the result of vibrational forces acting on that medium. Sound possessesthe attributes of wave phenomena, as do light and radio signals. But unlike itselectromagnetic counterparts, sound cannot travel through a vacuum. In SylvaSylvarum written in the early seventeenth century, Sir Francis Bacon deemed soundto be “one of the subtlest pieces of nature,” but he complained, “the nature of soundin general hath been superficially observed.” His accusation of superficiality fromthe perspective of the modern viewpoint was justified for his time, not only foracoustics, but also for nearly all branches of physical science. Frederick V. Hunt(1905–1967), one of America’s greatest acoustical pioneers, pointed out that “theseeds of analytical self-consciousness were already there, however, and Bacon’slibel against acoustics was eventually discharged through the flowering of a clearercomprehension of the physical nature of sound.”Modern acoustics is vastly different from the field that existed in Bacon’s timeand even 20 years ago. It has grown to encompass the realm of ultrasonics andinfrasonics in addition to the audio range, as the result of applications in materialsscience, medicine, dentistry, oceanology, marine navigation, communications,petroleum and mineral prospecting, industrial processes, music and voice synthesis,animal bioacoustics, and noise cancellation. Improvements are still being madein the older domains of music and voice reproduction, audiometry, psychoacoustics,speech analysis, and environmental noise control.This text—aimed at science and engineering majors in colleges and universities,principally undergraduates in the last year or two of their programs and graduationstudents, as well as practitioners in the field—was written with the assumption thatthe users of this text are sufficiently versed in mathematics up to and including thelevel of differential and partial differential equations, and that they have taken thesequence of undergraduate physics courses that satisfy engineering accreditationcriteria. It is my hope that a degree of mathematical elegance has been sustainedhere, even with the emphasis on engineering and scientific applications. Whilethe use of SI units is stressed, very occasional references are made to physicalvii

xPrefaceReferencesBacon, Sir Francis (Lord Veralum). 1616 (published posthumously). Sylva Sylvarum. InThe Works of Sir Francis Bacon, vol. 2. 1957. Spedding, Ellis, R. L., Heath, D. D., et al.(eds.). London: Longman and Co. 1957.Hunt, Frederick Vinton. 1992. Origins in Acoustics. Woodbury, NY: Acoustical Society ofAmerica.

ContentsPrefacevii1. A Capsule History of Acoustics 12. Fundamentals of Acoustics 133. Sound Wave Propagation and Characteristics 314. Vibrating Strings 715. Vibrating Bars 896. Membrane and Plates 1117. Pipes, Waveguides, and Resonators 1318. Acoustic Analogs, Ducts, and Filters 1519. Sound-Measuring Instrumentation 17310. Physiology of Hearing and Psychoacoustics 21311. Acoustics of Enclosed Spaces: Architectural Acoustics 24312. Walls, Enclosures, and Barriers 28113. Criteria and Regulations for Noise Control 31914. Machinery Noise Control 35715. Underwater Acoustics 409xi

xiiContents16. Ultrasonics 44317. Commercial and Medical Ultrasound Applications 47918. Music and Musical Instruments 50919. Sound Reproduction 56920. Vibration and Vibration Control 58521. Nonlinear Acoustics 617Appendix A. Physical Properties of Matter 629Appendix B. Bessel Functions 633Appendix C. Using Laplace Transforms to Solve DifferentialEquations 637Index 649

1A Capsule History of AcousticsOf the five senses that we possess, hearing probably ranks second only to sightin regular usage. It is therefore with little wonder that human interest in acousticswould date to prehistoric times. Sound effects entailing loud clangorous noiseswere used to terrorize enemies in the course of heated battles; yet the gentler aspectsof human nature became manifest through the evolution of music during primevaltimes, when it was discovered that the plucking of bow strings and the pounding ofanimal skins stretched taut made for rather interesting and pleasurable listening.Life in prehistoric society was fraught with emotion, just as in the present time,so music became a medium of expression. Speech enhanced by musical inflectionbecame song. Body motion following the rhythm of accompanying music evolvedinto dance. Animal horns were fashioned into musical instruments (the Bibledescribed the ancient Israelites’ use of shofarim, made from horns of rams orgazelles, to sound alarms for the purpose of rousing warriors to battle). Ancientshepherds amused themselves during their lonely vigils playing on pipes and reeds,the precursors of modern woodwinds.Possibly the first written set of acoustical specifications may be found in the OldTestament, Exodus XXVI:7:And thou shalt make curtains of goats’ hair for a tent over the tabernacle ....The length ofeach curtain shall be thirty cubits and the breadth of each curtain shall be four cubits....Additional specifications are given in extreme detail for the construction andhanging of these curtains, which were to be draped over the tabernacle walls toensure that the curtains would hang in generous sound-absorbing folds. More finedetails on the construction of the tabernacle followed. Absolutely no substitutionof materials nor deviation from prescribed methods was permitted.With the advent of metal forming skills, newer wind instruments were constructedof metals. The march evolved from ceremonial processions, on grandmilitary and ceremonial occasions. Patriotic fervor often was elevated to a state ofhigher pitch by the blare of martial music, indeed to the point of sheer madness onthe part of the citizenry, even in modern times as epitomized during the 1930s bythe grandiose thunder of Nazi goose-stepping marches through Berlin’s boulevardsto the accompaniment of the crowds’ roar.1

2 1. A Capsule History of AcousticsWith sound as a major factor affecting human lives, it was only natural for interestin the science of sound, or acoustics, to emerge. In the twenty-seventh centuryBCE, Lin-lun, a minister of the Yellow Emperor Huangundi, was commissionedto establish a standard pitch for music. He cut a bamboo stem between the nodesto make his fundamental note, resulting in the “Huang-zhong pipe”; the othernotes took their place in a series of twelve standard pitch pipes. He also tookon the task of casting twelve bells in order to harmonize the five notes, so asto enable the composing of regal music for royalty. Archeological studies of theunearthed musical instruments attested to the high level of instrument design andthe art of metallurgy in ancient China. Approximately 2000 bce, another Chinese,the philosopher Fohi, attempted to establish a relationship between the pitch of asound and the five elements: earth, water, air, fire, and wind. The ancient Hindussystematized music by subdividing the octave into 22 steps, with a large wholetone containing four steps, a small tone assigned three, and a half tone containingtwo such steps. The Arabs carried matters further by partitioning the octave into17 divisions. But the ancient Greeks developed musical concepts similar to thoseof the modern Western world. Three tonal genders—the diatonic, the chromatic,and the enharmonic—were attributed to the gods.Observation of water waves may have influenced the ancient Greeks to surmisethat sound is an oscillating perturbation emanating from a source over large distancesof propagation. It cannot have failed to attract notice that the vibrations ofplucked strings of a lute can be seen as well as felt. The honor of being the earliestacousticians probably falls to the Greek philosopher Chrysippus (ca. 240 bce),the Roman architect-engineer Vitruvius (also known as Marcus Vitruvius Pollio,ca. 25 bce), and the Roman philosopher Severinus Boethius (480–524). Aristotle(384–322 bce) stated in rather pedantic fashion that air motion is generated bya source “thrusting forward in like movement the adjoining air, so that soundtravels unaltered in quality as far as the disturbance of the air manages to reach.”Pythagoras (570–497 bce) observed that “air motion generated by a vibrating bodysounding a single musical note is also vibratory and of the same frequency as thebody;” and it was he who successfully applied mathematics to the musical consonancesdescribed as the octave, the fifth and the fourth, and established the inverseproportionality of the length of a vibrating string with its pitch. The forerunner ofthe modern megaphone was used by Alexander the Great (400 bce) to summonhis troops from distances as far as 15 km.The principal laws of sound propagation and reflection were understood bythe ancient Greeks, and the echo figured prominently in a number of classicaltales. Quintillianus demonstrated with small straw segments the resonance of astring in air. Vitruvius, after making use of the spread of circular waves on awater’s surface as an example, went on to explain that true sound waves travelin a three-dimensional world not as circles, but rather as outwardly spreadingspherical waves. He also described the placement of rows of large empty vasesfor the purpose of improving the acoustics of ancient theaters. While there may besome question if such vases have actually been employed in these theaters (sincearcheological excavations have failed to disclose their shards), it does presage

4 1. A Capsule History of AcousticsRobert Boyle (1626–1691) with the help of his assistant Robert Hooke (1635–1703) performed a classic experiment (1660) on sound by placing a ticking watchin a partially evacuated glass chamber. He proved that air is necessary, either for theproduction or emission of sound. In this respect he disproved Athenasius Kircher’s(1602–1680) negative experiment in which the latter enclosed a bell in a vacuumcontainer and excited the bell magnetically from the exterior. Kirchner’s resultswere erroneous because he did not take the precaution to prevent the conduction ofsound through the bell’s supports to the surroundings. Francis Hausksbee (1666–1713) repeated the Boyle experiment (in a modified form) before the Royal Society.Mention should be made here of Joseph Sauveur (1653–1713) who suggestedthe term acoustics (from the Greek word for sound) for the science of sound. Indescribing his research on the physics of music at the College Royal in Paris, heintroduced terms such as fundamental, harmonics, node, and ventral segment. 1 Itis also an interesting footnote to history that Sauveur may have been born withdefective hearing and speaking mechanisms; he was reported to have been a deaf–mute until the age of 7. He took an immense interest in music even though he hadto rely on the help of his assistants to compensate for his lack of keen musicaldiscernment in conducting acoustic experiments.Franciscus Mario Grimaldi (1613–1663) published Physicomathesis de lumine,coloribus et tride, which dealt with experimental studies of diffraction, much ofwhich was to apply to acoustics as well as to light, and in 1678 Hooke announced hislaw relating force to deformation, which established the foundations of vibrationand elasticity theories.Kircher’s publication Phomugia, die neue Hall- und Tonkunst (The New Art ofSound and Tone), issued in 1680, provides us a rather amusing insight into the worldof misconception, nostrums, and plain scientific hokum that were prevalent at thetime. While delving into the phenomena of echoes and whispering galleries, thetext recommended music as the only remedy against tarantula bites and provideda discourse on wines. In the chapter on wines, Kircher claimed that old wine haspurified itself and acquired a deep soul. If old wine is poured into a glass, whichis then struck, a sound will emanate. On the other hand, new wine was deemedto be “jumpy” as a child and bereft of a sound. Hence, recent wine in a glasswill not sound. Another misconception widely believed at the time was that soundcould be trapped in a little box and preserved indefinitely, the idea of attenuationor absorption of sound being completely alien then. It was even proposed by aProfessor Hut of the music academy at Frankfurt that a communications tube beconstructed to transmit speech over long distances.Ernst F. F. Chladni (1756–1827), author of the highly acclaimed Die Akustik,isoften credited for establishing the field of modern experimental acoustics through1 Nearly 20 years earlier, in 1683, Narcissus Marsh, then the Bishop of Ferns and Leighlin in theProtestant Church, published an article “An Introductory Essay to the Doctrine of Sounds, ContainingSome Proposals for the Improvement of Acousticks” in the Philosophical Transactions of the RoyalSociety of London. He was using the term “acousticks” to denote direct sound as distinguished fromreflected and diffracted sound.

1. A Capsule History of Acoustics 5his discovery of torsional vibrations and measurements of the velocity of soundwith the aid of vibrating rods and resonating pipes. The dawn of the eighteenthcentury saw the birth of theoretical physics and applied mechanics, particularlyunder the impetus of archrivals Isaac Newton (1642–1726) and Gottfried WilhelmLeibniz (1646–1716). Newton’s theoretical derivation of the speed of sound (inthe Principia) motivated a spate of experimental measurements by Royal Societymembers John Flamsteed (1646–1719), the founder of the Greenwich Observatoryand the first Astronomer Royal, and his eventual successor (in 1720),Edmund Halley (1656–1742); also by Giovanni Domenico Cassini (1625–1712),Jean Picard (1620–1682), and Olof Römer (1644–1710) of the French Acadêmiedes Sciences; and nearly half century later in 1738 by a team led by César FrançoisCassini de Thury (1714–1762), a grandson of the aforementioned G. D. Cassiniwho headed the earlier 1677 measurement team.Newton’s estimate was found to be in error, for in his observations he had erredby assuming an isothermal (rather than an isentropic) process as being the prevalentmode for acoustic vibrations. 2 Temperature was found to influence the speed ofsound in independent separate experiments by Count Giovanni Lodovico Bianoni(1717–1781) of Bologna and Charles Marie de la Condamine (1701–1773). Otheracoustic developments included the evolution of the exponential horn by RichardHelsham (1680–1758); this device loads the sound source heavily, thus causingthe source to concentrate its energy more than it could without the horn and directsthe output more effectively. Real understanding of this phenomenon did not comeabout until John William Strutt, Lord Rayleigh (1845–1919) treated the problemof source loading, and Arthur Gordon Webster (1863–1923) the theory of horns.Each of the optical phenomena of refraction, diffraction, and interference waselucidated during the seventeenth century. But all of these phenomena were soonrealized to apply to acoustics as well as to light. Willbrod Snell (or Snellius)(1591–1626) composed an essay in 1620 treating the refraction of light rays in atransparent medium such as water or glass, but he somehow neglected to publishhis manuscript which was later unearthed and used by Christian Huygens (1629–1695) in his own works, which secured posthumous fame for Snell, in spite of apublication of the same law by the stellar René Descartes (1596–1650) who, itturned out, had made two erroneous assumptions, which were corrected by Pierrede Fermat (1601–1665). Fermat’s principle derives from the assumption that thelight always travels from a source point in one medium to a receptor point in thesecond medium by the path of least time. Diffraction was first observed by the Jesuitmathematician Francesco Maria Grimaldi (1618–1663) of Bologna. His experimentswere repeated by Newton, Hooke, and Huygens; and soon this phenomenonthat light does not always travel in straight lines but can diffuse slightly around cornersconstituted a core issue in the controversy between the wave and corpusculartheories of light. But it took nearly 200 years following Newton’s era to resolve the2 Actually, what Newton really did was to assume that the “elastic force” of the fluid is proportionalto its condensation, which is now realized, in the context of modern thermodynamics, to be theequivalence of the isothermal process.

6 1. A Capsule History of Acousticsconflict by embracing elements of both theories. Newton essentially squelched thewave theory until its revival by Thomas Young (1773–1829) and Augustin JeanFresnel (1788–1817), both of whom, independently of each other, elucidated theprinciple of interference. On his analysis of diffraction, Fresnel drew heavily onHuygen’s principle in which successive positions of a wavefront are establishedby the envelope of secondary wavelets.Armed with the analytical tools afforded by the advent of calculus by Newton andLeibniz, the French mathematical school treated problems of theoretical mechanics.Among the major contributors were Joseph Louis Lagrange (1736–1813), theBernoulli brothers James (1654–1705) and Johann (1667–1748), G. F. A. l ′ Hôpital(Marquis de St. Mesme) (1661–1704), Gabriel Cramer (1704–1752), LeonhardEuler (1707–1783), Jean Le Rond d’Alembert (1717–1783), and Daniel Bernoulli(1700–1783). And the next generation provided a further flowering of genius:Joseph Louis Lagrange (1736–1813), Pierre Simon Laplace (1749–1827), AdrianMarie Legendre (1736–1833), Jean Baptiste Joseph Fourier (1768–1830), andSiméon Denis Poisson (1781–1840). The nineteenth century was also dominatedby discoveries in electricity and magnetism by Michael Faraday (1791–1867),James Clerk Maxwell (1831–1879), Heinrich Rudolf Hertz (1857–1894), and bythe theory of elasticity, principally developed by Clause L. M. Navier (1785–1836), Augustin Louis Cauchy (1789–1857), Rudolf J. E. Clausius (1822–1888),and George Gabriel Stokes (1890–1909).These developments constituted the foundation for understanding the physicaland eventually the physiologcial aspects of acoustics. In the attempt to grasp thenature of musical sound, Simon Ohm (1789–1854) advanced the hypothesis thatthe ear perceived only a single, pure sinusoidal vibration and that each complexsound is resolved by the ear into its fundamental frequency and its harmonics.Hermann F. L. von Helmholtz (1821–1894) arguably deserves the credit for layingthe foundations of spectral analysis in his classic Lehre von den Tonempfindungen(Sensation of Sound). The monumental two-volume Theory of Sound, released in1877 and 1878 by the future Nobel laureate, Lord Rayleigh, laid down in a fairlycomplete fashion the theoretical foundations of acoustics.When the newly constructed Fogg Lecture Hall was opened in 1894 at HarvardUniversity, its acoustics was found to be so atrocious so as to render that facilityalmost useless. This prompted Harvard’s Board of Overseers to request thephysics department that something be done to rectify the situation. The task wasassigned to a young Harvard researcher, Wallace Clement Sabine (1868–1919),and he discovered soon enough that excessive reverberations tend to mask the lecturer’swords. In a series of papers (1900–1915) evolving from his studies of thelecture hall, he almost single-handedly elevated architectural acoustics to scientificstatus. Sabine helped establish the Riverbank Acoustical Laboratories 3 at Geneva,Illinois. Just prior to his scheduled assumption of his duties at Riverbank, Sabinesuccumbed at the young age of 50 to cancer. His distant cousin, Paul Earls Sabine3 Riverbank is possibly the first research facility set up specifically for study and research in acoustics.

1. A Capsule History of Acoustics 7(1879–1958), also a Harvard physicist, took on the task of running the laboratory.The development of test procedures, methodology, and standardization in testingthe acoustical nature of products arose from the pioneering efforts of the youngerSabine. A third member of the family, Paul Sabine’s son Hale Johnson Sabine(1909–1981), began his career in architectural acoustics at the tender age of 10by assisting his father at Riverbank, and his efforts centered on control of noisein industry and institutions. Both father and son, Paul and Hale, served terms aspresident of the Acoustical Society of America.The genesis of ultrasonics occurred in the nineteenth century with James P.Joule’s (1818–1889) discovery in 1847 of the magnetostrictive effect, the alterationof the dimensions of a magnetic material under the influence of a magneticfield, and in 1880 with the finding by the brothers Paul-Jacques (1855–1941) andPierre (1859–1906) Curie that electric charges result on the surfaces of certaincrystals subjected to pressure or tension. The Curies’ discovery of the piezoelectricelectric effect provided the means of detecting ultrasonic signals. The inverseeffect, whereby a voltage impressed across two surfaces of a crystals give riseto stresses in the materials, now constitutes the principal method of generatingultrasonic energy.The study of underwater sound stemming from the necessity for ships to avoiddangerous obstacles in water supplied the impetus for development of ultrasonicapplications. Until the early part of the twentieth century ships were warned ofhazardous conditions by bells suspended from lightships. Specially trained crewmembers listened for these bells by pressing microphones or stethoscopes againstthe hulls. In the effort to counteract the German submarine threat during WorldWar I, Robert Williams Wood (1868–1955) and Gerrard in England and PaulLangevin (1872–1946) in France were assigned the task of developing countersurveillance methods.The youthful Russian electrical engineer, Constantin Chilowsky (1880–1958),collaborated with Langevin in experiments with an electrostatic (condenser)projector and a carbon-button microphone placed at the focus of a concave mirror.In spite of troubles encountered with leakages and breakdowns due to the highvoltages necessary for the operation of the projectors, Langevin and Chilowskywere able by 1916 to obtain echoes from the ocean bottom and from a sheetof armor plate at a distance of 200 m. A year later Langevin came up with theconcept of using a piezoelectric receiver and employed one of the newly developedvacuum-tube amplifiers—the earliest application of electronics to underwatersound equipment—and Wood constructed the first directional hydrophone gearedto locate hostile submarines. The first devices to generate directional beams ofacoustic energy also constitute the first use of ultrasonics. Reginald A. Fessenden(1866–1932), a Canadian engineer, working independently, developed a movingcoil transducer operating at frequencies in the range of 500–1000 Hz to generateunderwater signals. In the course of their underwater sound investigations, Woodand his co-worker Alfred L. Loomis (1887–1975), who also was a trained lawyer,and Langevin observed that small water creatures could be stunned, maimed, oreven destroyed by the effects of intense ultrasonic fields.

8 1. A Capsule History of AcousticsWorld War I ended before underwater echo-ranging could be fully deployed tomeet the German U-boat threat. The years of peace following World War I witnesseda slow but nevertheless steady advance in applying underwater sound todepth-sounding by ships. Improvements in electronic amplification and processing,magnetostrictive projectors, piezoelectric transducers provided refinements inecho-ranging. The advent of World War II heightened research activity on bothsides of the Atlantic, and most of the present concepts and applications of underwateracoustics traced their origins to this period. The concept of target strength,noise output of various ships at different speeds and frequencies, reverberation inthe sea, and evaluation of underwater sound through spectrum analysis were quantitativelyestablished. It was during this period that underwater acoustics becamea mature branch of science and engineering, backed by vast literature and historyof achievement.The invention of the triode vacuum tube and the advent of the telephone and radiobroadcasting served to intensify interest in the field of acoustics. The developmentof vacuum tube amplifiers and signal generators rendered feasible the design andconstruction of sensitive and reliable measurement instruments. The evolution ofthe modern telephone system in the United States was facilitated by the progressof communication acoustics, mainly through the remarkable efforts of the BellTelephone Laboratories.The historic invention of the transistor (1949) at the Bell Laboratories in MurrayHill, New Jersey, gave rise to a whole slew of new devices in the field of electronics,including solid state audio and video equipment, computers, spectrum analyzers,electric power conditioners, and other gear too numerous to mention here.Experiments and development of theory in architectural acoustics were conductedduring the 1930s and the 1940s at a number of major research centers,notably Harvard, MIT, and UCLA. Vern O. Knudsen (1893–1974), eventually thechancellor of UCLA, carried on Sabine’s work by conducting major research onsound absorption and transmission. The most notable of his younger associateswas Cyril M. Harris (b. 1917), who was to become the principal consultant onthe acoustics of the Metropolitan Opera House in New York, the John F. KennedyCenter in the District of Columbia, the Powell Symphony Hall in St. Louis, and anumber of other notable edifices.Sound decay, in terms of reverberation times, was discovered to be a decisivefactor in gauging the suitability of enclosed areas for use as listening chambers.The impedance method of rating acoustical materials was established to predict theradiative patterns of sonic output, and prediction of sound attenuation in ducts wasestablished on a scientific footing. The architectural acoustician now has a widearray of acoustical materials to choose from and to tailor the walls segmentwisein order to effect the proper acoustic environment.Acoustics also engendered the science of psychoacoustics. Harvey Fletcher(1884–1990) led the Bell Telephone Laboratories in describing and quantifyingthe concepts of loudness and masking, and there, many of the determinants ofspeech communication were also established (1920–1940). Fletcher, now regardedas “the father of psychoacoustics,” worked with the physicist Robert Millikan at

1. A Capsule History of Acoustics 9the University of Chicago, on the determination of the electron charge. Fletcherindeed performed much of the famed oil drop experiment, to the extent that manyphysicists feel that the student should have shared the 1923 Nobel Prize in physicswith his professor who received the award for this effort. At Bell Labs, Fletcher alsodeveloped the first electronic hearing aid and invented stereophonic reproduction.Sound reproduction also constituted the domain of Harry F. Olson (1902–1982),who directed the Acoustical Laboratory at RCA and developed modern versionsof loudspeakers. Warren P. Mason’s (1900–1986) major work in physical acousticsessentially laid down the modern foundations of ultrasonics, and Georg vonBekésy (1849–1972) earned the Nobel Prize for his research on the mechanics ofhuman hearing. Acoustics penetrated the fields of medicine and chemistry throughthe medium of ultrasonics: ultrasonic diathermy became established and certainchemical reactions were found to become accelerated under acoustic conditions.The outbreak of World War II served to greatly intensify acoustics research atmajor laboratories in Western Europe and in the United States, particularly in viewof the demand for sonar detection of stealthily moving submarines and for reliablespeech communication in cacophonous environments such as propeller aircraftand armored vehicles. This research not only has reached great proportions, it hascontinued unabated to this day, at major universities and government institutions,among them being the U.S. Naval Research Laboratory, Naval Surface WarfareCenter, MIT, Purdue University, Georgia Institute of Technology, and PennsylvaniaState University.Prominent among the researchers were Richard Henry Bolt (1911–2002) andLeo L. Beranek (b. 1914) who teamed up after World War II to found a majorresearch corporation, Bolt, Beranek & Newman (now BBN Technologies);Phillip M. Morse of MIT [who authored and co-authored with Karl Uno Ingard(b. 1921) major texts in physical acoustics]; R. Bruce Lindsay (1900–1985) ofBrown University; and Robert T. Beyer, who contributed to nonlinear acoustics,also at Brown. In 1947 Eugen Skudrzyk (1913–1990) began research in nearly allareas of acoustics at the Technical University of Vienna and went on to PennsylvaniaState University in the United States, he wrote possibly the best comprehensivetext on physical acoustics since Lord Rayleigh’s Theory of Sound.Karl D. Kryter (b. 1914) of California dealt with the physiological effects of noiseon humans, and Carleen Hutchins (b. 1911) is still providing great insight into thedesign and construction of musical string instruments, in her dual role as investigatingacoustician and craftsperson seeking to emulate the old Cremona masters in herhometown of Montclair, New Jersey. Laser intereferometry was applied by Karl H.Steson (b. 1937) and by Lothar Cremer (1905–1990) to visualize vibrations of theviolin body. Sir James Lighthill (1924–1998), who held the Lucasian chair (onceoccupied by Newton) in mathematics at Cambridge University, laid down the foundationsof modern aeroacoustics, building on the foundations of Lord Rayleigh’searlier research. UCLA’s Isadore Rudnick (1917–1997) performed major experimentsin superfluid hydrodynamics, involving sound propagation in helium at cryogeneictemperatures and also conducted studies of acoustically induced streamingmodes of vibrations of elastic bodies and attenuation of sound in seawater. At

References 11characterization of materials, better surveillance methodologies, and improvedmanufacturing techniques.And what does the future hold in acoustics? The continuing miniaturization ofelectronic circuitry is now resulting in digitized hearing aids that can circumventthe “cocktail party effect” (the tendency of background noise to make it difficultfor the sensorneurally impaired listeners to focus on a conversation). Even newerdiagnostic and therapeutic processes entailing acoustical signals are being developedand tested at major medical centers. More sensitive and versatile transducersthat can withstand harsher environments lead to new acoustical devices such assonic viscometers, undersea probes, and portable voice-recognition devices. Andif we can gain a greater understanding of how cetaceans make use of their naturalsonars to assess the submarine environment and perhaps to communicate witheach other, we could be well on the way to constructing far more sophisticatedmegachannel acoustical analyzers. The generation of acoustical waves in the gigahertzrange can rival or exceed the optical microscope for resolution with greaterpenetrating power. The repertoire of what is to come should truly constitute anamazing cornucopia of beneficence to humanity.ReferencesBeranek, Leo J. 1995. Harvey Fletcher: Friend and scientific critic, Journal of the AcousticalSociety of America 97(5):3357.Beyer, Robert T. 1995. Acoustic, acoustics. Journal of the Acoustical Society of America98(1): 33–34.Beyer, Robert T. 1999. Sounds of Our Times. New York: Springer-Verlag. [A fascinatinghistory of acoustics over the past 200 years, with many allusions to even earlier history.This text picks up where Frederick Vinton Hunt left off in his unfinished, meticulouslyresearched work which was published posthumously (seen below).]Blake, William K. 1964. Aero-hydroacoustics for Ships, 2 Vols. Bethesda, MD: DavidTaylor Basin publication DTNSRDC-84/010, June 1964.Bobber, Robert J. 1970. Underwater Electroacoustic Measurements. Washington, DC:Naval Research Laboratory.Chladni, E. F. F. 1802. Die Acustik. Leipzig: Breitkopf & Hartel.Clay, Clarence S. and Medwin, Herman. 1977. Acoustical Oceanography: Principles andApplications. New York: John Wiley & Sons.Fletcher, Steven Harvey. 1995. Harvey Fletcher: A son’s reflections. Journal of the AcousticalSociety of America 97(5 Pt. 2): 3356–3357.Galileo, Galilei. 1638 (translation published in 1939). Dialogues Concerning Two New Sciences,Translated by Crew, H. and De Salvio, A. Evanston, IL: Northwestern UniversityPress.Hanish, Sam. 1981. A Treatise on Acoustic Radiation. Washington, DC: Naval ResearchLaboratory.Harris, Cyril M. 1995. Harvey Fletcher: Some personal recollections. Journal of the AcousticalSociety of America 97(5 Pt. 2): 3357.Helmholtz, Hermann F. L. von. 1877. Lehre con den Tonempfindungen. Braunschweig,Wiesbaden: Vieweg.

12 1. A Capsule History of AcousticsHertz, J. H. (ed.). 1987. The Pentateuch and Haftorahs. London: Soncino Press.Hunt, Frederick V. 1992 (reissue). Origins in Acoustics. Woodbury, NY: Acoustical Societyof America. (Although left incomplete by the author at the time of his death, this text isone of the most definitive accounts by one of the great modern acoustical scientists ofthe history of acoustics leading up to the eighteenth century.)Junger, Miguel C., Feit, David. 1986. Sound, Structure, and Their Interaction. Cambridge,MA: The MIT Press.Kopec, John W. 1994. The Sabines at Riverbank. Proceedings, Wallace Clement SabineCentennial Symposium. Woodbury, NY: Acoustical Society of America, pp. 25–28.Lindsay, R. Bruce. 1966. The story of acoustics. Journal of the Acoustical Society ofAmerica 39(4): 629–644.Lindsay, R. Bruce (ed.). 1972. Acoustics: Historical and Philosophical Development.Benchmark Papers in Acoustics. Stroudsburg, PA: Dowden, Hutchinson & Ross, Inc.(A most interesting compendium of selected papers by major contributors to acousticalscience, ranging from Aristotle to Wallace Clement Sabine. A must-read for the seriousstudent of the history of acoustics).Lindsay, R. Bruce. 1880. Acoustics and the Acoustical Society of America in HistoricalPerspective. Journal of the Acoustical Society of America 68(1): 2–9.Mersenne, Marin. 1636. Harmonie universelle. Paris: S. Cramoisy; English translation:Hawkins, J. 1853. General History of the Practice and Science of Music. London: J. A.Novello, pp. 600–616, 650 ff.Newton, Sir Isaac. 1687. Philosophiae Naturalis Principia Mathematica. London: JosephStreater for the Royal Society.Pierce, Allan D. 1989 (reissue). Acoustics: An Introduction to its Physical Principles andApplication. Woodbury, NY: Acoustical Society of America.Raman, V. V. 1973. Where credit is due: Sauveur, the forgotten founder of acoustics.Physics Teacher pp. 161–163.Shaw, Neil A., Klapholz, Jesse, Gander, Mark R. 1994. Books and Acoustics, especiallyWallace Clement Sabine’s Collected Papers on Acoustics. Proceedings, WallaceClement Sabine Centennial Symposium. Woodbury, NY: Acoustical Society of America,pp. 41–44.Skudrzyk, Eugen. 1971. The Foundations of Acoustics—Basic Mathematics and BasicAcoustics. New York: Springer-Verlag. (A text of classic proportions. Nearly one quarterof this volume lays the mathematical foundations requisite to analysis of acousticalphenomena.)Strutt, John William (Lord Rayleigh). 1877. Theory of Sound. London: Macmillan & Co.Ltd. 2nd edition revised and enlarged 1894, reprinted 1926, 1929. Reprinted in twovolumes, New York: Dover, 1945. (These volumes should be in every acoustician’slibrary.)Wang, Ji-qing. 1994. Architectural Acoustics in China, Past and Present. Proceedings,Wallace Clement Sabine Centennial Symposium. Woodbury, NY: Acoustical Society ofAmerica, pp. 21–24.Webster, Arthur G. 1919. Proceedings of the National Academy of Science 5:275.

2Fundamentals of Acoustics2.1 Wave Nature of Sound and the Importance of AcousticsAcoustics refers to the study of sound, namely, its production, transmission throughsolid and fluid media, and any other phenomenon engendered by its propagationthrough media. Sound may be described as the passage of pressure fluctuationsthrough an elastic medium as the result of a vibrational impetus imparted to thatmedium. An acoustic signal can arise from a number of sources, e.g., turbulenceof air or any other gas, the passage of a body through a fluid, and the impact of asolid against another solid.Because it is a phenomenon incarnating the nature of waves, sound may containonly one frequency, as in the case of a pure steady-state sine wave, or manyfrequency components, as in the case of noise generated by construction machineryor a rocket engine. The purest type of sound wave can be represented by a sinefunction (Figure 2.1) where the abscissa represents elapsed time and the ordinaterepresents the displacement of the molecules of the propagation medium or thedeviation of pressure, density, or the aggregate speed of the disturbed moleculesfrom the quiescent (undisturbed) state of the propagation medium.When the ordinate represents the pressure difference from the quiescent pressure,the upper portions of the sine wave would then represent the compressivestates and the lower portions the rarefaction phases of the propagation. A sine waveis generated in Figure 2.2 by the projection of the trace of a particle A traveling ina circular orbit. This projection assumes the pattern of an oscillation, in which theparticle A’s projection or “shadow” A ′ onto an abscissa moves back and forth at aspecified frequency. Frequency f is the number of times the sound pressure variesfrom its equilibrium value through a complete cycle per unit time. Frequency isalso denoted by the angular (or radian) frequencyω = 2π f = 2π T(2.1)expressed in radians per second. The period T is the amount of time for a singlecycle to occur, i.e., the length of the time it takes for a tracer point on thesine curve to reach a corresponding point on the next cycle. The reciprocal of13

14 2. Fundamentals of AcousticsFigure 2.1. Plot of a sine wave y(t) = sin 2π sin ft over slightly more than two periodsof T = 1/f s, where f is the frequency of the sine wave. y(t) may be the displacementfunction x/x 0 , velocity ratio v/v 0 , pressure variation p/p 0 , or condensation variation s/s 0 ,where the subscript 0 denotes maximum values.Figure 2.2. The oscillation of a particle A ′ in a sinusoidal fashion is generated by thecircular motion of particle A moving in a circle with constant angular speed ω. A ′ isthe projection of Acos ωt = Acos θ onto the diameter of the circle which has a radiusA. The projection of point A to the right traces a sine wave over an abscissa representingtime t. The projections for three points at times t 1 , t 2 , and t 3 are shown here. The amplitudeof the oscillation is equal to the radius of the circle, and the peak-to-peak amplitude isequal to the diameter of the circle.

2.1 Wave Nature of Sound and the Importance of Acoustics 15period T is simply the frequency f . The most common unit of frequency usedin acoustics (and electromagnetic theory) is the hertz (abbreviated Hz in the SIsystem), which is equal to one cycle per second. An acoustic signal may or maynot be audible to the human ear, depending on its frequency content and intensity.If the frequencies are sufficiently high (>20 kilohertz, which can be expressedmore briefly as 20 kHz), ultrasound will result, and the sound is inaudible to thehuman ear. This sound is said to be ultrasonic. Below 20 Hz, the sound becomestoo low (frequency-wise) to be heard by a human. It is then considered to beinfrasonic.Sound in the audio frequency range of approximately 20 Hz–20 kHz can be heardby humans. While a degree of subjectivity is certainly entailed here, noise conveysthe definition of unwanted sound. Excessive levels of sound can cause permanenthearing loss, and continued exposure can be deleterious, both physiologically andpsychologically, to one’s well-being.With the advent of modern technology, our aural senses are being increasinglyassailed and benumbed by noise from high-speed road traffic, passing ambulancesand fire engine sirens, industrial and agricultural machinery, excessively loud radioand television receivers, recreational vehicles such as snowmobiles and unmuffledmotorcycles, elevated and underground trains, jet aircraft flying at low altitudes,domestic quarrels heard through flimsy walls, and so on.Young men and women are prematurely losing their hearing acuity as the resultof sustained exposure to loud rock concerts, discotheques, use of personalcassette and compact disk players and mega-powered automobile stereo systems.In the early 1980s, during the waning days of the Cold War, the Swedish navyreported considerable difficulty in recruiting young people with hearing sufficientlykeen to qualify for operating surveillance sonar equipment for trackingSoviet submarines traveling beneath Sweden’s coastal waters. Oral communicationcan be rendered difficult or made impossible by background noise; andlife-threatening situations may arise when sound that conveys information becomesmasked by noise. Thus, the adverse effects of noise fall into one ormore of the following categories: (1) hearing loss, (2) annoyance, and (3) speechinterference.Modern acoustical technology also brings benefits: it is quite probable that theavailability (and judicious use) of audiophile equipment has enabled many of us,if we are so inclined, to hear more musical performances than Beethoven, Mozartor even the long-lived Haydn could have heard during their respective lifetimes.Ultrasonic devices are being used to: dislodge dental plaque; overcome the effectsof arteriosclerosis by freeing up clogged blood vessels; provide noninvasive medicaldiagnoses; aid in surgical procedures; supply a means of nondestructive testingof materials; and clean nearly everything from precious stones to silted conduits.The relatively new technique of active noise cancellation utilizes computerizedsensing to duplicate the histograms of offending sounds but at 180 degrees outof phase, which effectively counteracts the noise. This technique can be appliedto aircraft to lessen environmental impact and to automobiles to provide quieterinteriors.

16 2. Fundamentals of Acoustics2.2 Sound Generation and PropagationSound is a mechanical disturbance that travels through an elastic medium at aspeed characteristic of that medium. Sound propagation is essentially a wave phenomenon,as with the case of a light beam. But acoustical phenomena are mechanicalin nature, while light, X rays, and gamma rays occur as electromagneticphenomena. Acoustic signals require a mechanically elastic medium for propagationand therefore cannot travel through a vacuum. On the other hand, thepropagation of an electromagnetic wave can occur in empty space. Other types ofwave phenomena include those of ocean movement, the oscillations of machinery,and the quantum mechanical equivalence of momenta as propounded by deBroglie. 1Consider sound as generated by the vibration of a plane surface at x = 0 as shownin Figure 2.3. The displacement of the surface to the right, in the +x direction,causes a compression of a layer of air immediately adjacent to the surface, therebycausing an increase in the density of the air in that layer. Because the pressure of thatlayer is greater than the pressure of the undisturbed atmosphere, the air moleculesin the layer tend to move in the +x direction and compress the second layer which,in turn, transmits the pressure impulse to the third layer and so on. But as the planesurface reverses its direction of vibration, an opposite effect occurs. A rarefactionof the first layer now occurs, and this rarefaction decreases the pressure to a valuebelow that of the undisturbed atmosphere. The molecules from the second layernow tend to move leftward, in the −x direction, and a rarefaction impulse nowfollows the previously generated compression impulse.This succession of outwardly moving rarefactions and compressions constitutesa wave motion. At a given point in the space, an alternating increase and decreasein pressure occur, with a corresponding decrease and increase in the density. Thespatial distance λ from one point on the cycle to the corresponding point on the nextcycle is the wavelength. The vibrating molecules that transmit the waves do not, onthe average, change their positions, but are merely moved back and forth under theinfluence of the transmitted waves. The distances these particles move about theirrespective equilibrium positions are referred to as displacement amplitudes. Thevelocity at which the molecules move back and forth is termed particle velocity,which is not to be confused with the speed of sound, the rate at which the wavestravel through the medium.The speed of sound is a characteristic of the medium. Sound travels far morerapidly in solids than it does in gases. At a temperature of 20 ◦ C sound moves atthe rate of 344 m/s (1127 ft/s) through air at the normal atmospheric pressure of1 The de Broglie theory assigns the nature of a wave to the momentum of a particle of matter in motionin the following way:mv = hνcwhere mv represents the moment of the particle, h Planck’s constant = 6.625 × 10 –27 erg s, c thevelocity of light = 3 × 10 8 m/s, and ν the radial frequency of the wave attributable to the particle.

2.2 Sound Generation and Propagation 17p, s. or Ψ10. 50-0.5-1λxVIBRATINGPLANEx = 0ξ10.50-0.5-1xFigure 2.3. Depiction of rarefaction and condensation of air molecules subjected to thevibrational impact of a plane wall located at x = 0. The degree of darkness is proportionalto the density of molecules. Lighter areas are those of rarefactions. Mini-plots of thelocal variations of molecular displacement Ψ, pressure p, condensation s = (ρ − ρ 0 )/ρ 0 ,and particle displacement ξ are given as functions of x for a given instant of the soundpropagation. Note that wavelength λ represents the distance between corresponding pointsof adjacent cycles.101 kPa (14.7 psia or 760 mmHg). Sound velocities are also greater in liquids thanin gases, but remain less in order of magnitude than those for solids. For an idealgas the velocity c of a sound wave may be computed fromc =√ γ pρ = √ γ RT (2.2)where γ is the gas constant equivalent to the thermodynamic ratio of specificheats, c p /c v , p the quiescent gas pressure, and ρ the density of the gas. R is thethermodynamic constant characteristic of the gas and T is the absolute temperature

18 2. Fundamentals of Acousticsof the gas. For air at 20 ◦ C, the sound propagation speed c is found fromc = √ γ RT = √ 1.4 [287 N · m/(kg K)](20 + 273.2) K = 343.2 m/sA simple relation such as Equation (2.2) does not exist for acoustic velocity inliquids, but the propagation velocity does depend on the temperature of the liquidand, to a lesser degree, on the pressure. Sound velocity is approximately 1461 m/sin deaerated water. For a solid the propagation speed can be found approximatelyfrom√Ec =(2.3)ρwhere E represents the Young’s modulus (or modulus of elasticity) of the materialand ρ the material density. As an example, considered cast iron with a specificgravity of 7.70 and a modulus of elasticity of 105 GPa. Applying Equation (2.3)and recalling that1Nisequal to 1 kg m/s 2 , we find that√105(10)c =9 N/m 2= 3692 m/s7700 kg/m 2which does represent the propagation speed of sound in that material. Appendix Alists the speed of sound for a variety of materials.The strength of an acoustic signal, as exemplified by loudness or sound pressurelevel (SPL), directly relates to the magnitudes of the displacement amplitudes andpressure and density variations, as we shall see later in Chapter 3.When the procession of rarefactions and condensations occurs at a steady sinusoidalrate, a single constant frequency f occurs. If the sound pressure of apure tone was plotted against distance for a given instant, the wavelength λ canbe established as being the peak-to-peak distance between two successive waves.The wavelength λ is related to frequency f by:λ = c f(2.4)where c represents the propagation speed. From Equation (2.4), it can be seenthat higher frequencies will result in shorter wavelengths in a given propagationmedium.2.3 Thermodynamic States of FluidsIn the treatment that follows this section, we eschew the details of molecularmotion and intermolecular forces by describing relevant effects in terms of macroscopicthermodynamic variables: pressure p, density ρ, and absolute temperatureT . These variables relate to each other through an equation of statep = p(ρ,T )

2.4 Fluid Flow Equations 19which is usually established experimentally. The implication of the equation ofstate is that only two of the variables are independent; this is to say if the valuesof two of the independent thermodynamic variables are given for a fluid, thespecific value of any other thermodynamic property is automatically established.The equation of state for an ideal gas,p= RT (2.5)ρcan be derived from simple kinetic theory. Here,R = gas constant, energy per unit mass per degreeR =R/MR=universal gas constant, energy per mole per degree= 8.314.3 kJ/kg mol K = 1545.5 ft lb f /lb mol R= 1.986 Btu/lb m mol RM = molecular weight of gas, kg/kg mol or lb m /lb m molEach kilogram-mole of the gas contains N 0 = 6.02 × 10 26 molecules. N 0 constitutesAvogadro’s number for the MKS system of dimensional units. With η ⌢ representingthe mass of a single-gas molecule, M = N 0 η, the number of molecules⌢per unit volume is N = ρ/ η. ⌢ The equation of state for the ideal gas can now berewritten as:p = N R N0T = NkTwhere k is the Boltzmann constant =R/N 0 = 1.38 ×10 –26 kJ/K.In liquids and gases under extreme pressures, the relationships between thethermodynamic variables p, T , ρ, X (here X is the quality or the fractional massof gas comprising a saturated liquid–gas mixture, e.g., X = 1.00 represents a fullysaturated gaseous state and X = 0 represents the fully saturated liquid state) arenot so simple, but the fact remains that these parameters are fully dependent uponeach other, and specifying two thermodynamic parameters (including enthalpy,entropy, etc.) will fully specify the thermodynamic state of the fluid.2.4 Fluid Flow EquationsIn the Eulerian description of fluid mechanics the field variables, such as pressure,density, momenta, and energy, are considered to be continuous functions of thespatial coordinates x, y, z and of time t. Because velocity has three componentsin three-dimensional space and only two independent thermodynamic variablesneed to be selected to fix the thermodynamic state of the fluid (we chose p andρ), we have a total of five field variables for which we need five independentequations. We can take advantage of conservation laws to establish these equations,namely the conservation of mass, which supplies one equation; the conservation ofmomentum along each of the three principal axes, which provides three equations;

20 2. Fundamentals of AcousticsQ z+Δz = (ρw) z+ΔzQ x = (ρu) xΔxQ y+ Δy = (ρv) y+ΔyQ y = (ρv) yΔzQ x+Δx = (ρu) x+ΔxzyQ z = (ρw) zΔyxFigure 2.4. Flow Q(v, t) into and out of a control volume V = xyz depicted forthe derivation of the equation of continuity.and the conservation of energy (or the equation of state, in the derivation of theactual wave equation) 2 that constitutes the fifth equation.2.5 Conservation of MassIn Figure 2.4, consider a parallelepiped serving as a control (or reference) volume,dV = dx dy dz, through which fluid flows. Conservation of matter dictates that thenet flow into this volume equals the gain or loss of fluid inside the volume, i.e.,( mm exit − m enter =t volume)V →dvLet the velocity V of the fluid resolve into u, v, w, the velocity components in thex, y, and z directions, respectively. In vector terminologyV = ui + vj + wk2 It can be argued that because the equation of state derives from the principles of conservation ofmomentum and energy in classic kinetic theory, it effectively becomes the equivalent of the energyconservation principle in the extraction of the acoustic wave equations for a fluid, in conjunction withthe equations of continuity and momentum.

2.5 Conservation of Mass 21where i, j, k represent the unit vectors along the x, y, z coordinates. The mass fluxQ(x, t) is defined as the flow of the mass of fluid per unit time per unit area, whichis represented byQ(x, t) = ρ(x, t) u(x, t)The rate of mass per unit time flowing into the control volume dV in the directionis given byṁ x = Q(x, t)dA x = (ρu) x,t dA x = (ρu) x,t dy dz (2.6)at position x and the rate of mass per unit time ṁ x+x leaving dV at x + x byṁ x+x = Q(x + x, t)dA x = (ρu) x,t dA x (ρu) x,t dydz (2.7)Then subtracting Equation (2.6) from Equation (2.7) yields the net flow in the xdirection∂(ρu)ṁ x+x − ṁ x = dA x [(ρu) x+x − (ρu) x ] = dA x∂xdx = ∂(ρu) dV (2.8)∂xSimilarly for mass flow in they and z directionsṁ y+y − ṁ y = ∂(ρv) dV (2.9)∂yṁ z+z − ṁ z = ∂(ρw) dV (2.10)∂zSumming the net mass flows Equations (2.8)–(2.10) and equating them to thechange of mass in the control volume:∂(ρu)∂x+ ∂(ρv)∂y+ ∂(ρw)∂z=− ∂ρ∂t(2.11)Equation (2.11) is the equation of continuity, a general statement of the conservationof matter for compressible fluid 3 flow. In vector notation Equation (2.11) maybe written as∂ρ−∇·(ρV) = 0 (2.12)∂twherein the gradient symbol represents∇=i ∂∂x + j ∂∂y + k ∂ ∂z3 If density ρ is constant, the fluid is said to be incompressible. Asρ is no longer a spatial or a timefunction, Equation (2.11) simplifies to:∂u∂x + ∂v∂y + ∂w∂z = 0

22 2. Fundamentals of Acousticsin the rectangular coordinate system. In the cylindrical coordinate system thegradient operator ∇ appears as∇= ∂∂r + 1 ∂r ∂φ + ∂ ∂zand in the spherical coordinate system as∇= ∂∂r + 1 r∂∂ϑ + 1r sin ϑ∂∂φ2.6 Conservation of MomentumIn order to develop the equations of momentum for a fluid, let us consider themotion of a fluid particle with a velocity field V p = V t (x, y, z, t). At a later timet + dt, the velocity becomes V p ′ = V t+dt (x + dx, y + dy, z + dz, t + dt). Thechange in velocity is given by:dV = (V p ′ − V p ) = ∂V∂xdx +∂V∂y∂V ∂Vdy + dz +∂z ∂t dtand the total acceleration of the particle is therefore expressed asa p = dVdt= ∂V dx∂x dt + ∂V dy∂y dt + ∂V dz∂z dt + ∂V∂t(2.13)But dx/dt = u, dy/dt = v, and dz/dt = w. Equation (2.13) now can be written as:a p = dVdtHere, the operator= u ∂V∂x + v ∂V∂y + w ∂V∂z + ∂V = DV∂t Dt(2.14)DDt = ∂ ∂t + u ∂∂x + v ∂∂y + w ∂ ∂zrepresents the total or convective derivative of fluid mechanics. While Equation(2.14) is a vector expression, we can rephrase it into scalar terms. With referenceto a rectangular coordinate system the scalar components of Equation (2.14) arewritten asa x = DuDta y = DvDta z = DwDt= ∂u∂t + u ∂u∂x + v ∂u∂y + w ∂u∂z= ∂v∂t + u ∂v∂x + v ∂v∂y + w ∂v∂z= ∂w∂t+ u ∂w∂x + v ∂w∂y + w ∂w∂zNo acceleration or deceleration of the fluid will occur unless forces are actingupon it. Two types of forces act on the fluid element as shown in Figure 2.5, namely,body forces and surface forces. Gravity constitutes a body force that pervadesthroughout the volume of the fluid. Surface forces include both normal forces

2.6 Conservation of Momentum 23⎛ ∂τzxτ zx + ⎞⎜ dz⎟dx ⋅ dy⎝ ∂z ⎠σ xx dy dzτ xy dy ⋅dzτ yz dx ⋅ dzτ xz dy ⋅ dz⎛ ∂σ zz ⎞⎜σzz+ dz⎟dx ⋅ dy⎛ ∂τzy⎞⎝ ∂z⎠⎜ τ zy + dz⎟dx ⋅dy⎝ ∂z⎠⎛ ∂τyxτ yx + ⎞⎜ dy⎟dx ⋅dz⎝ ∂y ⎠⎛ ∂σ yyσ yy + ∂ydy ⎞⎜⎟ dx ⋅dz⎝⎠⎛ ∂τ⎜τyz⎝+ yz ⎞dy⎟dx ⋅dz∂y ⎠σ yy dx ⋅ dzτ yx dx ⋅ dzτ zy dx ⋅ dy⎛ ∂τ⎜τxy⎝+ xy ⎞dx⎟dy ⋅dz∂x ⎠⎛ ∂σ xxσ xx + ⎞⎜dx⎟dy ⋅dz⎝ ∂x⎠⎛ ∂τ⎜τd⎝ xz + xz∂x dx ⎞⎟ dz⋅dy⎠zyσ zz dx ⋅ dyzx dx ⋅ dyxFigure 2.5. A fluid element acted on by normal and tangential stresses.(pressure) and tangential (shear) forces. A normal force is denoted by the symbolσ mm , where m denotes the direction of the normal. Because σ mm is dimensionallyexpressed in force per unit area, it must be multiplied by the area normal to it inorder to obtain the force.A shear stress acts along the plane of the surface. It is represented by the symbolτ mn , where the force produced by the shear is normal to coordinate m and parallelfor coordinate n, and either m or n may represent the principal coordinate x, y,orz, provided that m ≠ n.Ifm = n, then τ mm really represents the normal force σ mmand thus is no longer a tangential force. The shear stress is multiplied by the areait is acting on to yield the shear force. For example, a shear τ xy multiplied by area(dx dy) represents the shear force normal to the x-axis and parallel to the y-axis,as shown in Figure 2.5 for a fluid element displayed in Cartesian coordinates.In order to determine the net force F x in the x-direction, all of the forces in thex-direction must be summed. From Figure 2.5 we can write(dF x = ρg x dxdydz + −σ xx + σ xx + ∂σ xx+which simplifies to(τ yx + ∂τ yx∂y dx − τ yxdF x =(ρg x + ∂σ xx∂x)dxdz ++ ∂τ yx∂y)∂x dxdydz (τ zx + ∂τ zx∂z dx − τ zx)dxdz+ ∂τ )zxdxdydz (2.15)∂z

24 2. Fundamentals of AcousticsWe can easily apply the force summation procedure to the other two principal axes:(dF y = ρg y + ∂τ xy∂x+ ∂σ yy∂y+ ∂τ )zydxdydz (2.16)∂zdF z =(ρg z + ∂τ xz∂x + ∂τ yz∂y + ∂σ zz∂z)dxdydz (2.17)From Newton’s second law of motion,dF = d(ma) = ρdV DVDtwe can now formulate the differential momentum equations by combining thescalar components of Equation (2.13) with Equations (2.15)–(2.17) with the followingresults:ρg x + ∂τ xz∂x + ∂τ yz∂y + ∂σ (zz ∂u= ρ∂z ∂t + u ∂u∂x + v ∂u∂y + ∂u )(2.18a)∂zρg y + ∂τ xy∂x+ ∂σ yy∂y+ ∂τ (zy ∂v= ρ∂z ∂t + u ∂v∂x + v ∂v∂y + w ∂v )(2.18b)∂zρg z + ∂τ xz∂x + ∂τ yz∂y + ∂σ (zz ∂w= ρ + u ∂w∂z ∂t ∂x + v ∂w∂y + w ∂w )(2.18c)∂zIn order to use Equations (2.18a)–(2.18c), the expressions for the stresses shouldbe stated in terms of the velocity field. If a Newtonian fluid is assumed, the viscousstresses are proportional to the rate of shearing strain (i.e., the rate of angular deformation).Without going into details, we express the stresses in terms of velocitygradients and viscosity coefficient μ as follows:( ∂vτ xy = τ yx = μ∂x + ∂u )(2.19a)∂y( ∂wτ yz = τ zy = μ∂y + ∂v )(2.19b)∂z( ∂uτ zx = τ xz = μ∂z + ∂w )(2.19c)∂xσ xx =−p − 2 μ · V + 2μ∂u3 ∂xσ yy =−p − 2 μ · V + 2μ∂v3 ∂y(2.19d)(2.19e)σ zz =−p − 2 μ · V + 2μ∂w(2.19f)3 ∂zHere the term p is the local thermodynamic pressure, which is essentially anisotropic parameter at any given point in the fluid. If we assume the fluid to befrictionless, then μ = 0, and we are left with Equations (2.19d)–(2.19f) in the

2.8 Derivation of the Acoustic Equations 25following format:σ xx = σ yy = σ zz =−pand, neglecting the gravitational body force ρg i (where i = x, y, z), we recastEquations (2.18a)–(2.18c) as− ∂p∂x = ρ DuDt− ∂p∂y = ρ DvDt− ∂p∂z = ρ DwDt2.7 Conservation of EnergyThe energy content W of a fluid is the sum of the macroscopic kinetic energyρ|V | 2 /2 and the internal energy ρE of the fluid. In a gas, the microscopic kineticenergy (i.e., the thermal energy of the molecules) comprises the major portionof the internal energy, so the potential energy between molecules is negligible incomparison. Denoting the energy flux by S we write equation for the conservationof energy as∂W+ ∂ S∂t ∂x = 0 (2.20)The internal energy of a volumetric element can be increased through heat flowfrom the surrounding fluid or from external sources and by the work of compression− ∫ pdV by the surrounding fluid pressure. This energy balance and the fact thatthe internal energy is a thermodynamic state that can be fully specified by twoindependent thermodynamic variables constitute the first law of thermodynamics.With the conservation equations discussed above and the equation of state, wehave all the necessary equations to obtain solutions for the three components ofvelocity V, ρ, p and absolute temperature T . Because the fluid equations are nonlinear,solutions are not easy to come by, even with the aid of supercomputers tomap the complex motions of atmospheric eddies, turbulent jet flows, capillary flow,and so on. Exact solutions exist principally for a few simple problems. Nevertheless,through the derivation of these equations, we have established the foundationfor the derivation of acoustic field equations for fluids.2.8 Derivation of the Acoustic EquationsWe begin with the following assumptions:(1) the unperturbed fluid has definite values of pressure, density, temperature, andvelocity, all of which are assumed to be time independent and denoted by thesubscript 0.

26 2. Fundamentals of Acoustics(2) the passage of an acoustic signal through the fluid results in small perturbationsof pressure, temperature, density, and velocity. These perturbations areexpressed as p 0 + p,ρ 0 + ρ,u, and so on. The unperturbed velocity u 0 is setto zero; the unperturbed fluid does not undergo macroscopic motion, and uconstitutes the perturbation velocity in the x-direction. Also, p ≪ p 0 , ρ ≪ ρ 0 ,and T ≪ T 0 .(3) the transmission of the sound through the fluid results in low values of spatialtemperature gradients at audio frequencies, resulting in almost no heat transferbetween warmer and cooler regions of the plane wave. Thus the ongoingthermodynamics process may be considered an adiabatic process (at ultrasonicfrequencies there is virtually no time for heat transfer to occur).Under the above conditions we obtain an expansion of the continuity equationin the x-direction as follows:∂[(ρ 0 + ρ)u]=− ∂(ρ 0 + ρ)= u ∂ρ∂x∂t ∂x + ρ ∂u0∂xwhich, by discarding second-order terms, reduces to∂ρ ∂u=−ρ 0 (2.21)∂t ∂xHere we consider ρ 0 ≈ ρ 0 + ρ, also recalling that the quiescent density ρ 0 doesnot vary in time and space. Treating in the same fashion the one-dimensionalmomentum equation(∂(p 0 + p)∂u= (ρ 0 + ρ)∂x∂t + u ∂u )∂xyields∂p∂x = ρ ∂u0(2.22)∂tIn an adiabatic process involving an ideal gas,pρ −γ = constant.Here γ represents a thermodynamic constant, characteristic of the gas, equal tothe ratio of the specific heats c p /c v . The numerator of this thermodynamic ratiois the specific heat at constant pressure, and the denominator, the specific heat atconstant volume. By differentiation,ρ −γ dp − γ pρ −γ −1 dρ = 0and rearrangingdpp = γ dρ ρwe have for this situationdp= γ dρp 0 ρ 0

2.8 Derivation of the Acoustic Equations 27The above expression can be differentiated with respect to time:1 ∂p= γ ∂ρp 0 ∂t ρ 0 ∂tCombining Equations (2.21) and (2.23),(2.23)∂p= γ p 0 ∂ρ ∂u= γ p 0∂t ρ 0 ∂t ∂xand then differentiating with respect to time t we obtain∂ 2 p∂t = γ p ∂ 2 u2 0∂t∂xDifferentiating Equation (2.22) with respect to x results in∂ 2 p∂x = ρ ∂ 2 u2 0∂x∂tEquating the above two cross-differential terms to each other, as we consider themto be equivalent regardless of their order of differentiation, we obtain the result∂ 2 p∂x = ρ 0 ∂ 2 p2 γ p 0 ∂t = 1 ∂ 2 p(2.24)2 c 2 ∂t 2wherec 2 = γ p 0= γ RTρ 0Here c, R, and T are respectively the propagation velocity of sound, the gasconstant, and absolute temperature of the (ideal) gas. In three-dimensional formthe wave equation (2.17) appears as∇ 2 p = 1 ∂ 2 p(2.25)c 2 ∂t 2We also could have eliminated p in favor of u by reversing the differentiationprocedure between Equations (2.22) and (2.23), in which situation we would get∂ 2 u∂x = 1 ∂ 2 u(2.26)2 c 2 ∂t 2for the one-dimensional situation, and∇ 2 V = 1 ∂ 2 V(2.27)c 2 ∂t 2in the three-dimensional case. It is also a straightforward matter to derive the waveequation in terms of density, resulting in the following expressions:∂ 2 ρ∂x = 1 ∂ 2 ρ(2.28a)2 c 2 ∂t 2for the one-dimensional case and∇ 2 ρ = 1 ∂ 2 ρ(2.28b)c 2 ∂t 2for three dimensions.

28 2. Fundamentals of AcousticsEquations (2.24)–(2.28) are second-order partial differential equations in x andt. Ordinarily we need two initial conditions and two boundary conditions for a fullydefined solution for each of the equations, but we need not define these conditionsin order to ascertain the nature of the general solutions. The general solution toEquation (2.24) may be written asp(x, t) = F(x − ct) + G(x + ct) (2.29)The function F(x–ct) represents waves moving in the positive x-direction andG(x + ct) represents waves moving in the opposite direction. All solutions toEquation (2.24) must be of the form represented in Equation (2.29); otherwise anyp that does not adhere to this form cannot constitute a solution. Because Equations(2.26) and (2.28a) are functionally the same as Equation (2.24), their respectivegeneral solutions take on the same cast as that of Equation (2.29):u(x, t) = Φ(x − ct) + Γ (x + ct) (2.30)ρ(x, t) = Θ(x − ct) + Y (x + ct) (2.31)The arbitrary functions F, G, Φ, Γ, Θ, Y can be assumed to have continuous derivativesof the first and second order. Because of the manner in which the constantc appears in relation to x and t inside these functions, it must have the physicaldimensions of x/t,soc must be a speed, which is indeed the experimentally determinedrate at which the sound wave is propagated through a medium. No matterhow it is shaped, the propagating wave (or its counterpart, the backward travelingwave) moves without changing its form. To prove this, consider the sound pressurelevel at x = 0 and time t = t 1 for a wave moving in the positive x-direction. Thusp = fá(t 1 ). At time = t 1 + t 2 , the sound wave will have traveled a distance x =ct 2 . The sound pressure will now be(p = f α (t 1 + t 2 ) = f α t 1 + t 2 − ct )2= f α (t 1 )t 2This means the sound pressure has propagated without change.ReferencesBeranek, Leo L. 1986. Acoustics. New York: American Institute of Physics. (An exceptionallyclear text in the field.)Crocker, Malcolm J. (ed.). 1997. Encyclopedia of Acoustics, Vol. 1. New York: JohnWiley & Sons, Chapters 1 and 2. (Sir James Lighthill compared the significance of thisfour-volume compilation with that of Lord Rayleigh’s The Theory of Sound, which isnot at all far-fetched considering that this encyclopedia contains contributions from aneditorial board whose members constitute a veritable Who’s Who in Acoustics. Handbookof Acoustics by the same editor and publisher (1998) is a truncated version of theEncyclopedia, containing approximately 75 percent of the chapters. Chapters 1 and 2are identical in both publications.)

Problems for Chapter 2 29Shapiro, Ascher H. 1953. The Dynamics and Thermodynamics of Compressible FluidFlow, Vol. 1. New York: The Ronald Press Co., Chapter 1. (In spite of its venerable age,it is still one of the best works on the topic of fluid dynamics.)Problems for Chapter 21. Write the expression for a simple sine wave having a frequency of 10 Hzand an amplitude of 10 −8 cm. What is the frequency expressed in radians persecond? Plot the expression on graph paper or, better yet, on a computer withthe aid of a program such as Excel © , Mathcad © , MathLab © , etc.Repeat the process for a frequency of 20 Hz and for 50 Hz.2. If the frequency of a pure cosine wave is 100 Hz and the velocity of the wavefront is 330 m/s, what is the wavelength of this signal? Express the frequencyin radians per second.3. Air may be considered to be a nearly ideal gas with the ratio of specific heatγ = 1.402. At 0 ◦ C its density is 1.293 kg/m 3 . Predict the speed of sound c forthe normal atmospheric pressure of 101.2 kPa (1 Pa = 1 N/m 2 ).4. Nitrogen is known to have a molecular weight of 28 kg/kg mol. Predict thespeed of sound at 0 ◦ C, 20 ◦ C, and 50 ◦ C, with the assumption that nitrogenbehaves as an ideal gas. Repeat the problem for pure oxygen which has amolecular weight of 32.5. Compute the speed of sound (in ft/s) traveling through steel that has a Young’smodulus of 30 × 10 6 psi and a specific gravity of 7.7. Why does it differ fromcast iron?6. A solid material is known to have a density of 8.5 g/cm 3 . Sound velocitytraveling through this material was measured as being 4000 m/s. Determinethe Young’s modulus in GPa for this material.7. Find the speed of sound (in m/s) traveling through aluminum that typicallyhas a Young’s modulus of 72.4 GPa and a specific gravity of 2.7.8. For distilled water, the speed of sound c in m/s can be predicted within 0.05%as a function of pressure P and temperature T from the experimentally determinedformulac(P, t) = 1402.7 + 488t − 482t 2 + 135t 3 + (15.9 + 2.8t + 2.4t 2 )(P a /100)where P a is the gauge pressure in bars (1 bar = 100 kPa) and t = 0.01T , withtemperature T in degrees Celsius. Find the speed of sound for the water at20 ◦ C and 1 bar. What will be the wavelength of a 200-Hz sine-wave signaltraveling through water? If the same signal travels through air at the speed of331 m/s, what will be the corresponding wavelength?9. Explain why density and pressure are in phase and that both are out of phasewith particle velocity.10. When does the maximum amplitude of a pure sine wave occur with respect tothe particle velocity and the instantaneous pressure?

30 2. Fundamentals of Acoustics11. A molecule exposed to a pure cosine sound wave undergoes a particle displacementy, with maximum amplitude A, according toy = A cos ωtFind the corresponding particle velocity and show that the expressions for bothdisplacement and velocity constitute solutions to a wave equation.12. Demonstrate that y(x, t) = A 1 cos(x − ct) + B 1 sin(x + ct) + A 2 cos 2 2(x −ct) + B 2 sin 2 3(x + ct) constitutes a solution to the wave equation. Whichportion of the solution represents wave travel in the +x direction and whichportion denotes propagation in the –x-direction?13. If the density of a medium subject to wave propagation varies in the followingmanner:ρ = ρ 0 e i(x−ct)express the corresponding pressure p(x,t) in terms of quiescent pressure p 0and density ρ 0 .

3Sound Wave Propagationand Characteristics3.1 The Nature of Sound PropagationWhen energy passes through a medium resulting in a wave-type motion, severaldifferent types of waves may be generated, depending upon the motion of a particlein the medium. A transverse wave occurs when its amplitude varies in thedirection normal to the direction of the propagation. This type of wave has beenused to describe the transmission of light and alternating electric current. But thesituation is almost completely different in the case of sound waves, which are principallylongitudinal, in that the particles oscillate back and forth in the directionof the wave motion, with the result the motion creates alternative compression andrarefaction of the medium particles as the sound passes a given point. The net fluiddisplacement over a cycle is zero, since it is the disturbance rather than the fluidthat is moving at the speed of sound. The fluid molecules do not move far fromtheir original positions.Additionally, waves may also fall into the category of being rotational or torsional.The particles of a rotational wave rotate about a common center; the curlof an ocean wave roaring onto a beach provides a vivid example. The particlesof torsional waves move in a helical fashion that could be considered a vectorcombination of longitudinal and transverse motions. Such waves occur in solidsubstances, and shear patterns often result. These are referred to as shear waves,which all solids support.3.2 Forward Propagating Plane WaveIn Equation (2.29), which is the general solution to the one-dimensional waveequation, we consider only the wave moving in the +x direction with the solutionfor a monofrequency wave represented byp(x, t) = F(x − ct) = p m cos k(x − ct) (3.1)where p m is the peak amplitude of the sound pressure; k, the wave number whichequals 2π/λ; and λ, the wavelength. Figure 3.1 shows the variation of sound31

32 3. Sound Wave Propagation and CharacteristicsFigure 3.1. Variation of pressure from quiescent state along the x-axis for time instantst = 0, t = T/4, t = T/2, t = 3T/4, where T = 1/f is the period for a complete cycle tooccur.

3.3 Complex Waves 33pressure p for different time intervals t = 0, t = T/4 = 1/(4 f ) = (1/2)π/ω,t =T/2 = 1/(2 f ) = π/ω, t = 3/(4 f ) = 3T/4 = (3/2)(π/ω), and t = T = 1/f =2π/ω.3.3 Complex WavesThe concept of simple sinusoidal waves lacks specificity to be of practical valuein noise control, but complex periodic waveforms can be broken into two or moresinusoidal harmonically related waves. In Figure 3.2, a complex waveform resolvesinto a sum of harmonically related waves. The harmonic relationship in thisexample is such that the frequency of one harmonic is twice that of the other. Thelowest-frequency sine term is the fundamental, and the next highest frequency thesecond harmonic, the next the third harmonic, and so on. Sound pressure waves radiatingfrom pumps, gears, and other rotating machinery are usually complex andperiodic, with distinguishable discrete tones or pure tones. These sinusoidal wavescan be broken down or synthesized into simple sinusoidal terms. In the analysisof the noise emanating from rotating machinery, there are often 8 to 10 harmonicspresent with frequencies which are integer-ordered multiples of the fundamentalfrequency. Even aperiodic sounds such as the hiss of a pressure valve of an autoclave,the broadband whine of a jet engine, or the pulsating sound of a jackhammercan be resolved and described in terms of sums of simple sinusoids. Integer harmonicrelationships associated with periodic sound waves do not occur in thesesounds, and the composition entails more than a simple series. But the principleof synthesis still applies.Figure 3.2. The resolution of a complex waveform into a set of harmonically relatedsinusoidal waves. The fundamental wave and the second harmonic sine wave add upalgebraically to form the complex wave.

34 3. Sound Wave Propagation and CharacteristicsA general equation can be written to incorporate the elements of the soundpressure level associated with complex periodic noise sources asp(t) = A1 sin(ωt + φ 1 ) + A2 sin(2ωt + φ 2 )+ A3 sin(3ωt + φ 3 ) +···+ An sin(nωt + φ n )n∑n∑= A n sin(nωt + φ n ) = C n e iωt (3.2)1whereA n = amplitude of the nth harmonicφ n = phase angle of the nth harmonicC n = complex amplitude of the nth harmonicEquation (3.2) constitutes a form of the Fourier series, an analytical tool developednearly two centuries ago by the French physicist Jean Baptiste Joseph Fourier tocharacterize complex functions and used by him to predict tides. Fourier’s conceptof complex wave synthesis constitutes one of the most powerful analytical anddiagnostic tools available to the present-day acoustician.When two or more sound waves become superimposed upon each other, theycombine in a linear manner, i.e., their amplitudes add algebraically at any point inspace and time. The superposition generally results in a complex wave that can besynthesized into basic sinusoidal spectrum components. Two special phenomenaresulting from superposition are of special interest, namely, beat frequency andstanding waves.Consider the superposition of two sound waves of equal amplitudes but slightlydiffering frequencies. With A 0 denoting the amplitude of each wave and ω 1 ≄ ω 2 ,the total superimposed pressure becomesp(t) = A0(sin ω 1 t + sin ω 2 t)Applying the trigonometric identitysin α + sin β = 2 cos1(α − β)2the total pressure assumes the formp(t) = 2 A0 cos (ω 1 − ω 2 )t2= 2 A0 cos 2π ( f 1 − f 2 )t2sin(α + β)2sin (ω 1 + ω 2 )t2sin 2π ( f 1 + f 2 )t2(3.3)where ω = 2π f .From Equation (3.3) the resultant wave may be considered a complex soundwave with a frequency of ( f 1 + f 2 )/2, as indicated by the sine factor and whichis the average of the two superimposed waves. The amplitude isp ′ (t) = 2 A0 cos 2π ( f 1 − f 2 )t2

3.3 Complex Waves 35Figure 3.3. Addition of wavesA and B with equal amplitudes but slightly differing frequencies.The sum of the two sine waves yields an envelope C which has a beat frequencyequal to the difference between the frequencies of the superimposed waves A and B.When the argument of the cosine assumes integer values of π, the amplitude ofthe complex wave is a maximum that is equal to 2A 0 . Continuing the reasoningfurther, it is established that the amplitude of the complex wave vanishes when theargument of the cosine takes on integer odd values of π/2, i.e.,2π ( f 1 − f 2 )t2=(2n − 1)π2(n = 1, 2, 3,...) (3.4)A graph of the envelope of this transient amplitude modulation is given inFigure 3.3. The modulation or beat frequency is simply the frequency difference( f 1 – f 2 ) between the two superposed waves. To demonstrate this, let us solveEquation (3.4) for those times t n when the amplitude of the superimposed soundpressure is zero,t n = 2n − 1 (n = 1, 2, 3,...)2( f 1 − f 2 )Now consider in a general fashion the time difference between two consecutivebeats, namely, the nth and the (n + 1)th:t n+1 − t n ==2(n + 1) − 12( f 1 − f 2 )− 2n − 12( f 1 − f 2 )1f 1 − f 2(3.5)

36 3. Sound Wave Propagation and CharacteristicsFigure 3.4. Addition of waves A and B with slightly different frequencies but unequalamplitudes. Envelope C that results from adding waves A and B shows beats in which theminimum strength is not zero.The time duration between beats is, by definition, the period T b of the beat frequency;and the reciprocal of the period defined in Equation (3.5) yields the beatfrequency f b :f b = f 1 − f 2 (3.6)In the more general case when the amplitudes of the superposed equations are notequal, the amplitude of the superposed wave varies between the sum and differenceof the component waves, as shown in Figure 3.4. The periodic variation in amplitudegenerates a rhythmic pulsating sound, and when the frequency difference isonly a few hertz, say 4 or 5, the human ear can readily discern the beat.3.4 Standing WavesWhen a sound wave is superposed upon another wave of the same frequency buttraveling in a different direction, a standing-wave sound field is generated. As anillustration, consider the superimposition of two sound waves traveling in oppositedirections as given byp 1 (t) + p 2 (t) = A1 sin(2π ft − kx) + A2 sin(2π ft + kx) (3.7)The first sine term in Equation (3.7) represents a sound wave traveling in thepositive x-direction with amplitude A 1 and frequency f . The second sine term

3.4 Standing Waves 37represents a sine wave traveling in the negative x-direction with amplitude A 2 andidentical frequency f . Using trigonometric identities for the sum and differenceof angles, we rewrite Equation (3.7) as follows:p 1 (t) + p 2 (t) = A1 sin 2π ft cos kx − A1 cos 2π ft sin kx+ A2 sin 2π ft cos kx + A2 cos 2π ft sin kxFor waves of equal amplitudes we obtain the following simplification:p 1 (t) =+p 2 (t) = 2 A1 cos kx sin 2π ft (3.8)Equation (3.8) may now be considered to be a simple sinusoidal function of timewhose amplitude depends on the spatial location x of the observer. When theargument of the cosine assumes odd integer values of π/2, i.e.,kx = π 2 , 3π 2 , 5π − 1)π,....,(2n (n = 1, 2, 3,...)2 2the sound pressure vanishes, and there are nodal points in space where no soundoccurs. Solution of the preceding equation for x n yields the spatial locations ofthese nodes:(2n − 1)πx n = (n = 1, 2, 3,...)2kand since k = ω/c = 2π/λ, we obtain(2n − 1)λx n = (n = 1, 2, 3,...)4We thus note that the location of the nodes is simply related to the wavelengthof the superimposed waves. By taking the difference between successive nodallocations, it can be demonstrated that the nodes occur every half wavelength, i.e.,x n+1 − x n =2[(n + 1) − 1]λ4−(2n − 1)λ4= λ 2We can also establish from Equation (3.8) that the antinodes or points of maximumsound pressure in the standing wave occur when the argument of the cosine assumesintegers values of π, i.e.,kx = nπ (n = 1, 2, 3,...)The amplitude of the antinodes is simply 2A 1 . These antinodes or points of maximumsound pressure are stationary, located halfway between the nodes and spacedone-half wavelength apart.Example Problem 1Consider a case where a hydraulic pump radiates a 600-Hz sound wave that isreflected back from a tile wall located at 1mawayfrom the wall. What is thespacing of the nodes or position of minimum sound for the fundamental tones andits second harmonic?

38 3. Sound Wave Propagation and CharacteristicsSolutionWe first calculate the wavelength of the first harmonic from the relationship λ =c/f , with the speed of sound being taken at 344 m/s at room temperature:λ = c = 344f 600 = 0.573 mThe spacing between the nodes occurs every half wavelength, or distance intervalsof 0.287 m. With the hydraulic pump positioned in close proximity to the wall,the amplitude of the antinodes will be nearly twice as large, as a result of thesummation of the radiated and the reflected waves. For the second harmonic at1200 Hz, the spacing of the nodes will be 0.143 m, since its wavelength is half thatof the fundamental. This example points out the necessity for caution in takingmeasurements in close proximity to highly reflecting surfaces, which can yieldhighly misleading results. In many situations reflective surfaces are not in closeproximity and the amplitudes of reflected waves are relatively small compared tothe original waves, so the variation in the amplitudes of the standing waves arecorrespondingly small and thus can be neglected.3.5 Huygens’ PrincipleWhile originally conceived in the seventeenth century to explain optical phenomena,Huygens’ principle applies equally to sound propagation. The principle statesthat advancing wavefronts can be considered to be point sources of secondarywavelets. Figure 3.5 illustrates the Huygens’ construction of a wave front at timeFigure 3.5. Construction of a wavefront at time t + t from its previous state at time t,according to Huygens’ principle.

3.6 The Doppler Effect 39t + t from a wavefront at earlier time t. The new wavefront is the envelope of radiict centered at points on the preceding wavefront. Thus, a plane wave remainsa plane wave and a spherical wave remains a spherical wave with ever-enlargingradius.3.6 The Doppler EffectWhen a sound source moves, the acoustic radiation pattern changes, thereby producingchanges to the generated frequencies as perceived by a stationary observer.When an acoustic source generating a frequency f approaches an observer at velocityv, then during a single period T (=1/f ) a signal emitted at the onset of theperiod travels a distance cT. But the signal is emitted at the end of the period fromthe source that is closer to the observer by a distance vT . The distance betweenthe crests, i.e., the wavelength λ, has been reduced toλ = cT − vT = c − v(3.9)fThe resulting frequency heard by the observer is not the source’s output frequencybut that increased by the resulting drop in the wavelength, that is to say,f d = c λ = fcc − v = f(3.10)1 − v/cHere f d is the frequency perceived by the observer. When the source approachesat a velocity v, the observer hears a higher-frequency sound which represents theoriginal frequency multiplied by a factor (1 – v/c) −1 . On the other hand, whenv assumes a negative value, which means the source is pulling away from theobserver, f d assumes a lower value than that of the source frequency f , since thevelocity v in Equation (3.10) assumes a negative value.Example Problem 2A train emits a 250-Hz signal while traveling at the rate of 200 km/h. What are theapparent frequencies in approaching the observer and retreating from the observerat the railroad crossing?SolutionFrom Equation (3.10)v = (200,000 m/h)/(3600 s/h) = 55.6 m/sf d = 250 Hz (1 − 55.6/344) −1 = 298 Hz for approachf d = 250 Hz (1 + 55.6/344) −1 = 215 Hz for retreatIf an observer stands on a line making an angle θ with a source’s direction of motionat speed V , the approach velocity of the source is v = V cos θ, and Equation (3.10)

40 3. Sound Wave Propagation and Characteristicsmodifies to:ff d =1 − V cos θ(3.11)cThe relative frequency increases or diminished, when θ exists as an acute or obtuseangle.3.7 ReflectionWhen sound impinges upon a surface, a portion of its energy is absorbed by thesurface and the remainder bounces back or becomes reflected from the surface. Aperfectly hard surface will reflect back all of the energy. A classic example of thereflection phenomenon is the echo which has intrigued and mystified humanity forcenturies.As waves impinges on a hard, smooth surface, the waves are reflected with shapeand propagation characteristics unaltered, in accordance with Huygens’ principle.Consider the impingement of a series of plane waves on reflecting surface A-A ′ inFigure 3.6. The arrows normal to the wavefronts, or rays, which represent the directionsof propagation, are drawn to represent the impingement and the consequentreflection of the wavefront. It follows from the application of Huygens’ principleand geometry that the angle of incidence is equal to the angle of reflection, wherethe angles are defined between a normal to the reflecting plane and the incident andthe reflected rays, respectively. In Figure 3.7, the geometric ray construction is renderedfor a diverging spherical wave incident upon a plane surface. The directionof the reflected sound can at least be qualitatively determined. It should be pointedout here that standing wave interference patterns will occur from these reflections.It is of interest to consider the sound field resulting from reflection. Consider thesound waves in Figure 3.6 to be sinusoidal. As the incident or reflected wavefrontFigure 3.6. Geometric depiction of a plane-wave reflection. Angle of reflection θ r is equalto angle of incidence θ i .

3.7 Reflection 41Figure 3.7. Construction for a spherical wave from point S incident upon plane surfaceα–α ′ . Point S ′ is an imaginary point that is the mirror image of point S on the other sideof plane α–α ′ .intersects any normal to the wavefront, a scissorlike effect occurs, not unlike oceanwaves breaking obliquely along a beach. The intersection of these waves alongthe normals constitutes a projection of the incident and reflected waves. From theconcept of wave motion, the distance between crests along the normal may be establishedas a projected wavelength λ ′ , which relates to the incident wave as follows:λ ′ =λ = λ sec ϑ 1 (3.12)cos ϑ 1In obeying the laws of reflection, the reflected wave also scissors back alongthe normal in the opposite direction, producing a traveling wave with a projectedwavelength also equal to λ ′ . Hence, there occurs along any normal line the superimpositionof two waves traveling in opposite directions with wavelength λ ′ .From the concept of standing waves it can be inferred that nodes and antinodesoccur along the normal line, and, moreover, the spacing between the nodes andantinodes needs only to be modified by the factor sec θ 1 of Equation (3.12).Consider a complex periodic wave that impinges upon a fully reflective planesurface. A standing wave sound field will exist. The distance d ′ between peaksalong the normal ensues from Equation (3.12) in the following manner:d ′ = λ′2 = λn sec θ1where λ n is the wavelength of the nth harmonic and θ i the angle of incidence ofthe propagating wavefronts. From the last equation it will be noted for the specialcase of θ i = 0 (normal incidence), the nodal spacing reduces to λ/2, accordingto Equation (3.12). As the angle of incidence increases, the spacing between thenodes likewise increases, and in the limit θ i = π/2, there is no reflected wave, andthus the standing wave field vanishes.

42 3. Sound Wave Propagation and CharacteristicsThe phenomenon of sound wave reflection finds many applications. The time ittakes for a sound wave pulse to travel from a transducer at sea level to the oceanbottom and for the echo to travel back gives a measure of the depth of the water.Further, comparison of the spectral characteristics of the reflected wave with thoseof the generated waves provides an ample measure of the geological compositionof the ocean bottom, for example, silt, rock, sand, coral, and so on. Reflected soundis also used in an analogous way by geologists to gauge the depth and compositionof stratified layers in the earth crust, to locate oil, natural gas, and mineral deposits.3.8 RefractionA phenomenon more familiar in optics than in acoustics is that of refraction, inwhich the direction of the advancing wavefront is bent away from the straightline of travel. Refraction occurs as the result of the difference in the propagationvelocity as the wave travels from one medium to a different medium.In the optical situation, refraction occurs suddenly when light waves cross thesharp interface between the atmosphere and glass at the surface of a lens, becauselight travels more slowly in glass than it does in air. At audible frequencies ofsound waves, the wavelengths are so long that the apparatus would have to beextremely large in order to render observable acoustic refractions. However, at ultrasonicfrequencies, which correspond to extremely short wavelengths, refractionconstitutes the operating principle of the acoustic microscope. The device functionsas indicated in Figure 3.8. A piezoelectric transducer P z , under the impetusMFigure 3.8. A schematic of an acoustic microscope. Voltage V causes the piezoelectriccrystal P t to launch a short train of waves into lens L. The propagation velocity of thewaves slows down in liquid medium M, and the waves are focused toward point S onthe surface of the specimen. Reflected waves follow the same paths in reverse, reachingthe piezoelectric transducer that now has been switched into the detector mode.

3.8 Refraction 43of an input voltage V to the opposite faces of the piezoelectric crystal, deliversa short train of ultrasonic waves into the lens L, which may consist of a smallblock of sapphire, which incorporates a spherical hollow in the face facing liquidmedium M. The waves travel much faster in the crystal than in the adjoining liquid.As a result the central portion of each wave is retarded relative to the outerparts. All of the wavefront enter the liquid at the same time, but the refractiveeffect causes almost all of the nearly spherical waves to focus at the central pointof curvature. The strength of the reflection depends on the nature of the specimensurface at the focal point S. The operating mode of the lens and transducernow changes from the role of an emitter to that of a receptor. The lens gathersthe reflected signal and the transducer detects the signal. In this fashion this deviceresembles radar and sonar systems. As information is obtained from onlyone point at a time, the specimen must be moved in a raster pattern on the focalplane of the microscope while an image is progressively accumulated in a computermemory.If water is used as a medium, a 3-MHz signal, with a propagation velocity of1480 m/s, would have a wavelength of 500 μm = 0.5 mm, which would amountto a rather coarse resolution. Clearly, higher-frequency signals are called for, butsuch signals become strongly absorbed in water. A medium with a lower value ofc and, more importantly, less absorption than water constitutes another possibility.An attractive choice turned out to be liquid helium, used with instruments thatgenerate signals up to the 8-GHz frequency range. The wavelengths are as smallas 0.03 μm.In Figure 3.9, a geometrical ray construction illustrates the refraction of soundpassing from one medium to another. Application of Huygens’ principle leads toFigure 3.9. A sound wave passing from medium 1 to medium 2. In this case the speed ofsound c 2 in medium 2 is greater than the speed of sound c 1 in medium 1.

44 3. Sound Wave Propagation and Characteristicsthe basic laws of refraction, the most useful of which issin θic 1=sin θrc 2(3.13)whereθ i = angle of incidenceθ r = angle of refractionc 1 , c 2 = speeds of sound in medium 1 and medium 2, respectively.Equation (3.13) should be recognized as the analog to the Snell law for lightrefraction. While the analysis of refraction does not figure prominently in noisecontrol, we cannot overlook the fact that zones of severe temperature differencesdo occur in the atmosphere and oceans. When sound travels from zone to zone,often across regions of severe temperature gradients, the direction of propagationchanges measurably to an extent that cannot be ignored.For example, the surface of the Earth heats up more rapidly on a sunny day thanthe atmosphere. Due chiefly to conduction, the temperature of the air close to theground rises correspondingly. Because the speed of sound is higher in the warmerlower layer, sound waves traveling horizontally are refracted upward. Similarly ona clear night the Earth’s crust cools more quickly, and a layer of cooler air formsand bends the sound waves downward toward the surface. Thus the noise from anindustrial plant would be refracted downward at night and would seem louder to ahomeowner residing near the plant than during the day (when upward refractionoccurs), which is often the situation.Nonuniform sound speed also constitutes a very important factor in underwateracoustics owing to the persistent presence of temperature and salinity and pressuregradients in the ocean. It is not unusual to find a minimum in c at some depth,usually in the order of 1 km, with higher values above and below that stratum.Interesting possibilities can occur, one of which is communication through soundchannels in which trapped signals traveling horizontally retain their strength moreeffectively than if they had been able to spread in all directions. Another is theexistence of shadow zones, where sound waves from a particular source neverarrive, so they provide good places for submarines to hide.3.9 DiffractionIn Figure 3.10, sound waves impinge upon a barrier. Some of the sound is reflectedback, some continues onward unimpeded, and some of the sound bends or diffractsover the top. The barrier does not provide a sharply delineated acoustical shadow.Another example of diffraction is bending of sound around a building corner. Weusually can hear voices on the other side of a wall that is approximately 3 mhigh.

3.10 Octave and One-Third Octave Bands in the Audio Range 45Figure 3.10. Impingement of sound waves on a partial barrier, with a resulting sounddiffraction and a shadow zone.The analytical treatment of sound barriers is covered in Chapter 12, but it sufficesfor now to say qualitatively that sound at lower frequencies tends to diffractover partial barriers more easily than sound at higher frequencies. Moreover, thesharpness and extent of the shadow zone behind the barrier depend on the relativepositions of the source and receiver. The closer the source is to the barrier, thelonger is the shadow zone on the other side of the barrier, i.e., the more soundreduction obtained.3.10 Octave and One-Third Octave Bandsin the Audio RangeFor analytical purposes, the audio range of frequencies are divided into 10 standardoctave bands with center frequencies f C = 31.5, 63, 125, 250, and 500 Hz, and 1,2, 4, 8, and 16 kHz. Each octave band-center frequency f C is double the precedingone and each bandwidth doubles the preceding one. The lower and upper limits of

46 3. Sound Wave Propagation and CharacteristicsTable 3.1. Octave Bands.Lower Band Limit Center Frequency Upper Band Limitf L f C f U22.4 Hz 31.5 Hz 45 Hz45 63 9090 125 180180 250 355355 500 710710 1 kHz 1.4 kHz1.4 kHz 2 2.82.8 4 5.65.6 8 11.211.2 16 22.4each octave band are given byBecausef L = f C√2f U = √ 2 f C (3.14)f C = √ f L f U (3.15)we see that the center frequency f C constitutes the geometric mean of the upperband limit f U and lower band limit f L . The bandwidth BW for full octaves isdefined by( ) √2 1BW = f U − f L = f C − √2 = f C√ (3.16)2and thus the ratio BW/f C is shown to constitute a constant. In order to avoid theuse of irrational numbers the octave bands have been standardized in the field ofacoustics, according to Table 3.1.One-third octave bands are formed by subdividing each octave band intothree parts. The successive center frequencies increase in intervals by cube rootof 2, and the upper and lower frequencies are related to the center frequency asfollows:f L = f C6√2, f U = 6 √2 fC , f C = √ f L f U (3.17)From Equations (3.16) and (3.17), the ratio BW/f C is also a constant for the thirdoctavebands,BW √−6√= 6 2 − 2 (3.18)f CTable 3.2 lists the standardized one-third octave limits and center frequencies.

3.11 Root-Mean-Square Sound Pressure and the Decibel 47Table 3.2. One-Third Octave Band.Lower Band Limit Center Frequency Upper Band Limitf L (Hz) f C (Hz) f U (Hz)18.0 20 24.422.4 a 25 28.028.0 31.5 a 35.535.5 40 45 a45 a 50 5656 63 a 7171 80 90 a90 a 100 112112 125 a 140140 160 180 a180 a 200 224224 250 a 280280 315 355 a355 a 400 450450 500 a 560560 630 710 a710 a 800 900900 1,000 a 1,1201,120 1,250 1,400 a1,400 a 1,600 1,8001,800 2,000 a 2,2402,240 2,500 2,800 a2,800 a 3,150 3,5503,550 4,000 a 4,5004,500 5,000 5,600 a5,600 a 6,300 7,1007,100 8,000 a 9,0009,000 10,000 11,200 a11,200 a 12,500 14,00014,000 16,000 a 18,00018,000 20,000 22,400 aa Octave marking points.While one-third octave bands generally suffice in providing adequate information,there are cases where one-tenth and even one-hundredth octaves are applied.For 1/nth octaves, successive center frequencies are related as follows:f n+1 = 2 1/n f n (3.19)3.11 Root-Mean-Square Sound Pressure and the DecibelSound consists of small positive pressure disturbances (compression) and negativepressure disturbances (rarefaction) measured as deviations from the equilibriumor quiescent pressure value. The mean-pressure deviation from equilibrium is

48 3. Sound Wave Propagation and Characteristicsalways zero, since the mean rarefaction equals the mean compression. A simpleway to measure the degree of disturbance is to square the values of the soundpressure disturbance over a period of time, thereby eliminating the counter-effectsof negative and positive disturbances by rendering them always positive. The rootmean-squaresound pressure p rms can be defined byp rms = √ √ ∫ τ(p) 2 0=p2 dt∫ τ0 dt (3.20)where τ is the time interval of measurement and p the instantaneous pressure. Fora simple cosine wave over an interval of period T = 2π/ω, there results√ ∫ Tp0rms =p2 m cos2 k(x − ct) dtT= p m√2. (3.21)The sound pressure as portrayed by the oscillation of the pressure above andbelow the atmospheric pressure is detected by normal human ear at levels as lowas approximately 20 μPa (the SI unit of pressure is the pascal, abbreviated Pa,equivalent to 1.0 N/m 2 ). 1 Because p rms could vary over a wide range of ordersof magnitude, it would be cumbersome to use it as the measure of loudness. Atthe threshold of pain, p rms would reach approximately 40,000,000 μPa! The blastoffpressure in the vicinity of the launching pad of a Titan rocket can exceedup to a thousandfold the threshold of pain (i.e., 40 kPa). It is therefore moreconvenient to use the decibel as the folding-scale measure of loudness. This unit isdefined by( ) 2 ( )prmsprmsL p = 10 log = 20 log(3.22)p 0 p 0whereL p = sound pressure level (dB)log = common (base-ten) logarithmp 0 = 20 × 10 −6 Pa = the reference pressure.From the context of Equation (3.22) it can be established that the doubling ofa root-mean-square pressure corresponds to approximately 6 dB increase in thesound pressure level. In order to determine the sound pressure level from a givenvalue of L p , Equation (3.22) can be rewritten asorp rmsp 0= 10 L p20p rms = 20 × 10 L p20 −61 One standard atmosphere equals 101.325 kPa.

3.12 Decibel Additions, Subtracting, and Averaging 49Figure 3.11. Sound pressure levels and corresponding pressures of various sound sources.Figure 3.11 illustrates the range of the decibel scale in terms of measured valuesof common sound sources. The audible range of sound, which encompasses musicand speech, is shown delineated in Figure 3.12 in terms of sound pressure levelsand frequencies.3.12 Decibel Additions, Subtracting, and AveragingMost sound pressure levels do not arise from single sources, nor do they remainconstant in time. Mathematical procedures must be used to add, subtract, andaverage decibels. From the definition of Equation (3.22) it is apparent that decibelsfrom single-noise sources do not add or subtract directly. If we wish to add sound

50 3. Sound Wave Propagation and CharacteristicsFigure 3.12. Sound pressure levels for audible range, music range, and the range of speechversus frequency. (Source: Brüel & Kjær Instruments, Inc.)pressure levels (abbreviated to SPLs) L p1 , L p2 , L p3 ,...L pn , we must obtain theantilogs of these SPLs to convert them into squares of rms pressures which canthen be added directly to yield the square of total rms pressure, that in turn yieldsthe total dB. Because( ) 2 ( )piL= log −1 pip ref 10the total sound pressure level L pt becomes[ n∑L pt = 10 logi=1(pip ref) 2](3.23)or, in terms of the sound pressure levels,[ n∑ ( ) ] (Ln∑L pt = 10 log log −1 pi= 10 log10i=1i=110 L pi/10)(3.24)Example Problem 3Find the total sound pressure level due to L p1 = 96 dB, L p2 = 87 dB, and L p3 =90 dB.

3.12 Decibel Additions, Subtracting, and Averaging 51SolutionApplying Equation (3.24) we have for this caseL pt = 10 log(10 96/10 + 10 87/10 + 10 90/10 )= 97 dBIn certain situations, it is desirable to subtract an ambient or background noisefrom the total sound pressure level L pt in order to establish the sound pressurelevel L ps due to a particular source. Subtraction of decibels is analogous to theprocedure for their addition. The total sound pressure level L pt is converted intothe mean-square pressure ratio, as in Equation (3.23), and the background noiselevel L pB is measured simply by turning off the noise source. The mean-squarepressure ratio due to the background noise is obtained from( ) 2 pBL pB = 10 logp refor( ) 2 ( )pBL= logp −1 pB= 10 L pB /10ref 10The sound pressure level L ps of the source is found fromL ps = 10 log ( 10 L pt/10 − 10 L pB/10 ) (3.25)Example Problem 4Measurements indicated L pt = 93 dB at a specific location with a lathe in operation.When the lathe is shut down the background noise measures at 85 dB. Whatis the sound level due to the lathe?SolutionAccording to Equation (3.25), the noise level due to the lathe isL ps = 10 log(10 93/10 − 10 85/10 )= 92 dBA requirement may arise occasionally to find the average decibels in order todetermine the average sound pressure level L p . In some situations we may wishto measure the SPL at a single location several times and determine an averagevalue for engineering evaluation purposes. The procedure for averaging decibels isbased on the same premise for the summation and subtraction of decibels, namely,the application of Equation (3.22). Equation (3.24) for the addition of decibels is

52 3. Sound Wave Propagation and Characteristicsmodified by dividing the sum by n, the number of levels taken into consideration,in order to obtain the average value of L p :( )1n∑L p = 10 log 10 L pi10(3.26)nExample Problem 5Determine the average sound pressure level L p for a series of measurements takenat different times: 96, 88, 94, 102, and 90 dB.SolutionUsing Equation (3.26) we write[ ]1L p = 10 log5 (1096/10 + 10 88/10 + 10 94/10 + 10 102/10 + 10 90/10 )= 97 dBn=13.13 Weighting Curves and Associated Sound LevelsHuman perception of loudness depends on the frequency of a sound. A noisehaving most of its energy concentrated in the middle of the audio spectrum (e.g.,in the region of 1 kHz) is perceived as being louder than noise of equal energybut concentrated either in the low-frequency region (say, 40 Hz) or in the highfrequencyregion (near 15 kHz). This frequency effect becomes more apparent withsoft sounds than is the case with loud sounds, which provides the raison d’êtrefor the presence of a loudness control on some audio amplifiers. This controlsupplies a loudness contour at low volumes, which applies greater amplificationto the high- and low-frequency contents of the program material relative to themiddle-frequency components.Frequency weighting takes typical human hearing response into account whenthe loudness generated by all of the audible frequency components present isto be represented by a single value. Rather than describing the sound level ineach frequency band, we can use the A-weighted sound level to report the overallloudness. The A-weighted sound level is obtained from the conversion chartof Table 3.3, which also lists the B and C weightings in 1/3-octave bands.A-weighting is almost exclusively used in measurements that entail human responseto noise. Sound level that is measured with A-weighting is reported interms of dB(A) or simply dBA rather than the generic decibel dB. Similarly,B-weighted and C-weighted measurements are designated as dB(B) (rarely used)and dB(C), respectively. In the conversion table, it will be noted that all of the

54 3. Sound Wave Propagation and CharacteristicsFigure 3.13. Frequency responses for the A-, B-, and C-weighting networks.Example Problem 6Find the total A-weight sound level L for the octave-band sound pressure levelsgiven below:Band-Center Frequency (Hz)Sound Pressure Level (Hz)31.5 7363 68125 72250 68500 801000 882000 954000 838000 9716,000 92

3.14 Performance Indices for Environmental Noise 55SolutionUse Table 3.3 to obtain the dB conversion from a flat response to dB(A) for eachof the octave bands. This results in73 dB at 31.5Hz= 73 − 39.4 = 33.6 dB(A)68 dB at 63 Hz = 68 − 26.2 = 41.8 dB(A)72 dB at125 Hz = 72 − 16.1 = 55.9 dB(A)68 dB at 250 Hz = 68 − 8.6 = 59.4 dB(A)80 dB at 500 Hz = 80 − 3.2 = 76.8 dB(A)88 dB at 1 kHz = 88 − 0 = 88 dB(A)95 dB at 2 kHz = 95 + 1.2 = 96.2 dB(A)83 dB at 4 kHz = 83 + 1.0 = 84 dB(A)97 dB at 8 kHz = 97 − 1.1 = 95.9 dB(A)92 dB at 16 kHz = 92 − 6.6 = 85.4 dB(A)The dB(A) values in each of the bands can be summed up for the total sound levelL p , through the use of Equation (3.24).3.14 Performance Indices for Environmental NoiseAs the result of the passage of the Noise Control Act of 1972 by the U.S. Congress,the Environmental Protection Agency (EPA) issued two major documents publishedin April 1974, in accordance with Section 5 of the Act. One document dealtprincipally with the criteria for time-varying community noise levels and the otherdocument is concerned with definitions of performance indices for noise levels.These indices are generally represented as single-number criteria, serving as internationallyrecognized, simple means of assessing the noise environment. Threeperformance indices are described in this section. Because they utilize A-weightedmeasurements, these three statistically based methods of quantifying noise exposurestend to have good correlation with human response. These indices are L N ,which presents the levels exceeded N percent of the measurement time; L eq , theequivalent continuous sound pressure level in dB(A); and L dn , the day–night soundlevel average in dB(A).L N may be measured with the use of an amplitude-distribution analyzer. Anoutput of the device can provide a histogram, an example of which is shownin Figure 3.14. The time in any chosen band can be read as a percentage of thetotal observation time. The cumulative distribution curve in the figure indicates theprobability of exceeding each range of decibel levels. In noise-abatement planning,criteria are often specified in terms of sound levels that are exceeded 10%, 50%,and 90% of the time. These levels are customarily represented as L 10 , L 50 , andL 90 , respectively.

56 3. Sound Wave Propagation and CharacteristicsFigure 3.14. A histogram showing probability of exceedance for plant noise.Example Problem 7From perusal of Figure 3.14, estimate the sound levels that are exceeded 10%,50%, and 90% of the time. Also establish the percentage of the total observationtime that the sound was between 65 and 67 dB(A).SolutionThe probability-of-exceedance curve in Figure 3.14 is read to yield ≈69 dB(A)(exceeded 10% of the time), L 50 ≈ 63 dB(A) (exceeded 50% of the time), andL 90 ·≈ 58 dB(A) (exceeded 90% of the time).Equivalent sound level L eq is the sound energy averaged over a given period oftime T , i.e., it is the rms or mean level of the time-varying noise. It is defined byL eq = 10 log( ∫1 TT 0p 2p 2 ref)dt(3.27)where p 2 = p 2 (t) is the mean-square (time-varying)sound pressure and p ref =20 μPa. Equation (3.27) can be more conveniently rewritten in terms of soundlevel L = L(t) using the relationship of Equation (3.22):( ∫ 1 T)L eq = 10 log 10 L p/10 dtT 0(3.28)In order to facilitate digital processing in the measurement of L eq through the useof an integrating sound level meter, the integral form of Equation (3.28) is replaced

3.14 Performance Indices for Environmental Noise 57by the equivalent summation:()1N∑L eq = 10 log 10 L n/10(3.29)Nn=1Here L n is acquired instrumentally for each of N equal intervals to yield L eq ,inthe course of digital processing of discrete samples.Example Problem 8Find L eq in the case where L n = 90.5, 95, 103, 88, and 98 dB(A) are obtained asthe respective average levels for five short, equal time intervals.SolutionFrom Equation (3.29)[ ]1Leq = log5 (1090.5/10 + 10 95/10 + 10 103/10 + 10 88/10 + 10 98/10 )= 97.9 dB(A)The day–night equivalent sound pressure level, L dn , essentially a modification ofL eq , was conceived for the purpose of evaluating community noise problems. Themodification consists of a nighttime penalty of 10 dB imposed on measurementsbetween 10 P.M. and 7 A.M. With time t given in hours, Equation (3.28) nowbecomes⎡ ⎛⎞⎤Ldn = 10 log ⎣ 110∫P.M.7A.M. ∫⎝ 10 L/10 dt + 10 (L+10)/10 dt⎠⎦ (3.30)247A.M.10 P.M.When the equivalent sound levels L eqd and L eqn are known for the day and nightperiods, respectively, the following version of Equation (3.30) can be used tocompute the day–night sound level:L dn = 10 logExample Problem 9[ 124(15 × 10 L eqd/10 + 9 × 10 (L eqn+10)/10 )] (3.31)Find L dn for the situation where the daytime equivalent sound level is 82 dB(A)and the nighttime equivalent sound level is 76 dB(A).SolutionInserting the appropriate values in Equation (3.31) we obtain[ ]182(76+10)Ldn = 10 log (15 × 10 10 + 9 × 10 10 ) = 84 dB(A)24

58 3. Sound Wave Propagation and Characteristics3.15 Particle Displacement and VelocityInvoking Equation (2.21) and inserting the wave equation solution (3.1), we obtainthe expression for the particle velocity u:u =− 1 ∫ ∂ρρ ∂x dt =−p mcos k(x − ct)ρcu =p(3.32)ρ cwhere ρ = quiescent density of air = 1.18 kg/m 3 at a normal room temperature of22 ◦ C and atmospheric pressure of 101.3 kPa and c = speed of sound = 344 m/s.The term ρc is the characteristic or acoustic impedance for a wave propagating inair in a free-field condition. The value of ρc at standard conditions of temperatureand pressure is 40.7 rayls or 407 MKS rayls. The dimensional unit rayl is definedas follows:1 rayl = 1.0 dyne s/cm 3The particle displacement x for a cosine wave function can be found by simplyintegrating Equation (3.32) with respect to time:x = p msin k(x − ct)ρc2 It is interesting to note that at 0 dB, the threshold of human hearing, the oscillationof an air molecule covers an rms amplitude that is approximately only one-tenththe diameter of a hydrogen atom.The particle acceleration is obtained from the differentiation of Equation (3.32)with respect to time, and for a cosine wave function the acceleration isdudt= p sin k(x − ct)ρ3.16 Correlated and Uncorrelated SoundCorrelated sound waves occur when they have a precise time and frequency relationshipbetween them. An example of correlated sound waves is the output oftwo identical loudspeakers located in the same plane, consisting of a pure tonesupplied by a single amplifier connected to both loudspeakers. Most of the soundwaves that we hear are generally uncorrelated.Consider two sound waves which are detected at a point in space:p 1 = P 1 cos (ω 1 t + φ 1 )p 2 = P 2 cos (ω 2 t + φ 2 )(3.33a)(3.33b)

or in terms of complex exponential functionsp 1 = Re [ P 1 e i(ω 1t + φ 1 ) ]p 2 = Re [ P i( ω 2 t+ φ 2 )]2where3.16 Correlated and Uncorrelated Sound 59p = instantaneous sound pressureP = amplitude of sound pressureω = angular frequencyφ = phase angle(3.33c)(3.33d)The instantaneous sound pressure resulting from the superimposition of the twowaves is given by the sum of the two instantaneous sound pressures, and theroot-mean-square sound pressure of the combined waves can be found fromp 2 rms = 1 T∫ T0(p 1 + p 2 ) 2 dt (3.34)where T represents the averaging time, which should be an integer number ofperiods at both frequencies. In real measurements it suffices to have the averagingtime cover many periods so that contributions from fractional periods becomeinsignificant. This condition is met if T ≫ 1/f lower , where f lower is the lower ofthe two frequencies. Inserting Equations (3.33a) and (3.33b) or Equations (3.33c)and (3.33d) into Equation (3.34) and integrating, we obtainprms 2 = P12 + P2 2= prms1 2 2+ prms2 2 for ω 1 ≠ ω 2 (3.35)p 2 rms = P12 + P2 2+ P1 P2 cos(φ21 φ 2 ) for ω 1 = ω 2 (3.36)Consider the case of two 4-kHz signals that are in phase at the receiving point.Each of the signals has a sound pressure level of 60 dB. From Equation (3.36) forsound waves of the same frequency, the root-mean-square pressure is given byprms 2 = P2 1 + P2 2+ P 1 P 2 cos 0 = 2P1 2 2= 4 prms2The increase in sound pressure level as the result of adding an identical in-phasepure tone is( p 2L P − L P1 = 10 log rmsp 2 ref= 10 log 4 ≈ 6dB)− 10 log( p 2) (rms14p2)= 10 log rms1p 2 refp 2 rms1The combined signals result in a 4-Hz signal with an SPL of 66 dB. If the amplitudeswould be out of phase by π radians (or 180 ◦ ) and the frequencies are equal, thesound pressure would theoretically be zero at the observation point.

60 3. Sound Wave Propagation and CharacteristicsFigure 3.15. Complex exponentials portrayed as rotating vectors.Let us now determine the effect of adding two 60-dB signals, which have frequenciesof 1000 and 1100 Hz, respectively. Because the frequencies are not equal,the root-mean-square sound pressure is double that of one wave. The sound pressurelevel increases by( p 2)10 log rmsprms12 = 10 log 2or approximately 3 dB. The combined SPL is 63 dB.We gain a further insight into the phenomenon of beat frequency if we visualizethe complex exponential functions (3.33c) and (3.33d) as rotating vectors inFigure 3.15. Without loss of generality we can define time t = 0 as the instantwhen both vectors lie along the positive real axis, resulting in the maximum soundpressure. The two vectors will be opposed along the real axis when (ω 2 − ω 1 )t =2π. The envelope of the pressure–time curve yields a period τ = 2π/(ω 2 − ω 1 ) =1/( f 2 – f 1 ). The term ( f 2 − f 1 ) is, of course, the beat frequency, which has beenpreviously discussed in Section 3.3.3.17 Sound IntensityAn acoustic signal emanates from a point source in a spherical pattern over anincreasingly larger area. When a closed surface completely surrounding the sourceis defined, the sound power W radiated by the source can be established from∫W = I · dS (3.37)whereI = sound intensity, W/m 2dS = element of surface area, m 2SS = surface area surrounding source

3.17 Sound Intensity 61Here the surface integral in Equation (3.37) is the integral of the sound intensity Inormal to the element dS of the surface area. The integration can be executed over aspherical or hemispherical surface enclosing the source. If the source of power W ismounted on an acoustically hard surface (i.e., a surface which is totally reflective),the sound waves expand within a hemisphere. Other surfaces, such as those of aparallelopiped (representing, for instance, the walls of a room), are often used inpractical applications. When the integration is performed over a spherical surfaceof radius r for a nondirectional source, sound intensity is related to sound powerbyI (r) = W S = W(3.38)4π r 2where S denotes the area of a sphere having radius r. Equation (3.38) constitutesthe inverse-square law of sound propagation, which accounts for the fact that soundbecomes weaker as it travels in open space away from the source, even if viscouseffects of the medium are disregarded. For sound radiation within a hemisphere,with the sound source mounted at the origin above a totally reflective surface,Equation (3.38) becomesI =W2π r 2Intensity, which represents the transfer of sound wave energy, equals the productof sound pressure and particle velocity,I = p · u (3.39)and for a simple cosine spherical wave, the pressure p(r, t) is given as a solutionto the spherical coordinate form of Equation (2.25)p(r, t) = A rcos k(r − ct)A is a constant amplitude with its physical units in N/m. From Equation (2.22) thevelocity isu(r, t) =− 1 ∫ [ Aρ r k sin k(r − ct) + A ]r cos k(r − ct) dt2=− 1 [− kA cos k(r − ct) −A]ρ kcr r 2 kc sin k(r − ct)oru(r, t) =A[ρcr cos k(r − ct) 1 + 1]kr tan k(r − ct)At large values of kr Equation (3.40) becomesu(r, t) ≈p(r, t)ρc(3.40)k 2 r 2 ≫ 1 (3.41)

62 3. Sound Wave Propagation and Characteristicsand for k 2 r 2 ≪ 1:u(r, t) ≈Ap(r, t)sin k(r − ct) ≈ ̸ 90 ◦ re p(r, t) (3.42)kρcr2ρckrThe difference between Equations (3.41) and (3.42) connotes, respectively, the farfield and near field effects of a spherical wave. As r approaches the center of thespherical source the sound pressure and particle velocity becomes progressivelymore out of phase, approaching 90 ◦ as the limit. In the near field the sound intensityis not simply related to the root-mean-square value of the sound pressure.A sound source is generally directional, and the sound intensity does not havethe same value at all points on the surface. In order to evaluate the integral ofEquation (3.37) it is necessary to execute an approximation by segmenting thesurface into a finite number of sub-elements, each subtending an area S i and toestablish the sound intensity on each sub-element in a direction normal to thatelement. A summation procedure over all of the surface sub-elements will yieldthe total sound powerW = ∑ iI i S i (3.43)whereI i = sound intensity averaged over the ith element of area S i , W/m 2S i = ith element of area, m 2Equation (3.43) can be expressed logarithmically as( ) ( )I SLw = 10 log + 10 logI0 S0( ) SLw = L I + 10 logS0whereL w = sound power level, dB re 10 −12 WL I = sound intensity level, dB re 10 −12 W/m 2S = area of surface, m 2S 0 = reference area = 1.0m 2I 0 = reference sound intensity, internationally set at 10 −12 W/m 23.18 The Monopole SourceA monopole can be described as an idealized point generating a spherical soundwave. A pulsating sphere can be considered a good approximation of a point sourcewhen its radius is small compared with the wavelength of the sound it generates.

3.18 The Monopole Source 63The three-dimensional Equation (2.25) is expressed in spherical coordinates, withoutangular dependence, as∂ 2 (p ∂ 2∂ t = p2 c2∂ r + 2 )∂p(3.44)2 r ∂rThe solution to Equation (3.44) must be of the formp = 1 r [F1 (ct − r) + F2(ct + r)]with the term F 1 (ct − r) describing waves moving away from the source. Wediscard the term F 2 (ct + r) which describes waves traveling toward the source.With time t = 0 in order to eliminate a phase angle, a harmonic solution to thespherical wave equation (3.44) is given byp = A rcos[k(ct − r)]where A is a constant and k the wave number equal to ω/c. The root-mean-squaresound pressure prms 2 at a distance r from the source is given by∫ Tp 2 rms = 1 p 2 dt =A2cos[k(ct − t)]dt = A2(3.45)T 0 r 2 T 02 r 2Here T = 1/f = 2π/ω or the period needed to complete one cycle. If many frequenciesare present, prms 2 and the root-mean-square pressure can be measuredfairly accurately if the integration time is sufficiently large compared with theperiod of the lowest frequency. The sound power of the spherical source can befound by the use of Equation (3.37) as follows:∫∫W = I · n dS (3.46)∫ TEquation (3.46) represents the sound power of a source wheres∫ TI = 1 ρu dt (3.47)T 0constitutes the vector sound intensity; u represents the particle velocity vector, Sany closed surface about the source, n the unit normal to surface S, and T theaveraging time. The surface S for a spherical wave is defined by a sphere of radiusr about the source, so that Equation (3.46) becomes∫∫W = Ir dS (3.48)The magnitude of the sound intensity I r is directed radially, i.e., it runs parallelto unit normal n. When sufficiently far from the source, the sound pressure andparticle velocity are in-phase. Applying Equation (3.39) yieldsIr = p rms u rms = p2 rmsρcS

64 3. Sound Wave Propagation and Characteristicswhere ρ is the mass density of the propagation medium. With the sound sourcebeing isotropic (i.e., omnidirectional, with no angular-dependent variations), andintegration of Equation (3.46) over 4π steradians, the sound power is given byW = Ir S = 4π r 2 Ir = 4πr 2 prms2 (3.49)ρcIf sound power is measured in half-space, i.e., if the source lies on a reflectivesurface, then integration occurs over 2π steradians andW = 2πr 2 prms2 (3.50)ρcHere, W denotes the sound power of the source in watts, I the sound intensity(W/m 2 ) in the direction of wave propagation, and r the distance from the centerof the source.For a spherical wave in full space, root-mean-square sound pressure and soundintensity in the direction of wave propagation are related to the sound power ofthe source byprms 2 = ρcW(3.51a)4π r 2I =W(3.51b)4π r 2The tendency of the sound intensity in Equation (3.51b) to decrease with increasingdistance from the source is called the inverse square law.3.19 The Spherical Wave: Sound Pressure Leveland Sound Intensity LevelCombining Equations (3.51a) and (3.51b) with the definition of sound pressurelevel given by Equation (3.22), we express the sound pressure Lp in terms of soundpower W of the source and distance r from the source:( p 2) ( )L p = 10 log rmsρcWpref2 = 10 log4πr 2 pref2= 10 log(ρcW) − 20 log r + 83 (3.52)where p ref = 20 μPa. The sound intensity level L I in the direction of sphericalpropagation is found from( ) ( )IWL I = 19 log = 10 logIref4πr 2 pref2= 10 log W − 20 log r + 109= L W − 20 log r − 11 (3.53)where I ref = 10 −12 W/m 2 and L W is the sound power level (dB re 1 pW).

3.21 Energy Density 653.20 The Hemispherical WaveIf an omnidirectional sound power source W is placed above an acoustically hard(i.e., totally reflective) surface, the sound waves expand within a half-space. Forsound pressure and particle velocity in phase, sound intensity in the direction ofpropagation is given byand the sound intensity level byI =W2πr 2 (3.54)L I = L W − 20 log r − 8 (3.55)A hemianechoic chamber is constructed by installing wedges of sound-absorbingmaterials on the ceiling and walls of a room with an acoustically hard floor. Incontrast, a full anechoic chamber is constructed with virtually all of its surfaces,including the floor, lined with sound absorption wedges. A mesh floor or gratingabove the bottom surfacing provides structural support to equipment and laboratorypersonnel. With either type of chamber, a free-field condition is generated within,in which the sound emitted by a source placed inside does not undergo reflection.3.21 Energy DensityIn the mathematical treatment of sound in enclosed spaces, we need to know theamount of energy per unit volume being transported from a source to differentparts of the room. Both kinetic energy and internal (potential) energy are involvedin sound propagation. An interchange between these two forms of energy occursfrom the compression/rarefaction process and the motion of the propagationmedium particles. The kinetic energy density e k is simply ρu 2 /2, and when thewave amplitude is small it will be fairly legitimate to assume the quiescent valueof ρ 0 instead of ρ. In general, we may write an expression for kinetic energy interms of displacement vector x:e k = ρ 02The three-dimensional version of Equation (2.24) is1∇ p =− ∂2 xρ 0 ∂t 2( ) ∂x 2(3.56)∂twhich leads to∫1ρ 0∇ pdt = ∂x∂t(3.57)

66 3. Sound Wave Propagation and CharacteristicsInserting Equation (3.57) into Equation (3.56) yieldse k = 1 (∫ ) 2∇ pdt2 ρ 0For a general case of a sine spherical wave described byp = A sin(r − ct)we havee k = 12 ρ 0 c 2 p2 (3.58)From elementary thermodynamics, the change in energy per unit volume V 0 associatedwith the variation of density is given for a volume V of the fluid by∫ V0e p =− 1 pdV (3.59)V0 VThe negative sign indicates that the potential energy increases when compressionoccurs (i.e., when density increases) and decreases with rarefaction (when densitydecreases) under the impetus of an acoustic signal. In order to perform the integrationwe need to express all variables in terms of one variable, namely, instantaneouspressure p. From conservation of mass p 0 V 0 = pV = constant, and differentiatingyieldsFrom Equation (2.23),dV =− V ρdρ ≈−V0ρ 0dρdpdρ = γ p 0ρ 0= c 2for an isentropic process in an ideal gas. Eliminating dρ between the precedingtwo equations givesdV =− V0ρ 0 c dp 2which now can be inserted into Equation (3.59), which is then integrated from 0to p to yielde p = 1 (3.60)2 ρ 0 c 2The sum of Equations (3.58) and (3.60) constitutes the total instantaneous energydensity denoted by ee = 1 )2 ρ 0(u 2 + p2ρ0 2 (3.61)c2Because the particle speed and acoustic pressure are functions of both timeand space, the instantaneous energy density is not constant throughout the fluidp 2

References 67medium. The time average E of e provides the energy density at any point in thefluidE = 1 T∫ T0edtwhere the time interval T represents one period of a harmonic wave. With the factthat p =∀ρ 0 cu, as manifested in Equation (3.32), Equation (3.61) becomese = ρ 0 u 2 = pu/cIf we now let p and u represent the amplitude of the pressure and particle velocity,respectively, then the time-averaged energy E is written asE = 1 2puc = p22 ρ 0 c 2 = 1 2 ρ 0 u 2 (3.62)For the cases of spherical or cylindrical waves or standing waves in a room, thepressure and particle velocity in Equation (3.61) must be the real quantities derivedfrom the superposition of all waves present. In these more complex cases, thepressure is not necessarily in phase with the particle speed, nor is the energydensity given by pu/2c. But E = pu/2c does constitute a good approximationfor progressive waves if the surfaces of constant phase approaches a radius ofcurvature much greater than a wavelength. This situation occurs for spherical andcylindrical waves at distances considerably far (i.e., many wavelengths) from theirsources.ReferencesFederal Register, May 28, 1975. v. 40: 23105.Federal Register, Oct. 29, 1974. v. 39: 38208.Federal Register, Oct. 30, 1974. v. 39: 38338.Harris, Cyril M. (ed.). 1991. Handbook of Acoustical Measurements and Noise Control,3rd ed. New York: McGraw-Hill, Chapters 1 and 2. (A most valuable reference manualfor the practicing acoustician.)Lord, Harold, Gatley, William S., and Evensen, H. A. 1980. Noise Control for Engineers.New York, NY: McGraw-Hill, pp. 7–30.Morse, Philip M. and Ingard, K. Uno K. Uno. 1968. Theoretical Acoustics. New York:McGraw-Hill. Chapter 1. (A classic text which probably contains more details thannecessary, with the result that the mathematics tends to obscure the physics ofacoustics.)Pierce, Alan D. 1981. Acoustics: An Introduction to Its Physical Principles andApplications. New York: McGraw-Hill, Chapters 1 and 2. (An excellent modern text bya contemporary acoustician.)

68 3. Sound Wave Propagation and CharacteristicsProblems for Chapter 31. A signal consists of the following components:y 1 = 10 sin 4t, y 2 = 6 sin 8t, y 3 = 4.3 sin 10tPlot each component and add them up to obtain a composite wave.2. A 300-Hz sound wave is propagating axially in a steel bar. Find the wavelengthand the wave number.3. Determine the wave number and the wavelength of a 30-Hz pure tone at20 ◦ C.4. Two sine waves have approximately the same amplitudes, but one is at afrequency of 135.3 Hz and the other 136.0 Hz. What is the beat frequency?What is the time duration between the beats?5. Show mathematically how noise cancellation can be effected by duplicatingan offending signal and changing its phasing. In real life situation, can thecancellation be a total one? If not, why not?6. It is desired to place a worker at a “quieter” location near a machine that putsout a steady 400 Hz hum. Where would this location be and where are thepoints where the noise would be greater?7. A train whistle is measured with its frequency at 250 Hz when the trainsapproaches the observer near the tracks at the rate of 125 km/h. Predict thefrequency when the train is pulling away from the observer.8. Train A traveling 60 km/h is approaching Train B traveling on a parallel trackat 85 km/h. Train A blows its whistle which has a fundamental of 255 Hz. Whatfrequency will the engine man at Train B hear? What will be the perceivedfrequency after the whistle on Train B passes the locomotive of Train A?Neglect the distance between the tracks.9. An observer stands 150 km from a railroad track. A train is 300 km on thenormal from the track to the observer. It is traveling at 80 km/h and approachingthis normal. Its whistle emits a fundamental of 300 Hz. What will be thefrequency of this signal perceived by the observer?10. Develop an equation for the rate of change of the frequency with respect toan observer subtending an angle θ with a source’s direction. The source ismoving at a velocity V .11. A boundary exists between two mediums, A and B, through which an acousticsignal travels. The signal traveling in Medium A impinges the boundary atan angle of 45 ◦ . In passing into Medium B on the other side of the boundarythe signal refracts at an angle of 55 ◦ . The velocity of sound in Medium A is450 m/s. Determine the velocity of sound in Medium B on the other side ofthe boundary.12. Given a 150-Hz signal expressed asp(x, t) = 35 sin 2.5(x − 344t)determine the followings:

Problems for Chapter 3 69(a) the wave number(b) the wavelength(c) the root-mean-square pressure13. Convert the following rms pressures into decibels:(a) 20 μPa(b) 150 μPa(c) 1 kPa(d) 50 kPa14. Convert the following values expressed in dB into rms pressure:(a) 20 dB(b) 60 dB(c) 90 dB(d) 130 dB15. Two machines are running. One machine puts out 95 dB and the other 98 dB.What will be the combined sound pressure level in dB?16. Three machines are operating simultaneously. Their combined noise level is115 dB. One machine is shut down and noise level drops to 110 dB. Theremaining two machines have identical noise output. What is the noise outputof these two machines?17. An octave-band analysis of a machine yields the following results:Band-center frequencySPL (Hz)31.5 7263 76125 77250 72500 691000 842000 924000 838000 8016,000 78Find the total A-weighted sound level, the total B-weighted sound level, andthe total C-weighted sound level.18. The following SPL readings were taken of a noisy electric generator:Amount of time, sUnweighted SPL reading, dB15 73.422 79.420 88.912 91.9Find the equivalent sound pressure level L eq .

70 3. Sound Wave Propagation and Characteristics19. At a property line the noise from a nearby machine shop was found over a24-h period to have the following averaged sound pressure levels.TimeNoise Level, dB(A)7 A.M.–12 noon 87.512 noon–4 P.M. 84.64 P.M.–9 P.M. 78.59 P.M.–3 A.M. 76.53 A.M.–7 A.M. 77.4Determine the L eq and L dn .20. Determine the particle velocity for air at 1 atm and 22 ◦ C.

4Vibrating Strings4.1 IntroductionIn dealing with vibrating systems it is commonly assumed that the entire mass ofthe system is concentrated at a single point and the motion of the system can bedescribed by giving the displacement as a function of time. This rather simplifiedapproach yields approximations rather than accurate closed-form solutions. Aspring, for example, certainly does not concentrate its mass at one end, nor cana loudspeaker be accurately depicted as being a massless piston engaged in anoscillating motion. The loudspeaker diaphragm consists of a considerable portionof its mass spread out over its surface, and each part of the diaphragm can vibratewith a motion that is different from those of other segments.The vibrational modes of a loudspeaker constitute a complex affair, so it wouldbehoove us to study simpler modes of vibration, say, those of a vibrating string orbar, so that we can readily visualize the transverse vibrations. Even in the simplestof cases, certain simplifying assumptions have to be made which cannot be fullyjustified in the real physical world.4.2 The Vibrating String: Basic AssumptionsConsider a long, heavy string stretched to a moderate tension between two rigidsupports. A momentary force is applied to the string that becomes displaced fromits equilibrium position. The displacement does not remain in the initial position;it breaks up into two separate disturbances that propagate along the string apartfrom each other as shown in Figure 4.1. The propagation velocity of all smalldisplacements depends only on the mass and tension of the string, not on the shapeand amplitude of the initial displacement. The wave generated by such a transverseperturbation is generally known as a transverse wave.4.3 Derivation of the Transverse Wave EquationIn Figure 4.2, a portion of a string under tension T and rigidly clamped at itsends is shown. The string has negligible stiffness and a uniform linear density δ.71

72 4. Vibrating StringsFigure 4.1. History of the propagation of a disturbance in a stretched string.Dissipation of vibrational energy is neglected. We let x represent the coordinate ofa point along the horizontal distance with the origin at the left clamp of the string.The y-coordinate represents the transverse displacement from the equilibriumposition. As the transverse displacements are defined as being small, tension Tcan be considered nearly constant (T cos θ is even more so). Let θ denote the anglebetween a tangent to the string and the x-axis. In the segment of the string shownFigure 4.2. String element under the influence of tension force T .

4.4 General Solution of the Wave Equation 73in Figure 4.2, the difference between the y-components of the tension at the twoends of element ds is the net transverse force given by Fy = (T sin θ ) x+x − (T sin θ ) x (4.1)Here (T sin θ) x+x is the value of T sin θ at x + x, and (T sin θ) x is the valueat x. Letting x → dx and applying the Taylor’s series expansionf (x + dx) = f (x) + ∂ f (x)∂xdxEquation (4.1) can be rewritten as∂(T sin θ)dF y = (T sin θ ) x + − (T sin θ )∂xx =∂(T sin θ)dx∂xAs the displacement y is assumed to be small, θ will be correspondingly small andthe relationship sin θ ≈ tan θ applies, with tan θ equal to y/x. The net transverseforce on the element ds then becomes[(∂ T ∂y )]∂xdF y =dx = T ∂2 y∂x∂ x dx 2The mass of the string element is δ dx. Applying Newton’s law F = ma, we get:T ∂2 y∂x dx = y2 δdx∂2 ∂t 2Setting√Tc =δ(4.2)the equation of string motion becomes∂ 2 y∂t 2= c2 ∂2 y∂x 2 (4.3)The constant c defined in Equation (4.2) represents the propagation velocity of thetransverse wave. Equation (4.3) is the wave equation representing the wave disturbancespropagated along the string. This equation was first derived by LeonhardEuler in 1748.4.4 General Solution of the Wave EquationThe second-order partial differential Equation (4.3) has the general solutiony = f (ct − x) + g(ct + x) (4.4)where the functions, f (ct – x) and g(ct + x), are arbitrary with arguments (ct ± x).The first term of the right-hand side of Equation (4.4) represents a wave moving

74 4. Vibrating Stringsto the right (in the positive x-direction) and the second term a wave moving tothe left (in the negative x-direction). While each of the two wave shapes remainsconstant as the initial perturbation propagates along the string, the actual fact isquite the opposite since the simplifying assumptions are not fully realized in realstrings. In relatively flexible strings with low damping, the rate of wave distortionis quite minimal as long as the initial perturbation is kept small. Large amplitudes,however, will result in a larger rate of change of the wave shapes.The functions f (ct − x) and g(ct + x) cannot be freely arbitrary; they are constrainedby initial and boundary conditions. The initial conditions, established attime t = 0, are dictated by the type and the location of application of the perturbingforce applied to the string. To cite a musical example, the initial wave shapegenerated by plucking the string of a banjo or a harp will be quite different fromthe wave shape created by bowing a violin string. The boundary conditions extantat the ends of a string further limit the wave function. Real strings always havefinite lengths and are fixed in some fashion at their ends. The displacement sumy = f + g of Equation (4.4) is constrained to have a zero value at all times at theclamping points. Also, when a string is sustained in a steady-state condition byperiodic external driving forces, the functions f and g will also have the samefrequency as the applied forces but the amplitudes of vibration are determined bythe point of application of the force and by boundary conditions at the ends of thestring.Example 1: String Clamped at Both Ends. Given a string of length P clampedrigidly at x = 0 and x = L. The solutions y 1 = f (ct − x) and y 2 = g(ct + x) areno longer arbitrary, and their sum must be zero at all times, i.e.,f (ct − 0) + g(ct + 0) = 0orf (ct − 0) =−g(ct + 0) (4.5)The two functions must be of the same form but with opposite signs; and we cannow rewrite Equation (4.5) asy(x, t) = f (ct − x) − f (ct + x) (4.6)The first term on the right-hand side of Equation (4.6) represents a wave traveling tothe right (in the positive x-direction) and the second term a wave moving leftward(in the negative x-direction).4.5 Reflection of Waves at BoundariesThe reflection process at the boundary x = 0 can be viewed as one in which asecond wave does not pass the boundary point but is considered to reflect back,generating a similarly shaped wave of opposite displacement traveling in the positivex-direction. The presence of a fixed point at x = L results in another reflection.

4.6 Simple Harmonic Solutions of the Wave Equation 75In this case, the wave traveling in the positive x-direction reflects back as a similarwave of opposite displacement moving in the negative x-direction. The majorresult of these two reflections is that the motion of the free vibration becomesperiodic. A pulse leaving x = 0 reaches x = L after an interval of L/c seconds.There, it is reflected and returns to the origin where it again undergoes a reflectionafter a time lapse of 2L/c seconds. The shape of the pulse after its second reflectionis identical with that of the original pulse. This periodicity has resulted from thespecified boundary conditions, i.e., fixed points at x = 0 and x = L.4.6 Simple Harmonic Solutions of the Wave EquationSimple harmonic vibrations frequently occur in nature, and we shall now considera simple harmonic motion (SHM) propagating along a string. Any vibration ofthe string, however complex, can be resolved into an equivalent array of simpleharmonic vibrations. This resolution of complex vibration into a series of SHMsis not a mere mathematical exercise but constitutes the phenomenal principle ofhow the ear functions. The ear breaks down a complex sound into its simpleharmonic components. This capability permits us to distinguish the differencesbetween different voices and musical instruments. A piano sounding a note willsound differently from the same note played by an oboe. If all the frequenciespresent in the sound consist of a fundamental tone plus its harmonics, they willsound more harmonious than in the situation where the frequencies are not relatedso simply to each other.The displacement of any point on the string exciting a SHM of angular frequencyω can be depicted by the special solution to Equation (4.3):y = a 1 sin(ωt − kx) + a 2 sin(ωt + kx) + b 1 cos(ωt − kx) + b 2 cos(ωt + kx)(4.7)where a 1 , a 2 , b 1 , b 2 are arbitrary constants and k is the wavelength constant givenbyk ≡ ω/cApplying the boundary condition y(0, t) = 0 (i.e., y = 0atx = 0), which describesa fixed point, Equation (4.7) reduces to(a 1 + a 2 ) sin ωt =−(b 1 + b 2 ) cos ωt (4.8)Because this equation applies to all values of t, the following relations betweenthe constants must exista 1 + a 2 = 0, b 1 + b 2 = 0 or a 1 =−a 2 , b 1 =−b 2The two limitations for the arbitrary constants of Equation (4.7) are equivalentto the single restriction of Equation (4.5) in the general solution of wave equation(4.3). The two waves must be of equal and opposite displacements and must

76 4. Vibrating Stringstherefore differ in phase by π radians at x = 0. With these restrictions Equation(4.7) becomesy = a 1 [sin(ωt − kx) − sin(ωt + kx)] + b 1 [cos(ωt − kx) − cos(ωt + kx)](4.9)Making use of trigonometric transformations for the sine and cosine terms wesimplify Equation (4.9) toy = [−2a 1 cos ωt + 2b 1 sin ωt] sin kxThus y is expressed as a product of a time-dependent term and a coordinatedependentterm.Applying the boundary condition of y(L, t) = 0 (i.e., the displacement is zeroat end point x = L) adds yet another restrictionsin kL = sin nπ where n = 1, 2, 3,...Consequently, the string cannot vibrate freely at any random frequency; it can onlyvibrate with a discrete set of frequencies given byω n = nπc/Lwhere n = 1, 2, 3,..., or, in terms of frequency:f n = nc/(2L) (4.10)4.7 Standing WavesThe boundary conditions at x = 0 and at x = L reduced the general SHM solutionEquation (4.7) to a pattern of standing waves on the string. At the lowest orfundamental frequency, where n = 1, the displacement is given byy 1 = (A 1 cos ωt + B 1 sin ωt) sin k 1 x (4.11)Here k 1 = π/L, and A 1 and B 1 are arbitrary constants of which numerical valuesare established by the initial conditions, i.e., the type of excitation imparted tothe string at t = 0. This fundamental mode of vibration is associated with thefundamental (or first harmonic) frequency f 1 = c/2L. The nth mode of vibrationcorresponding to the nth harmonic frequency is represented byy n = (A n cos ω n t + B n sin ωt) sin k x (4.12)and the frequency is f n = nc/2L, i.e., n times the fundamental frequency. Theconstants A n and B n are determined by the initial excitation.In evaluating the term sin k n x = sin nπ x/L, we recognize that displacementy n = 0 occurs for all values of x when sin nπ x/L = 0, i.e.,nπ x/L = mπ where m = 0, 1, 2, 3,....

4.7 Standing Waves 77Figure 4.3. Different modes of vibrations for a string for the fundamental and the firsttwo harmonics.The cases for which m = 0 and m = n correspond to the boundary conditions atthe fixed points at the two ends of the string. However, there are additional (n − 1)locations, called nodal points or nodes, where the displacement produced by thenth harmonic mode of vibration remains at zero, as illustrated in Figure 4.3. Thissituation may be viewed as one in which the harmonic wave moving in the positivex-direction cancels precisely at the nodal points at all times t the harmonic wavemoving in the opposite direction. Because the points of zero displacement remainfixed, the resultant wave pattern constitutes what is known as standing waves. Thedistance between nodal points for the nth harmonic mode of operation is L/n,andthe points of maximum vibrational amplitudes are referred to as antinodes or loops.Let us take a “snapshot” of the vibrating string at a particular time for the sixthharmonic mode (Figure 4.4). The displacement of the string frozen in time occursas a sinusoidal function of x.This function repeats itself every length 2L/n of thex-coordinate, which, in turn, is equal to the wavelength λ n of the harmonic waves.In this case of the sixth harmonic mode, the repetition occurs every L/3 of thestring length. The wavelength is related to the velocity of propagation byλ n = c/f n

78 4. Vibrating StringsFigure 4.4. The sixth harmonic mode of a vibrating string stretched between x = 0 andx = L.where f n is the frequency of vibration in the nth mode and the wavelength constantby k n = 2π/λ n . From Equation (4.10) we deriveλ n = 2L/n (4.13)that is, the wavelength is twice the nodal distance of the associated wave pattern.4.8 The Effect of Initial ConditionsThe complete general solution to the general harmonic wave equation for a freelyvibrating string rigidly clamped at its ends contains all the individual modes ofvibration described by Equation (4.12). It is expressed as∞∑y n = (An cos ω n t + Bn sin ω n t) sin k n t (4.14)n=1where A n and B n are the amplitude coefficients dependent on the method of excitingthe string to vibrate. The actual amplitude of the nth mode is√a n = A 2 n + B2 nConsider the initial condition at t = 0 at the time when the string is displaced fromits normal linear configuration so that the displacement y(x,t) at each point of thestring is given by the functiony(x, 0) = y 0 (x)The corresponding velocity v(x,t) is given for t = 0byv(x, 0) =∂y(x, 0)∂t= v 0 (x)

4.8 The Effect of Initial Conditions 79In order that Equation (4.14) represents the string at all times, it also must describethe displacement at t = 0 and therefore is written asy 0 (x) = y(x, 0) =∞∑An sin k n x (4.15)The derivative of y with respect to time must also represent the velocity at t = 0,v 0 (x) = v(x, 0) =n=1∞∑ω n B n sin k n x (4.16)We apply Fourier’s theorem 1 to Equation (4.14) in order to obtain∫ LAn = 2 y 0 (x) sin k n xdx (4. 17a)L 0and then apply the theorem to Equation (4.16) to getn=1Bn = 2ω n L∫ L0v 0 (x) sin k n xdx(4.17b)Example 2: String Pulled and Suddenly Released. Consider a string that isplucked by pulling it at its center a distance d and then is suddenly released atinstant t = 0. In such a case, v 0 (x) = 0 and all the coefficients B n will be zero.The coefficients A n are given byA n = 2 L[ ∫ L202dxL= 8dn 2 π 2 sin nπ 2∫ Lsin k nxdx+ 2 dL L (L − x) sin kn dx2](4.18)1 The theorem states that a complex vibration of period T can be represented by a displacement x = f (t)written in terms of a harmonic seriesx = f (t) = A 0 + A 1 cos ωt + A 2 cos 2ωt +···A n cos nωt +···+ B 1 sin ωt+ B 2 sin 2ωt +···B n sin nωt + ...where ω = 2π/T and the constants are given byA 0 = 1 ∫ Tf (t) dtT 0A n = 2 ∫ Tf (t) cos nω dtT 0B n = 2 ∫ Tf (t) sin nω dtT 0

80 4. Vibrating StringsTherefore, all even modes n = 0, 2, 4,...haveA 2 = A 4 = A 6 =···=0and the odd modes result in non-zero A n coefficients:A 1 = 8dπ , A 2 3 =− 8d9π , A 2 5 = 8d ,25π2etc. (4.19)The amplitudes of the various harmonic modes are given by the numerical valuesofA n . In general, it may be observed that no harmonics are generated having a nodeat the point of the string initially plucked. As the nodal number n increases, theassociated amplitudes decrease from the value of the fundamental amplitude, i.e.,the fundamental A 1 is 9 times larger than A 3 and 25 times larger than A 5 , and so on.Example 3: Sharp Blow Applied to String. If the string is struck a sharp blow (asopposed to being plucked, as described above), v 0 (x,0) has nonzero values but noinitial displacement exists. Then all the coefficients A n are zero and the coefficientsB n are given by Equation (4.17b). A common example of a struck string is theimpact of a piano hammer striking a string. It is interesting to note that pianos aredesigned in such a way that the impact point of the hammer is one-seventh of thedistance from one end of the string, thus eliminating the seventh harmonic (whichwould have produced a discordant sound).4.9 Energy of Vibrating StringIn any nondissipative system the total energy content remains constant, equal to thevalue of the maximum kinetic energy. For the nth mode of vibration the maximumvalue of the kinetic energy of a segment of length dx isdE n = ω nδ (A22 n + Bn2 )sin 2 k n xdx (4.20)which is established by simply applying the relation dE = (mv)dv/2 in conjunctionwith Equation (4.6) and the fact that the mass of the string element is givenbym = δ dxwhere δ is the linear density. With integration over the variable x from 0 to L, themaximum kinetic energy of the string isE n = ω2 n δ (A22 n + Bn) 2 L2 = m (4 ω2 n A2n + Bn)2Here m is the total mass of the string and (A 2 n + B2 n ) is the square of the maximumdisplacement of the nth harmonic. In a conservative system (which describes thedissipationless vibrating string) the maximum potential energy is also equal to themaximum kinetic energy of the system. From Equation (4.20) the energy of the

4.10 Forced Vibrations in an Infinite String 81nth mode of vibration isE n = m 4 ω2 n A2 n = m 4( nπcL) ( ) 2 8d 2= 16md2 c 2n 2 π 2 n 2 π 2 L 2It becomes apparent that as n increases, the energy of the nth mode lessens. Forexample, the energy of the third harmonic is one-ninth that of the fundamentalmode.4.10 Forced Vibrations in an Infinite StringWhile this may appear to be a purely academic exercise, the simple case of atransverse sinusoidal force on an idealized string of infinite length can provideinsight into the forced vibrations of finite strings and the transmission of acousticwaves.An ideal string of infinite length subject to a tension T receives a transversedriving force T cos ωt at the string end x = 0. The end at x = 4 is rigidly clampedbut the point x = 0 is rigid only in the x-direction, being free to move in they-direction, a support that can be approximated by a pivoted lever as shown inFigure 4.5. Thus, the driving force can move the lever as well as the string. We shallneglect the mechanical impedance of the pivoted lever (or hinge) that is deemedto have no friction and no stiffness. No waves are reflected from the far end x = 4,and hence no waves travel in the negative x-direction. The displacement of thestring can now be described by the general solution containing only the expressionfor a harmonic wave traveling in the positive x-direction:or in a complex format:y = a 1 sin(ωt − kx) + b1 cos(ωt − kx)y = A e i(ωt−kx) (4.21)Figure 4.5. Forces acting at one end of an infinite string.

82 4. Vibrating Stringswhere A is a complex constant of which magnitude equals the displacement amplitudeof the wave motion and whose phase angle renders the difference in phasebetween the motion of the string and the driving force.In complex format the harmonic driving force can be written asf = Fe iωt (4.22)In Figure 4.5, the driving force is shown being applied to the string at an angle θthe string makes with the horizontal. This angle is given by( ) ∂ytan θ =∂xThe force exerted in the horizontal direction at the support at the end of the stringis −T cos θ. Because the displacements are assumed small, cos θ ≈ 1, and themagnitude of this force in the horizontal direction is essentially equal to tensionT in the string. From similar considerations, the transverse force exerted by thesupport on the string is −T sin θ, approximated by( ) ∂yf =−T sin θ =−T(4.23)∂x x=0Equation (4.23) indicates that for any applied transverse force the shape of thestring at x = 0 will vary. Inserting f and y from Equations (4.21) and (4.22) intoEquation (4.23) yields for this boundary conditionorx=0Fe iωt =−T (−ik)Ae i[ωt−k(0)]A =F(4.24)ikTThe term F/kT represents the magnitude of this complex amplitude A. InsertingEquation (4.24) into Equation (4.21) and then differentiating with respect to timeresults in the complex velocity vv = F( cT)e i(ωt−kx)The mechanical (or wave) impedance Z s of the string is defined as the ratio of thedriving force to the transverse velocity of the string at x = 0:Z s = T c = √ T δ = δcIt turns out that Z s is a real quantity, with no imaginary load. The mechanicalload presented by the string to the driving force is purely resistance. The inputimpedance exists as a function of the linear density δ and the tension appliedto the string, and it does not depend on the applied driving force; this means itis a property characteristic of the string, not the wave propagation in the string.The input, often termed characteristic or mechanical impedance (or resistance)

4.11 Strings of Finite Lengths: Forced Vibrations 83is analogous to the characteristic electrical impedance of an infinite transmissionline.The average power input to the string is found from the average value of theinstantaneous power W = fv evaluated at x = 0, orW = F 22δc = δcV 022where V 0 is the velocity amplitude at x = 0.4.11 Strings of Finite Lengths: Forced VibrationsIn the case of a finite string, the reflections from the far end generate frequenciesthat cause the input impedance to change greatly with the frequency of the drivingforce. If the support at finite x = L is fully rigid and no dissipative forces occurin the string, the input impedance becomes a pure reactance, and no power isconsumed in the string.Because the complex expression for transverse waves on a finite string now needsa term descriptive of the reflected wave, Equation (4.21) needs to be rewritten asy = Ae i(ωt−kx) + Be i(ωt+kx) (4.25)for all times t. The boundary condition at x = 0is( ) ∂yFe iωt =−T∂x x=0for all values of t. Inserting Equation (4.25) into Equation (4.26) yields(4.26)F =−T (−ikA + ikB) (4.27)Applying y(L, t) = 0 for the rigid clamp at x = L, Equation (4.25) becomes0 = Ae −ikL + Be ikL (4.28)Solving the Equations (4.27) and (4.28) for A and B results inA =FikT ·e ikLe ikL + e −ikL =Fe ikL2ikT cos kLandB =− FikT · e −ikL=−Fe−ikLe ikL + e−ikL 2ikT cos kLSubstituting the above constants into Equation (4.26) yieldsy = FeiωtikT · eik(L−x) − e ik(L−x)= Feiωt2 cos kL kT· sin k(L − x)cos kL(4.29)The real portion of Equation (4.29) graphs a pattern of standing waves on thestring with nodes occurring at those points where sin(L − x) = 0, in addition to

84 4. Vibrating Stringsy(0, L) = 0. The displacement at x = 0, however, has an amplitudey 0 = (F tan kL)/kT (4.30)Singularities of Equation (4.30) occur when cos kL = 0, i.e.,orandkL = ωLc(2n − 1)π= , n = 1, 2, 3,...2ω n =(2n − 1)πc2L(2n − 1)f n = c4LThese singularities connote infinite amplitudes which, of course, do not occurin real strings, because dissipative forces neglected in the foregoing analysis doactually exist. But the amplitudes do achieve maximum values at these frequencies.In a similar fashion we can ascertain the minimum amplitudes from the conditionkL =±1, i.e.,andkL = ω n L/c n = 1, 2, 3,....ω n = nπcLorf n = nc2LThe minimum amplitudes decrease progressively with increasing frequencies.In fact, on comparing with Equation (4.10) it is noted that the frequencies ofminimum amplitudes are identical to those of the free-string vibration, and theterm antiresonance has been applied to describe those frequencies. Differentiationof Equation (4.29) with respect to time t yields the complex velocity v of thestring,v = FeiωtT/c· sin k(L − x)cos kLThe input mechanical impedance then becomesZ s = feiωtv= T iccos kLsin kL=−iδc cot kLwhich exists as a pure reactance, with no power absorbed by the string. The amplitudeof vibration is a maximum at cot kL = 0, which occurs at the frequencygiven by f n = nc/2L. For extremely low frequencies the input impedance has thelimitsZ s =− iδckL =−iT ωL

References 854.12 Real Strings: Free VibrationReal strings manifest some degree of stiffness, causing the observed frequenciesto be higher than the theoretical values for idealized strings. This results from thepresence of elastic boundary forces augmenting the action of tensile forces consideredpreviously, with the net effect of increased restoring forces. The presenceof stiffness exerts a greater influence with increasing frequency, and the overtonesof a stiff string no longer constitute an exact harmonic series.We must also be attentive to the fact that clamping at the ends of the string maynot be exactly rigid and that yielding can occur at these points. The wave impedanceat the ends will constitute the transverse mechanical impedance of the supports.Consider a case where the left end of a finite string at x = 0 is attached to apivot representing the slightly loose clamp. The transverse force f 0 exerted by thestring on the hinge is( ) ∂yf 0 = T sin θ ≈ T∂x x=0Here y is the complex expression for the transverse wave on the string, as givenby Equation (4.25). Because the motion of the swivel must match that of the endof the string, the velocity is( ) ∂yv 0 = = iωy 0∂t x=0Let z 0 denote the transverse mechanical impedance of the hinge at x = 0( ) ∂yz 0 = f 0= Tv 0 iω · ∂x x=0and hence the boundary condition at x = 0 becomesy 0y 0 = T (∂y/∂x) x=LiωZ Lwhere Z L is the mechanical impedance of the swivel located at x = L. Ifthesupports were truly rigid, Z 0 = Z 1 = 4, and the boundary conditions reduce toy 0 = y L = 0.ReferencesFletcher, Neville H. and Rossing, Thomas D. 1998. The Physics of Musical Instruments,2nd ed. New York: Springer-Verlag, Chapter 2. (This text provides an excellent expositionon the physical principles of musical instruments.)Jean, Sir James. 1968. Science and Music. New York: Dover Publications, Chapter 3.(Although rather skimpy on the mathematical details, the exposition provides a goodinsight into the physics of vibrating string to the reader.)

86 4. Vibrating StringsKinsler, Lawrence E., Frey, Austin R., Coppens, Alan B., and Sanders, James V. 1999.Fundamentals of Acoustics, 4th ed. New York: John Wiley & Sons, Chapter 2. (Still agood textbook which dates back to 1950.)Morse, Philip M. 1982 (reprint from 1948 edition published by McGraw-Hill). Vibrationand Sound. Woodbury, NY: Acoustical Society of America, Chapter 3.Morse, Philip M. and Ingard, K. Uno. 1968. Theoretical Acoustics. New York: McGraw-Hill, Chapter 4.Reynolds, Douglas R. 1981. Engineering Principles of Acoustics, Noise and VibrationControl. Boston: Allyn and Bacon, pp. 213–224.Problems for Chapter 41. Show by direct substitution that each of the following expressions constitutessolutions of the wave equation:(a) f (x − ct)(b) ln [ f (x − ct)](c) A(ct − x) 3(d) sin [A(ct − x)]2. Show which of the followings are solutions and not solutions to the waveequation:(a) B(ct − x 2 )(b) C(ct − c)t(c) A + B sin(ct + x)(d) A cos 2 (ct − x) + B sin(ct + x)3. Plot (by computer if possible) the expression y = Ae −B(ct−x) for times t =0 and t = 1.0, with A = 6 cm, B = 4cm −1 , and c = 3cm/s. Discuss thephysical significance of these curves.4. Consider a string of density 0.05 g/cm, in which a wave form y = 4 cos (5t −3x) is propagating. x and y are expressed in centimeters, and time t in seconds.(a) Determine the amplitude, phase speed, frequency, wavelength, and thewave number.(b) Find the particle speed of the string element at x = 0 at time t = 0.5. A string is stretched with tension T between two rigid supports located atx = 0 and x = L. It is driven at its midpoint by a force F cos ωt.(a) Determine the mechanical impedance at the midpoint.(b) Establish that the amplitude of the midpoint is given by F tan (kL/2)/(2kT).(c) Find the amplitude of displacement at the quarter point x = L/4.6. Determine the mechanical impedance with respect to the applied force drivinga semi-infinite string at a distance L from the rigid end. What is the significanceof the individual terms in the expression for mechanical impedance?7. Consider a string of density 0.02 kg/m that is stretched with a tension of 8 Nfrom a rigid support to a device producing transverse periodic vibrations at theother end. The length of the string is 0.52 m. It is noted that for a specific drivingfrequency, the nodes are spaced 0.1 m apart and the maximum amplitude is0.022 m. What are the frequency and the amplitude of the driving force?

Problems for Chapter 4 878. A device that has a constant speed amplitude u(0,t) = U 0 e iωt , where U 0 is aconstant, drives a forced, fixed string.(a) Find the frequencies of the maximum amplitude of the standing wave.(b) Repeat the problem for a constant displacement amplitude y(0, t) =Y 0 e iωt .(c) Compare the results of (a) and (b) with the frequencies of mechanicalresonances for the forced fixed string. Does the mechanical amplitudecoincide with the maximum amplitude of the motion?9. Consider a string fixed at both ends, with specified values of ρ L, c, L, f, andT . Express the phase speed c ′ in terms of c and the fundamental resonancef ′ in terms of f if another string of the same materials is used but(a) the length of the string is doubled.(b) the density per unit length is doubled.(c) the cross-sectional area is doubled.(d) the tension is reduced by half.(e) the diameter of the string is doubled.10. Consider a string of length L that is plucked at the location L/3 by producingan initial displacement δ and then suddenly releasing the string. Findthe resultant amplitudes of the fundamental and the first three harmonicovertones. Draw (through computer techniques, if possible) the wave formsof these individual waves and the shape of the string occurring from the linearcombination of these waves at t = 0. Redo this problem for time t = L/c,where c represents the transverse wave velocity of the string.11. A string of length 1.0 m and weighing 0.03 kg has a mass of 0.15 kg hangingfrom it.(a) Find the speed of transverse waves in the string (Hint: neglect the weightof the string in establishing the tension in string).(b) Determine the frequencies of the fundamental and the first overtonemodes of the transverse vibrations.(c) For the first overtone of the string, compare the relative amplitude of thestring’s displacement at the antinode with that of the mass.12. A string having a linear density of 0.02 kg/m is stretched to a tension of 12 Nbetween rigid supports 0.25 m apart. A mass of 0.002 kg is loaded on thestring at its center.(a) Find the fundamental frequency of the system.(b) Find the first overtone frequency of the system.13. A standing wave on a fixed–fixed string is given by y = 3 sin (π x/4) sin 2t.The length of the string is 36 cm and its linear density 0.1 gm/cm. The unitsof x and y are in centimeters, and t is given in seconds.(a) Find the frequency, phase speed, and wave number.(b) Determine the amplitude of the particle displacement at the center of thestring and at x = L/4 and x = L/3.(c) Find the energy density for those points, and determine how much energythere is in the entire string?

5Vibrating Bars5.1 IntroductionThe theory underlying the physical operation of vibrating bars is of great interest toacousticians, because a number of acoustic devices employ longitudinal vibrationsin bars and frequency standards are established by producing sounds of specificpitches in circular rods of different lengths. The analysis of vibrating bars facilitatesour understanding of acoustic waves through fluids, for the mathematicalexpressions governing the transmission of acoustic plane waves through fluid mediaare similar to those describing the travel of compression waves through a bar.Moreover, if the fluid is confined inside a rigid pipe, the boundary conditions beara close correlation to those of a vibrating bar. An example of devices falling intothe category of vibrating bars include piezoelectric crystals which are cut so thatthe frequency of the longitudinal vibration in the direction of the major axis of thecrystal may be used to monitor the frequency of an oscillating electric current orto drive an electroacoustic transducer.The principal mode of sound transmission in bars is through the propagationof longitudinal waves. Here the displacement of the solid particles in the baroccurs parallel to the axis of the bar. The lateral dimensions of a bar are smallcompared with the length, so the cross-sectional plane can be pictured as movingas a unit. In reality, because of the Poisson effect that generally occurs in solidmaterials, the longitudinal expansion of the bar results in a lesser degree of lateralshrinkage and expansion; but this lateral motion can be disregarded in very thinbars.5.2 Derivation of the Longitudinal Wave Equation for a BarIn Figure 5.1, a bar of length L and uniform cross-sectional area Â is subjectedto longitudinal forces which produce a longitudinal displacement ξ of each ofthe molecules in the bar. This displacement in long, thin bars will be the same ateach point in any specific cross section. If the applied longitudinal forces vary ina wavelike perturbative manner, the displacement ξ is a function of both x and t89

90 5. Vibrating BarsFigure 5.1. A bar undergoing longitudinal strain in the x-direction.Figure 5.2. An element of the bar undergoing compression.and is fairly independent of lateral coordinates y and z. Thus,ξ = ξ(x, t)The x-coordinate of the bar is established by placing the left end of the bar at x =0,with the right end terminating at x = L. Consider an incremental element formedby dx of the unstrained bar positioned between x and x + dx as shown in Figure 5.2.The application of a force in the positive x-direction causes a displacement of theplane at x by a distance ξ to the right and the plane at x + dx by a distance ξ + dξalso to the right. A force acting in the opposite direction will likewise causecorresponding negatively valued displacements to the left. Because the element dxis small, we can represent the displacement at x + dx by the first two terms of aTaylor series expansion of ξ about x:( ) ∂ξξ + dξ = ξ + dx∂xThe left end of the element dx has been displaced a distance ξ and the right end adistance ξ + dξ, thus yielding a net increase dξ in the length of the element given by( ) ∂ξ(ξ + dξ) − ξ = dξ = dx∂xIn solid mechanics the strain ε of an element is defined as the ratio of the changeof its length to the original length, i.e.,( ) ∂ξε =∂xdxdx = ∂ξ∂x(5.1)

5.2 Derivation of the Longitudinal Wave Equation for a Bar 91In the situation in which static forces are applied to a uniform bar, the strain is thesame for each point and is time-independent. But we are considering a dynamiccase in which the strain in the bar varies with coordinate x and with time t. Thistype of variation generates a longitudinal wave motion in the bar in a manneranalogous to the transverse waves in a string. When a bar undergoes strain, elasticforces are generated inside the bar. These forces act across each cross-sectionalplane and essentially constitute reactions to longitudinally applied forces. Welet F x = F x (x, t) denote these longitudinal forces and adopt the convention thatcompressive forces are represented by positive values of F x and tensile forces bynegative values of F x . The stress σ in the bar is defined byσ = F x /ÂHere Â is the cross-sectional area of the bar. We can now apply Hooke’s lawσ = F xÂ=−Eε =−E∂ξ∂x(5.2)where E is the elastic constant or Young’s modulus, a property characteristic ofthe material constituting the bar. Table A in Appendix lists the values of Young’smoduli for a number of commonly used materials. Because E must always have apositive value, a negative sign is introduced in Equation (5.2) to accommodate thefact that a positive stress (compression) results in a negative strain, and a negativestress (tension) in a positive strain. We rewrite Equation (5.2) to express force F xat point x as follows:F x =−E Âε =−E Â ∂ξ(5.3)∂xUnlike the static case where the strain ε = ∂ξ/∂x and hence the force F x , remainsconstant throughout the bar, both the strain and F x vary in the dynamic case,and a net force acts on element dx. F x represents the internal force at x, and soF x + (∂ F x /∂x)dx constitutes the force at x + dx. The net force acting to theright becomes(dF x = F x − F x + ∂ F )x∂x dxCombining Equations (5.3) and (5.4) results in=− ∂ F xdx (5.4)∂xdF x = E Â ∂2 ξdx (5.5)∂x2The volume of the element dx is given by Âdx, and therefore the mass is ρ Âdx,where ρ denotes the density (kg/m 3 ) of the bar material. Applying Newton’sequation of motion, with acceleration ∂ 2 ξ/∂t 2 , to Equation (5.5), we obtainρ Âdx ∂2 ξ∂t 2= E Â ∂2 ξ∂x 2 dx

92 5. Vibrating BarsSettingc 2 = E/ρ (5.6)we now obtain the one-dimensional longitudinal wave equation:∂ 2 ξ∂t 2= c2 ∂2 ξ∂x 2 (5.7)Equation (5.7) corresponds to Equation (4.3) for the transverse motion of thestring, with the longitudinal displacement ξ assuming the role of the transversedisplacement y. We note that Equation (5.7) is identical to Equation (2.3), andwe have derived in this section the wave equation (5.7) that applies to acousticpropagation in a linearly elastic solid.5.3 Solutions of the Longitudinal Wave EquationThe format of the general solution to Equation (5.7) is identical with that of thesolution to Equation (4.3), i.e.,ξ = f (ct − x) + g(ct + x) (5.8)The square root of Equation. (5.6) gives us the wave propagation velocity c:√Ec =(5.9)ρwhich indicates that c is a property of the bar material.Let us write the solution (5.8) in the form of a complex harmonic solutionξ = Ae i(ωt−kx) + Be i(ωt+kx) (5.10)where A and B represent complex amplitude constants and k = ω/c is the wavenumber. We now assume that the bar is rigidly fixed at both ends; the boundaryconditions ξ(x, t) becomes ξ = 0atx = 0 and at x = L at all times t. Applyingthe condition ξ(0, t) = 0 yields A =−B, and Equation (5.10) revises toThe stipulation ξ(L, t) = 0 results inor equivalentlywhich means thatξ = Ae iωt(e−ikx −e ikx )e −ikt − e ikt =2 sin kLisin kL = 0= 0k n L = nπ, n = 1, 2, 3,...(5.11)

5.4 Other Boundary Conditions 93The allowed modes of vibration possess the radial frequencies ω n and the correspondingcyclic frequencies f n given byω n = nπc/L or f n = nc/2L, n = 1, 2, 3,...In simplifying Equation (5.11), the complex displacement ξ for the nth mode ofvibration isThe real part of Equation (5.12) isξ n = iA n e iω nt sin k n x (5.12)ξ n = sin k n x(A n cos ω n t + B n sin ω n t) (5.13)where the real amplitude constants A n and B n are related to the complex constantA n as follows:2A n = B n + iA nThe full solution to Equation (5.7) consists of the sum of all of the individualharmonic solutions, i.e.,∞∑ξ = sin k n x(A n cos ω n t + B n sin ω n t) (5.14)n=1The constants A n and B n can be evaluated by using the Fourier analysis describedin the last chapter, provided the initial conditions are known with respect to thedisplacement and the velocity of the bar.5.4 Other Boundary ConditionsIt should be understood that the boundary conditions corresponding to rigid supportsare difficult to realize in practice. The free-end condition, on the other hand,can be simulated by supporting the bar on extremely pliant supports placed somedistance inward from the ends. The end of the bar can now move freely and nointernal elastic force exists at that location. We now apply Equation (5.3), settingF x = 0; this gives rise to the condition ∂ξ/∂x = 0 at the free end. If the bar is freeto move at both ends (this is termed the free–free bar), the condition ∂ξ/∂x = 0applied to x = 0 in the wave equation, solution (5.10) yieldswith the resultA = Bξ = Ae iωt (e −ikx + e ikx ) (5.15)Inserting the condition ∂ξ/∂x = 0 into the above Equation (5.15) for the locationx = L yields−e −ikL + e ikL = 0 or sin kL = 0

94 5. Vibrating BarsThe allowable frequencies for a free–free bar are the same as those for the barfixed at both ends ( fixed–fixed bar). There will be, however, major differences inthe respective wave patterns of the free–free bar and the fixed–fixed bar. RecastingEquation (5.15) by making use of the relation2 cos kx = e −ikL + e ikLwe can express the complex displacements corresponding to the nth mode ofvibration asξ = 2 A n e iω nt cos k n tThe real part of the preceding equation gives the tangible vibrations described byξ n = cos k n x(A n cos ω n t + B n sin ω n t) (5.16)By comparing Equation (5.13) for the fixed–fixed bar with Equation (5.14) for thefree–free bar, it will be seen that antinodes exist at the end points for the latterbar in contrast with the nodes that must exist at the end points of the fixed–fixedbar. The nodal patterns for both bars are shown in Figure 5.3. It is of interest toobserve that when an antinode exists at the center of the bar, the vibrations areFigure 5.3. Typical standing waves for the first three modes of vibration in a fixed–fixedbar and in a free–free bar.

5.5 Mass Concentrated Bars 95symmetrical with respect to the center; otherwise a node in the center correspondsto asymmetrical vibration.As with case of the vibrating string, we can rigidly clamp a bar at any one ofits nodal positions without affecting the modes of vibrations which have a nodeat this position. But the vibrations that do not have a node at this position willbecome suppressed. The nature of the vibration of the free–free bar is such that itis not possible to clamp it at any position that will not eliminate at least some ofthe allowed modes of vibrations.The case of a free-fixed bar also makes for an interesting study. One end remainsfree at x = 0, and the other is rigidly clamped at x = L. The first condition∂ξ/∂x = 0atx = 0 leads again to Equation (5.15), while the second conditionξ = 0atx = L yieldse −ikL + e ikL = 0 or cos kL = 0which means that the allowable frequencies must satisfyork n L = ω nc L = π (2n − 1), n = 1, 2, 3,...2ω n = (2n − 1)π c2L , f n = (2n − 1) c 4lThe fundamental frequency is half that for an otherwise identical free–free bar, andonly odd-numbered harmonic overtones exist. The quality of the sound producedby an oscillating free-fixed bar will thus differ from that of a free–free bar, becauseof the absence of the even harmonics.5.5 Mass Concentrated BarsIn practical situations a vibrating bar is not truly clamped totally nor is it completelyfree to move at its ends. It may incorporate some type of mechanical impedance,most commonly as the result of concentrating a certain amount of mass at a certainlocation. An example is a diaphragm represented as a distributed mass located atone end of a vibrating tube inside a sonar transducer.As an example let us consider a bar that is unfettered at x = 0 and has a loadingconsisting of a mass m concentrated at x = L. The mass is depicted as a pointmass so that it does not move as a unit and thus merely sustain waves propagatingthrough it. The boundary condition ∂ξ/∂x = 0atx = 0 again leads us to Equation(5.17), as the result of A = B. For the boundary condition at x = L we again invokeNewton’s law of motion:( ∂ 2 ξF x (L, t) = m(5.17)∂t)x=L2A positive value of F x , which compresses the bar, will result in acceleration of themass in the positive x-direction. Because the mass is rigidly coupled to the bar,

96 5. Vibrating Barsthe accelerations of the mass and of the end of the bar should be identical. But ifthe mass had been concentrated at x = 0, a positive (compression) force wouldcorrespond to a reaction force to the left on the mass. The appropriate boundarycondition for this case would be( ∂ 2 ξ−F x (0, t) = m(5.18)∂t 2 )x=0Incorporating the boundary condition Equation (5.18) into Equation (5.16) resultsinwhich rearranges to−E Âe iωt (−ike −ikt + ike ikt ) = mAe iωt (−ω 2 )(e −ikL + e ikL )kEÂ sin kL =−mω 2 cos kLortan kL =− ωmc(5.19)E ÂBecause Equation (5.19) is a transcendental equation, no explicit solution exists.However, if the mass m is very small, m ≈ 0 and hence tan kL ≈ 0 and kL ≈ nω,both of which constitute the allowed conditions for a free–free bar. This is a resultthat should occur, since light loadings render a bar nearly free at both ends. At theother extreme, for very heavy mass loadings, the mass behaves very nearly like arigid support, and the allowed frequencies will approximate those of a free-fixedbar.In the more general case of intermediate mass loading, it is rather cumbersometo solve by hand the transcendental equation (5.19) through graphic means. However,computer programs such as Mathcad r○ , MathLab r○ , Mathematica r○ ,orevenaprofessional-level spread sheet for IBM compatible and Macintosh personal computerscan be used to facilitate solutions. Eliminating Young’s modulus in Equation(5.19) by applying E = ρc 2 from Equation (5.9) and recognizing that the mass ofthe bar is given by m b = ρ ÂL, we can rewrite Equation (5.19) astan kL=− m (5.20)kL m bThe right-hand side of Equation (5.20) is fixed by amount of mass m b in the barand the loading mass m located at x = L. An example of the solution to Equation(5.20) is given in Figure 5.4 for the case of a steel bar with a mass loading m/m bof 20%. The longitudinal velocity of sound propagation of steel is 5050 m/s. Thefundamental frequency f 1 is found from k 1 L through the relationf 1 = k 1L2πand the higher frequencies are similarly established fromf n = k n L2πcLcL

5.5 Mass Concentrated Bars 97Figure 5.4. Plot of 0.2 × kL (Curve a) and tan kL (Curve b) vs. kL. The intersections of the two sets of curves as shownabove provide the first five solutions kn L(kL = 0 being trivial) of the transcendental equation (5.20) for the fundamentaland overtones of a vibrating bar with 20% concentrated loading at location x = L. The fundamental occurs at k1L = 1.571,the first overtone at k2L = 3.792, the second overtone at k3L = 4.712; and neither of the latter two are harmonics of thefundamental.

98 5. Vibrating BarsFigure 5.5. Location of node in a free mass-loaded bar and fundamental mode of vibration.But overtones corresponding to the higher frequencies are not harmonics of thefundamental. In the example of Figure 5.4, in which the plots of (m/m b )kL and tankL vs. kL yield intersections which constitute the solutions to Equation (5.20), theratio of the first overtone to the fundamental is 3.792/1.571 = 2.414, not the valueof 2.0 that would have specified the overtone to be a harmonic of the fundamental.Solutions exist where the occurrence of non-harmonics overtones could be useful:for example, in a properly mass-loaded loudspeaker, a pure monofrequency inputwould not result in harmonics that could arise from the driving signal, if at all.The nodes of the vibration in the bar exist at locations where cos kx = 0. Thefundamental mode where kL = 3.792 engenders a node at 3.792x/L = π/2, orx = 0.414L, not x = 0.5L that would normally occur in a second harmonic. Incontrast of the free–free bar, the node in a free mass-loaded bar is no longerat the center—it shifts toward the loading mass as shown in Figure 5.5. In thisparticular case, the bar could be supported at this nodal position without affectingthe fundamental mode of vibration.5.6 General Boundary Conditions for a FreelyVibrating BarConsider a freely vibrating bar with arbitrary loadings at each end. We shall establishthe normal modes of vibration in terms of the mechanical impedances at bothends of the bar. Designating the mechanical impedance of the support at x = 0byZ mi0 we express the force acting at this support due to the bar asf 0 =−Z mi0 u(0, t)Here the minus sign is introduced to indicate that a positive (compressive) forcegenerates an acceleration of the support in the negative x-direction. But a positivecompressive force at the end x = L causes an acceleration of the other support tothe right, in the positive x-direction; the force acting on this support isf L =+Z miL u(L, t)where Z miL represents the mechanical impedance of the support at x = L. Thepreceding two equations can be restated in terms of particle displacements by

5.6 General Boundary Conditions for a Freely Vibrating Bar 99applying Equation (5.3) to supplant the compressive forces and by expressing theparticle velocity as u = ∂ξ/∂t:( ) ∂ξ= Z ( )mio ∂ξ(5.21)∂x x=0 ρ L c 2 ∂t x=0( ) ∂ξ= Z ( )miL ∂ξ(5.22)∂x x=L ρ L c 2 ∂t x=Lwhere ρ L = ρ Â is the linear density (kg/m) of the bar.If the loads Z mio and Z miL are purely reactive, there is no transient or spatialdamping, and hence no loss of acoustical energy occurs. Equation (5.16) constitutesa proper solution. And because no loss of acoustical energy occurs, a wave travelingin the +x-direction must equal the energy of a wave moving in the oppositedirection. The absolute magnitudes of the complex wave amplitudes must thereforebe equal, i.e., |A| =|B|. The boundary conditions (5.21) and (5.22) establish thephase angles of the complex amplitudes.But if the mechanical impedances contain some measure of resistive components,a solution more general than that of Equation (5.16) needs to be applied. Asin the case of a freely vibrating string terminated by a resistive support, transient(or temporal) damping has to occur in the presence of resistance. The transientbehavior of the bar is characterized by a complex angular frequency ω = ω + iβ.The real portion of this frequency is the angular frequency ω; the imaginary partrepresents the transient absorption coefficient β. But no internal losses occur inthe bar, so wave equation (5.7) still applies, and we infer the solutionξ(x, t) = (Ae −ikx + Be ikx )e iωt (5.23)where ω 2 = c 2 k 2 . If the losses are quite small we can use the approximationω ≈ ck to simplify the solution. Applying boundary conditions (5.21) for x = 0and (5.22) for x = L to (5.23) and making use of the approximation we obtain thefollowing pair of equationsA −ikLeA − B =− Z mi0(A + B)ρ L c− B ikLe= Z miL (A−ikLeρ L c+ BeikL )Solution of these preceding two equations by elimination of A and B results in thetranscendental equationtan kL = iZ mi0ρ L c + Z miLρ L c1 + Z mi0ρ L cZ miLρ L cThe characteristics of the vibration are determined from the complex impedancesZ mi0 and Z miL . The solution of the preceding transcendental equation is rendered

100 5. Vibrating Barsmore difficult by the presence of any resistive component in either Z mi0 or Z miL ,which produces a complex argument of the tangent.5.7 Transverse Vibrations of a BarA bar not only vibrates longitudinally; it can also vibrate transversely, which isusually the case because the strains engendered by longitudinal motion give avirtually automatic rise to transverse strains as the result of the Poisson effect.A hammer blow aimed along the axis of a long, thin bar supported at its centerwill usually result in principally transverse vibrations rather than the expectedlongitudinal effects, because it is not easy in the real world to avoid a slighteccentricity in applying the blow.In our derivation of the transverse wave equation, consider a straight bar of lengthL with a uniform bilaterally symmetric cross-section Â. In Figure 5.6, a segment dxof the bar is shown bent (the bending is exaggerated to better illustrate the effect).The x-coordinate lies along the axis of the bar and the y-coordinate measures thetransverse displacements of the bar from its unperturbed configuration. The barbehaves as a beam, i.e., the upper part of the cross section stretches under tensionand lower part becomes compressed. A neutral axis NN ′ , whose length remainsunchanged, comprises the line of demarcation between compression and tensionin the bar. If the cross section of the bar is symmetrical about a horizontal plane,the central axis of the bar will coincide with the neutral axis. The bending of thebar is gauged by the radius of curvature R of the neutral axis. Consider the lengthincrement δx = (∂ξ/∂x)dx due to the bending of a filament in the bar located ata distance r from the neutral axis. The longitudinal force df is found fromdf = EdÂ δxdx= EdÂ∂ξ∂x(5.24)where dÂ is the cross-sectional area of the filament. Above the neutral axis NN ′ thevalue of δx is positive, so the force df becomes negative and thus is a tension. Forfilaments positioned below the neutral axis, δx is negative, resulting in a positivecompressive force. From geometry we note thatdx + δxR + r= dxRand this leads to δx/dx = r/R. Equation (5.24) now becomesdf =− E R rdÂThe negative forces above the neutral axis cancel out the positive forces below theneutral axis, and hence the total longitudinal force f = Idf equals zero. But abending moment M occurs in the bar, i.e.,∫M = rdf =− E ∫r 2 d ÂR

5.7 Transverse Vibrations of a Bar 101From classical mechanics we recognize the radius of gyration κ of the crosssectionalarea Â, as defined by∫rκ 2 2 d Â=ÂWe now obtainM = κ2 E Â(5.25)R(For a bar with a rectangular cross section, κ = t/ √ 12 where t denotes the thicknessof the bar; for a circular rod of radius a, the radius of gyration is given by κ = a/2.)The radius of curvature R varies along the neutral axis, but the mathematicscan be simplified by assuming the displacements y of the bar to be quite small,∂y/∂x ≪ 1, which permits the use of the approximationEquation (5.25) now modifies toR = [1 + (∂y/∂x)2 ] 3/2∂ 2 y/∂x 2≈1∂ 2 y/∂x 2M =−κ 2 E Â ∂2 y(5.26)∂x 2The curvature shown in Figure 5.5 has a negative ∂ 2 y/∂x 2 , and the bending momentis consequently positive. In order to get this type of curvature the torque must beapplied to the left end of the segment in a counterclockwise (or positive angular)direction and the torque at the right end of the segment must be clockwise (or inthe negative angular direction).Shear forces as well as bending moments arise when a bar becomes distorted.In Figure 5.7, a shear force F y (x) acts upward (in the positive sense) on the leftend of the element dx. An opposing shear force −F y (x + x) acts downward atFigure 5.6. An element of a bar showing bending stresses and strains.

102 5. Vibrating BarsFigure 5.7. An element of the bar showing shear forces and bending moments.the right end of the element. To sustain static equilibrium in the bent bar, the shearforces and torsions acting on the element must counterbalance each other so thatthere is no net turning momentum. As shown in Figure 5.7, taking the left end ofthe element as the reference pivot point we obtainM(x) − M(x + dx) = F y (x + dx) (5.27)The terms M(x + x) and F y (x + x) are now expanded in a Taylor’s seriesabout point x, with the result that Equation (5.27) becomesF y =− ∂ M∂x = κ2 E Â ∂3 y(5.28)∂x 3In Equation (5.28) the second-order and higher terms in dx have been discarded.In undergoing transverse vibrations the bar is in dynamic rather than static equilibrium.This requires that the right-hand side of Equation (5.27) must equal the rateof increase of the angular momentum of the segment. But as long as the displacementand the slope of the bar remain small, the variations in angular momentumcan be disregarded, and Equation (5.28) should serve as a good approximationof the correlation between the displacement y and the acting force F y . The netupward force dF y in element dx is given bydF y = F y (x) − F y (x + x) =− ∂ F y∂x =−κ2 ÊA ∂4 y∂x dx 4The element undergoes an upward acceleration under the impetus of the force, andthe equation of motion for the mass of the element, ρ Âdx, may now be written as:ρ Âdx ∂2 y∂t 2=−κ2 E Â ∂4 y∂x 4 dxSetting c = (E/ρ) 1/2 , as in the case of longitudinal waves, the last equation changesto∂ 2 y∂t 2=−κ2 c 2 ∂4 y∂x 4 (5.29)

5.7 Transverse Vibrations of a Bar 103Equation (5.29) for the transverse wave differs from Equation (5.7) for the longitudinalwave principally in the presence of the fourth partial derivative with respectto x. Solutions in the functional form of f (ct − x) donot apply to transversewaves, as can be readily proved by direct substitution into Equation (5.29). Thismeans that transverse waves do not travel in the x-direction with constant speed cand unchanging shape.Equation (5.28) can be solved by separation of variables by setting the complextransverse displacement y asy = Ψ(x)e iωt (5.30)and inserting into Equation (5.29). This yields a new total differential equation inwhich Ψ exists as a function of x only:Settingd 4 Ψdx 4 = ω2κ 2 c 2 Ψthe fourth-order differential equation becomesv = √ κωc (5.31)d 4 Ψdx 4= ω4v 4 Ψ (5.32)The function Ψ may be assumed as an exponential of the form Ψ(x) = Ae γ x andsubstituted into Equation (5.32). The result isγ 4 = (ω/v) 4Four values of γ occur: ±(ω/v) and ±(iω/v). The complete solution to Equation(5.32) consists of the sum of the four solutions:Ψ = Ae ωx/v + Be −ωx/v + Ce i(ωx/v) + De −i(ωx/v)wherein A, B, C, D constitute complex amplitude constants. From Equation (5.30)the solution for the displacements y can be written asy = e iωt (Ae ωx/v + Be −ωx/v + Ce i(ωx/v) + De −i(ωx/v) ) (5.33)It should be noted that none of the terms in Equation (5.33) contains a wavemoving with a velocity c. The third term inside the parenthesis of Equation (5.33)represents a wave disturbance moving to the left and the fourth term represents awave moving in the positive x-direction. The phase speed v is itself a function ofthe frequency, as attested by Equation (5.31), so waves of differing frequencies willtravel with different phase speeds. Higher-frequency waves will outpace the lowerfrequencywaves, and accordingly a complex wave containing a number of differentfrequencies will alter its shape along the x-axis. Each frequency component of acomplex wave travels at its own speed v, which gives rise to a situation analogous tothe transmission of light through glass, in which different component frequencies of

104 5. Vibrating Barsthe light beam travel with different speeds thereby causing dispersion. A vibratingbar thus acts as a dispersive medium for transverse waves.The real part of Equation (5.33) constitutes the actual solution of Equation(5.29). We make use of the following hyperbolic and trigonometric identities:sin y = (e iy − e −iy )/2, cos y = (e iy + e −iy )/2sin(iy) = i sinh y, sinh(iy) = i sin ycos(iy) = cosh y, cosh(iy) = cos yto recast Equation (5.33) asy = cos(ωt + φ)(A cosh ωx ωx ωx+ B sinh + C cosv v vωx)+ D sinv(5.34)Here A, B, C, D are real constants that occur from the rearrangement of the originalcomplex constants A, B, C, D. The intricate relationships between the real set ofconstants and the set of complex constants are not really of much concern to us,because it is the application of the initial and boundary conditions that provides theevaluation of these constants. However, there are twice as many arbitrary constantsin the transverse equation (5.34) as in the longitudinal wave equation (5.7), due tothe fact that the former is of the fourth-differential order rather than the seconddifferentialorder. Therefore, twice as many boundary conditions are required, andthis can be satisfied by specifying pairs of boundary conditions at the ends of thebars. The nature of the supports establishes the boundary conditions that generallyfall into the categories of free and clamped ends.5.8 Boundary Conditions for Transverse Vibrations1. If the bar is rigidly clamped at one end, the both the displacement and the slopemust be zero at that end at all times, and the boundary conditions are expressed as:y = 0, ∂y/∂x = 0 (5.35)2. On the other hand, neither an externally applied moment nor a shear force mayexist at a free end of a vibrating bar. But the displacement and the slope of the barat a free end are not constrained, excepting for the mathematical stipulation theyremain small. From Equations (5.26) and (5.28) the boundary conditions become∂ 2 y∂x 2 = 0,Case 1: Bar Clamped at One End∂ 3 y∂x 3 = 0 (5.36)Consider a bar of length L that is rigidly clamped at x = 0 but is free at x = L.At x = 0, the two conditions of Equation (5.35) apply, so A =−C and B =−D.The general solution (5.35) reduces toy = cos(ωt + φ)[A(cosh ωxv− cosωxv) (+ B sinh ωxvωx)]− sinv

5.8 Boundary Conditions for Transverse Vibrations 105Applying free-end condition Equation (5.36) at x = L yields the following twosets of equations:(A cosh ωL ) (ωL+ cos =−B sinh ωL )ωL+ sinv vv v(A sinh ωLv− sin ωL ) (=−B cosh ωLvv+ cos ωL )vBoth of the preceding two equations cannot hold true for all frequencies. In order todetermine the permissible frequencies, one equation is divided into the other, thuscanceling out the constants A and B. Ridding the resulting equation of fractionalexpressions by cross-multiplication and using the identities cos 2 θ + sin 2 θ = 1and cosh 2 θ + 1 = sinh 2 θ, we obtaincosh ωLvcosωLv =−1We can alter the last equation by using the identitiestan θ √1 − cos θ2 = 1 + cos θ , tanh θ √cosh θ − 12 = cosh θ + 1and we now obtaincot ωL2v=±tanhωL2v(5.37)The frequencies which correspond to the allowable modes of vibration can befound through the use of a microcomputer program which determines the intersectionsof the curves of cot ωL/2v and ± tanh ωL/2v, as shown in Figure 5.8.The frequencies of the permissible modes are given byωL2v = ζ π (5.38)4where ζ = 1.194, 2.988, 5, 7,... with ζ approaching whole numbers for thehigher allowed frequencies. Inserting v = (κωc) 1/2 into Equation (5.38), squaringboth sides, and solving for frequencies f , we obtainf = ζ πκc8L 2The constraint imposed by the boundary conditions leads to a set of discreteallowable frequencies, but the overtone frequencies are not harmonics of thefundamental. When a metal bar is struck in such a manner that the amplitudes ofthe vibration of some of the overtones are fairly strong, the sound produced hasa metallic cast. But these overtones rapidly die out, and the initial sound soonevolves into a mellower pure tone whose frequency is the fundamental. This isa characteristic of the behavior of a tuning fork that emits a short metallic soundupon being struck before emitting a pure tone.The distribution of the nodal points along the transversely vibrating bar is quitecomplex, with three distinct types of nodal points being identified mathematically.

106 5. Vibrating BarsFigure 5.8. Trigonometric functions, used in Equations (5.37) and (5.38), plotted as functionsof (ωL/2v).The clamping point of the bar constitutes one type, with conditions y = 0 and∂y/∂x = 0 at all times. Another group of points called true nodes is characterizedby y = 0 and ∂y/∂x ≈ 0, and they are found near points of inflections on the bar.The spacing between these true nodes is very nearly (but not quite) to λ/2. The thirdtype of nodal point occurs at the node very near the free end of the bar, where y = 0,but the corresponding point of inflection where ∂ 2 y/∂x 2 ≈ 0 does not coincidewith that point but it is moved out to the free end. The vibrational amplitudes donot equal each other at the various antinodes but the greatest vibrational amplitudeis that of the free end.Case 2: Free–Free BarIn the case of a bar that is free to move at both ends, the boundary conditionsat x = 0 are satisfied by A = C and B = D as the result of applying Equation(5.36). The same set of boundary conditions applied to the other end x = Lyieldstan ωL2v=+tanhωL2v

5.8 Boundary Conditions for Transverse Vibrations 107Figure 5.9. The first four transverse modes of a vibrating bar.which in turn gives a discrete set of allowable frequencies of the transverse vibration.The frequencies are given byf = ξ πκc8L 2where ξ = 3.0112 2 , 5 2 , 7 2 , 9 2 ,....The overtones are not harmonics of the fundamental.Figure 5.9 illustrates the transverse modes of a clamped-free bar and a free–freebar. In the free–free bar the modes correspond to the fundamental frequency f 1and all additional odd-numbered frequencies. The odd-numbered frequencies f 3 ,f 5 , and so on, are symmetric about the center of the bar. The slope ∂y/∂x is alwayszero at the center, which is a true antinode. But the even-numbered frequencies f 2 ,f 4 , f 6 ,...yield asymmetric modes of vibrations with respect to the center. In allmodes the nodal points are found to be distributed symmetrically about the center.A bar may therefore be supported on a knife edge (or held by knife-edged clamps)at any nodal point without affecting the mode of vibration having a node at thatpoint. A knife edge or a knife-edged set of clamps disallows displacement but nota change in the slope that occurs at the node.The xylophone consists of metal bars that are supported at the nodal locationsof the fundamental. But the nodes of the associated overtones are unlikely to belocated at the same points, and the overtones quickly die out, leaving the pure toneof the fundamental. The concept of a free–free bar applies to a tuning fork whichis essentially a U-shaped bar attached to a stem (Figure 5.10). The geometry of thefork and the mass-loading effect of the stem cause the nodes of the fundamental tobe spaced closely near the stem. When the tuning fork is struck, overtones damp

108 5. Vibrating BarsFigure 5.10. The tuning fork, essentially a U-shaped bar attached to a stem.out in a very short time, leaving only the pure sinusoidal fundamental, much inthe same manner as a xylophone. Because the stem shares the antinodal motionof the center of a free–free bar, the radiation efficiency of a tuning fork becomesgreatly increased by touching the stem to a surface of a large area such as acounter top.ReferencesFletcher, Neville H. and Rossing, Thomas D. 1991. The Physics of Musical Instruments.New York: Springer-Verlag, pp. 53–60.Kinsler, Lawrence E., Frey, Austin R., Coppens, Alan B., and Sanders, James V. 1982.Fundamentals of Acoustics, 3rd ed. New York: John Wiley & Sons, Chapter 3.Morse, Philip M. and Ingard., K. Uno. 1968. Theoretical Acoustics. New York: McGraw-Hill, Chapter 5.Reynolds, Douglas R. 1981. Engineering Principles of Acoustics, Noise and VibrationControl. Boston, MA: Allyn and Bacon, pp. 224–234.Wood, Alexander. 1966. Acoustics. New York: Dover Publications, pp. 384–386.Problems for Chapter 51. Show that a bar of length L, that is rigidly fixed at x = 0 and totally free atx = L will have only odd integral harmonic overtones.2. Determine the fundamental frequency of the bar in Problem 1 if L = 0.60 mand the bar is made of steel. If a static force F is applied to the free end, so

Problems for Chapter 5 109that the bar displaces a distance δ, and then is suddenly released, demonstratethat the amplitudes of the subsequent longitudinal vibrations are given byA n = 8δn 2 π sin nπ 2 2Find these amplitudes for this steel bar with a cross-sectional area of5.0(10) −5 m 2 , under the effect of a force of 4500 N.3. A steel bar is free to move at x = 0. It has 0.0002 m 2 cross-sectional area and0.35 m length. A 0.20 kg load is placed at x = 0.35 m.(a) Determine the fundamental frequency of the longitudinal vibration of themass-loaded bar.(b) Establish the position at which the bar may be clamped so as to minimizeinterference with the fundamental mode.(c) Find the ratio of the displacement amplitude of the free end to that of themass-loaded end for the first overtone of the bar.4. A steel bar having a mass of 0.05 kg and 0.25 m length is loaded at one endwith 0.028 kg and 0.056 kg at the other end. Find the fundamental frequencyof the system’s longitudinal vibration and determine the location of the nodein the bar. Also compute the ratio of the displacement amplitudes at the twoends of the bar.5. Redo Problem 4 for an aluminum bar of the same length but with a mass of0.03 kg, subjected to the same end loadings.6. Consider a thin bar of length L and mass M that is rigidly fixed at one endand free at the other. What mass m must be affixed to the free end in order tolower the fundamental frequency of longitudinal vibration by 30% from thefixed-free value?7. A fixed-free bar of length L and mass m has a mechanical reactance, or stiffness,equal to −is/ω in the fixture. Develop an expression for the fundamentalfrequency of the longitudinal vibration.8. A longitudinal force F cos ωt drives a long thin bar at x = 0. The bar is freeto move at x = L.(a) Obtain the equation for the amplitude of the standing waves occurring inthe bar.(b) Obtain the expression giving the input mechanical impedance of the barof length L.(c) Derive the expression for the input mechanical impedance of the same barhaving an infinite length.(d) For the case of part (a) plot the amplitude of the driven end of the baras a function of frequency over the range 100 Hz–3 kHz, if the materialof the bar is aluminum, its length is 1.5 m, its cross-sectional area is1.5 (10) −4 m 2 , and the amplitude of the driving force is 12 N.9. Demonstrate that the dimensional units of v = √ κωc are those of speed. Atwhat frequency will the phase speed of transverse vibrations of a steel rod

110 5. Vibrating Barsof 0.005 m diameter equal the phase speed of longitudinal vibrations in therod?10. Given an aluminum rod of 0.010 radius and length 0.4 m, what will be thefundamental frequency of free–free vibrations? Predict the displacement amplitudeof the free ends if the displacement amplitude of the rod at its centeris 2.5 cm.

6Membrane and Plates6.1 IntroductionIn this chapter we are now applying two-dimensional wave equations to membranesand plates. In this group of physical applications, three dimensions arereally involved in the theory that applies to two-dimensional surfaces such asdrumheads and diaphragms of microphones or loudspeakers. Two spatial coordinatesare required to locate a point on a vibrating surface, but the displacementgenerally occurs along the third spatial coordinate. However, the expansion of thegeneral one-dimensional solution to the wave equation for a string or a bar caneasily be extended to two dimensions. Again, boundary conditions determine thediscreteness of the vibrational frequencies, but the peripheral geometry of a membraneconstitutes a factor additional to the effects expected for different types ofsupport. For certain membrane or plate geometries, the choice of appropriate referencecoordinate systems to match the contour of the subject surface can greatlyfacilitate the solution of the wave equation.In the following sections the wave equation for a membrane under tension is derived,and solutions are developed for various geometries and supports. The classiccase of a kettledrum is included as a practical example that includes the effect ofdamping, and forced vibrations are also taken into additional consideration.The number of easily solvable two-dimensional problems is limited by thenarrow choice of coordinate systems, and more complex cases can be treated bycomputerization, most notably through the application of finite-element methods.6.2 Derivation of the Wave Equationfor a Stretched MembraneIn order to develop a viably simple equation of motion for a vibrating membranestretched under tension, the assumption is made that the extremely thin membraneis uniform and the vibrational amplitudes remain small. The membrane is alsodeemed to be perfectly elastic, without any damping. Designate T as the tension111

112 6. Membrane and PlatesFigure 6.1. An element of the vibrating element.of the membrane applied as a uniform force per unit length (e.g., in N/m) andρ S the surface density of the membrane as mass per unit area (kg/m 2 ). Thus, inFigure 6.1, an essentially two-dimensional membrane is stretched outward towardopposing sides of an element (each side of length dL) under the influence of atension force TdL. A cartesian coordinate system is adopted, with the expanse ofthe quiescent membrane lying in the x −y plane and the transverse displacementof a point at (x, y) occurring in the z-direction as z = z(x, y, t). In Figure 6.1 anelement of area dS = dx dy undergoes the effect of transverse forces acting in thex- and y-directions along the peripheral lengths of the element. The net force inthe x-direction is given by[( ) ( ) ]∂z ∂zT− dy = T ∂2 zdx dy∂x x+dx ∂x x ∂x2and similarly, the net force in the y-direction is T (∂ 2 z/∂y 2 ) dx dy. These twoterms add up to contribute to the acceleration ∂ 2 z/∂t 2 of the element’s massρ S dx dy, in accordance with the Newton’s law of motion:( ∂ 2 )zT∂x + ∂2 zdydy = ρ 2 ∂y 2 S dx dy ∂2 z∂t 2and setting√Tc =(6.1)ρ Swe obtain the classic two-dimensional wave equation∂ 2 z∂x 2 + ∂2 z∂y 2 = 1 c 2 ∂ 2 z∂t 2 (6.2)

6.3 Rectangular Membrane with Fixed Edges 113We can recast the wave equation (6.2) in the more general Laplacian format:∇ 2 z = 1 ∂ 2 z(6.3)c 2 ∂t 2Equation (6.2) is suitable for treatment of rectangular membranes, but Equation(6.3) should be expressed in terms of polar coordinates to facilitate mathematicaltreatment of circular membranes:∂ 2 z∂r + 1 ∂z2 r ∂r + 1 ∂ 2 zr 2 ∂θ = 1 ∂ 2 z(6.4)2 r 2 ∂t 2For normal vibrational modes it is the standard mathematical procedure to assumethat the solution to Equation (6.3) consists of a spatially dependent functionΨ and a strictly time-dependent function e iωt (we dispense with the other functione −iωt as being superfluous for the current physical applications):z = Ψe iωt (6.5)Inserting the above expression into Equation (6.3) and setting k = ω/c yields thetime-independent Helmholtz equation:∇ 2 Ψ + k 2 Ψ = 0 (6.6)whose solutions upon insertion into Equation (6.5) yield normal modes of vibrationsin a membrane of a given geometry and boundary conditions.6.3 Rectangular Membrane with Fixed EdgesConsider a stretched rectangular membrane that is fixed at its four edges x = 0,x = L x , y = 0, and y = L y . The boundary conditions may be expressed asz(0, y, t) = z(L x , y, t) = z(x, 0, t) = z(x, L y , t) = 0 (6.7)In the Cartesian format the solution z(x, y, t) = Ψ(x, y)e iωt to Equation (6.2) mustderive from the Helmholtz equation given below for Cartesian coordinates:∂ 2 Ψ∂x + ∂2 Ψ2 ∂y + 2 k2 Ψ = 0 (6.8)But Ψ(x, y) can be stated as the product of two singly dimensioned functions X(x)and Y (y) so thatand then Equation (6.8) transforms toΨ(x, y) = X(x)Y(y)1 ∂ 2 XX ∂x + 1 ∂ 2 Y2 Y ∂y + 2 k2 = 0 (6.9)

114 6. Membrane and PlatesBecause the three terms of Equation (6.9) cannot all sum to zero and the first andsecond terms are wholly independent of each other, we separate Equation (6.9)into two separate differential equations, one wholly dependent on x and the otheron y:where kx 2 and k2 y are constants related by1 ∂ 2 XX ∂x + 2 k2 x = 0(6.10)1 ∂ 2 YY ∂y + 2 k2 y = 0k 2 x + k2 y = k2The solutions to the equation set (6.10) consists of sinusoids, with the resultz(x, y, t) = α sin(k x x + φ x ) sin(k y y + φ y )e iωt (6.11)Here α represents the maximum displacement of the membrane in the transversedirection, and φ x and φ y are determined by boundary conditions. With the firstand the third of the boundary condition set (6.7) we find that φ x = φ y = 0 and theremaining conditions necessitate that sin k x L x = 0 and sin k y L y = 0. Therefore,the normal modes occur fromz(x, y, t) = αe iωt sin k x x sin k y y (6.12)for which k x and k y turn out to be discrete values established byk x = nπ/L x n = 1, 2, 3,...k y = mπ/L y m = 1, 2, 3,...The frequencies of the allowed modes of vibrations are found from√ (f nm = ω nm2π = c ) n 2 ( ) m 2+(6.13)2 L x L yEquation (6.13) constitutes a fairly simple extension of the allowable frequenciesof a idealized free vibrating string to two-dimensional status.The fundamental frequency is found by merely setting n = m = 1 in Equation(6.12). The overtones corresponding to m = n > 1 will be harmonics of the fundamentalfrequency, but those in which m ≠ n (with either m or n > 1) may notnecessarily be so. A number of possible modes in a rectangular membrane areillustrated in Figure 6.2. The shaded areas vibrate π radians out of time phase withthe unshaded areas. Each normal mode is designated by an ordered pair (n, m),and the nodal lines are those with zero displacement at all times. In theory, rigidsupports could be placed along these lines without affecting the nodal pattern forthe associated specific frequency.

6.4 Freely Vibrating Circular Membrane with Fixed Rim 115Figure 6.2. Four modes of a vibrating membrane. Lines located within the borders of themembranes constitute nodal loci where displacements are zero for these respective modes.6.4 Freely Vibrating Circular Membrane with Fixed RimAs mentioned in the foregoing it is preferable to adopt the polar coordinate version(6.4) of the wave equation (6.3) to treat the case of a circular membrane that isfixed at its rim. Accordingly, the zero displacement of the membrane’s boundaryat radius r = a gives the boundary conditionz(a,θ,t) = 0The harmonic solution to Equation (6.4) can be represented as the product of threeterms, each of which is functions of only one variable:z(r,θ,t) = R(r)(θ)e iωt (6.14)The boundary condition stipulated at the rim of the circular membrane now becomesR(a) = 0and insertion into Equation (6.4) yields the polar coordinate version of theHelmholtz equation: d2 Rdr + dR2 r dr + R d 2 r 2 dθ + 2 k2 R = 0 (6.15)where k = ω/c, as before. Equation (6.15) is then rearranged to effect the separationof two variables, so we obtainr 2 ( d 2 RR dr + 1 )dR+ (kr) 2 =− 1 d 2 (6.16)2 r dr dθ 2

116 6. Membrane and PlatesThe left-hand side of Equation (6.16) is solely a function of r while the rightdepends only on θ. In order for the equality of Equation (6.16) to prevail, bothsides of the equation must be set equal to a constant m 2 . Then we obtain from theright-hand side of Equation (6.16)d 2 dθ + 2 m2 = 0which in turn yields the harmonic solution(θ) = cos(mθ + ε)Here ε is the phase angle. The azimuthal coordinate is of periodic nature, repeatingitself every 2π radians. In order that the displacement z be single-valuedfunction of position, z(r,θ,t) must then equal itself every 2π radians, that is,z(r,θ,t) = z(r,θ + 2π, t), with the result that the constant m is constrained tointegral values m = 1, 2, 3,.... The left-hand side of Equation (6.16) thereforebecomes the Bessel’s differential equation:d 2 Rdr + 1 ( )dR2 r dr + k 2 − m2R = 0 (6.17)r 2The solution of Equation (6.17) isR(r) = AJ m (kr) + BY m (kr) (6.18)where J m (kr) and Y m (kr) are, respectively, the transcendental Bessel functions ofthe first and the second kind, each of the order m. Bessel functions are oscillatingfunctions with diminishing amplitudes for increasing kr. Y m (kr) approachesinfinity as kr → 0. The numerical values of Bessel functions and their propertiesare given in standard tables and advanced mathematical computer programs.An abbreviated set of Bessel formulas and tables art given in Appendix B. Becausethe circular membrane includes the origin r = 0 and the displacement of themembrane must remain finite at that point, it is necessary to set 1 B = 0, reducingEquation (6.18) toR(r) = AJ m (k)Applying the boundary condition R(a) = 0 requires that J m (ka) = 0. Let us designateby q mn those values of the argument ka at which the mth order Bessel functionJ m equals zero. From J m (q mn ) = 0 we can find those discrete values of k whichare given byk mn = J mn(q mn )a= J mna1 On the other hand, if an annular membrane stretched over region a < r < b (thus excluding the originr = 0) is considered, both Bessel functions must be retained in Equation (6.18) to provide the twoarbitrary constants needed for the boundary conditions at the inner and outer borders.

6.4 Freely Vibrating Circular Membrane with Fixed Rim 117A number of values of q mn that yield zeros of the Bessel functions are given inAppendix B. We can therefore write the normal modes of vibration asz mn (r,θ,t) = A mn J n (k mn r) cos(mθ + ε mn )e iω mnt(6.19)with k mn a = q mn and the natural frequencies found fromf mn = 1 q mn c(6.20)2π aThe real part of Equation (6.19) describes the physical displacement in thenormal mode (m, n) as follows:z mn (r,θ,t) = A mn J m (k mn r) cos(mθ + ε mn ) cos(ω mn t + φ mn )where A mn = A mn = e iφmn . The arbitrary constant ε mn is an azimuthal phase angle.For each normal mode, this constant of integration defines the directions alongwhich the radial nodal lines of zero displacement occur, but the value of ε mndepends on the value of the azimuthal angle at which the membrane is excited att = 0. A number of the first few (and simpler) modes of vibration with ε mn = 0 areshown in Figure 6.3. Each mode is designated by the ordered pair of integers (m, n).Integer m governs the number of radial nodal lines, whereas integer n determinesthe number of azimuthal nodal circles. Since mode (0, 0) would obviously be trivial,Figure 6.3. Vibration modes in a circular membrane fixed at its perimeter. A number ofsimpler modes are shown here.

118 6. Membrane and PlatesTable 6.1. Relative Frequencies for Various Vibrational Modes.f 01 = 1.0 f 01 f 11 = 1.593 f 01 f 21 = 2.135 f 01f 02 = 2.295 f 01 f 12 = 2.917 f 01 f 22 = 3.500 f 01f 03 = 3.598 f 01 f 13 = 4.230 f 01 f 23 = 4.832 f 01the case of n = 1 constitutes the least allowed value of n, and this corresponds toa mode of vibration where one azimuthal nodal circle occurs only at the rim of themembrane where r = a. Virtually the entire membrane vibrates in the z-directionin axisymmetric unison, with the maximum amplitude at r = 0 tapering off to zeroat the boundary r = a.For each non-zero value of m there exists a chain of allowed radial vibrationmodes of increasing frequency, as illustrated in Figure 6.3. At m = 0, J 0 (k 0n a)equals zero, which sets the conditions for the allowed frequencies. For m = 1,J 1 (k 1n a) = 0 provides the allowed frequencies; and for m = 2, J 2 (k 2n a) = 0 suppliesthe corresponding frequencies for that radial mode. Table 6.1 lists the frequenciesf mn of the circular membrane relative to the fundamental frequency f 01obtained from Equation (6.20). From Table 6.1 it is observed that none of theovertones exists as a harmonic of the fundamental.6.5 Case Study: Symmetric Vibrations of a CircularMembrane Fixed at its PerimeterThe case of symmetric vibrations of a circular membrane fixed at its rim has manyapplications, so this situation holds great interest for us. Vibrational symmetryimplies a solution to the wave equation (6.4) that is independent of θ, and we limitthe solutions for this case toz 0n = A 0n J 0 (k 0n r)e iω 0nt(6.21)and because only m = 0 applies, we discard this subscript and retain only theindex n. The fixedness of the circular perimeter forecasts the boundary conditionz = 0atr = a. Therefore, J 0 (ka) = 0, with the zeroes occurring as k n a =2.405, 5.520, 8.654, 11.792,....Thefundamental frequency is derived fromf 1 = ω 12π = k 1c2π = 2.4052πa√Tρ s(6.22)with the (non-harmonic) overtones to the fundamental frequencies obtained as( ) 5.520f 2 = f 1 = 2.295 f 12.405f 3 = 3.58 f 1 ,f 4 = 4.90 f 1 ,...,etc.

6.5 Symmetric Vibrations of a Circular Membrane Fixed at its Perimeter 119For the real part of z 1 = A 1 J 0 (k 1 r)e iωt the general expression for the membrane’sdisplacement in the fundamental mode can be written as( ) 2.405rz 1 = A 1 cos(ω 1 t + φ 1 )J 0 (6.23)rawhere A 1 is the maximum absolute value of the complex amplitude A 1 at thecenter r = 0 of the membrane. To completely describe the symmetric vibrationsthe complete solution must be expressed asz = ∑ A n cos(ω n t + φ n )J 0 (k n r) (6.24)For the symmetric modes of vibration above the fundamental, nodal circles willexist at the inner radii at which J 0 (k n r) vanishes. For example, the first overtoneJ 0 (k 2 r) = 0 when k 2 r = 5.520 r/a = 2.405, or r = 0.436a. In Figure 6.3, thismode of vibration is shown as an annulus surrounding the central circle. Whenthe central circle of the membrane moves up, the outer annulus moves down, andvice-versa. Because portions of the membrane are moving upward, at the sametime the remaining areas move downward, the efficiency of the sound output of adrum is low for the overtone frequencies.A measure of the sound output of each mode is the average displacement amplitudeof the surface when it is vibrating in that mode. For the nth symmetric mode,we can apply Equation (6.24) to determine the average displacement amplitude〈Ψ n 〉 over surface S as follows:〈Ψ n 〉= 1 Aπa∫S2 n J 0 (k n r) dS= 1πa 2 ∫ a0A n J 0 (k n r)2πr dr= 2A nk n a J 1(k n a) (6.25)For all nonsymmetric modes, we note that the angular dependence cos (mθ + ε) ensuresthat the average displacement is zero. Thus, using the prior double-subscriptnotation, we can state that 〈Ψ n 〉 = 0 for all m ≠ 0.It is of interest to find the value of average displacement amplitude 〈Ψ n 〉 forthe fundamental mode and to compare it with 〈Ψ n 〉 for the first overtone. FromEquation (6.25),〈Ψ 1 〉= 2A 1k 1 a J 1(k 1 a) = 2A 12.405 = 0.432A 1The motion of the membrane can be equated to the displacement of a rigid flatpiston of radius a moving with an amplitude of 0.432A 1 . We can also applyEquation (6.25) to find that 〈Ψ n 〉 =−0.123A 2 . The negative sign denotes that theaverage displacement amplitude is opposite in direction to the displacement at thecenter. The fundamental node of vibration is thus seen to be more than three timesas effective for displacing air as for the first overtone.

120 6. Membrane and PlatesIn real applications such as loudspeakers, the amount of air displaced by themembranes, rather than the exact shape of the moving surface, determine theprinciple characteristics of generated sound waves. The radiating source can bedepicted by an equivalent simple piston of area S eq , and this piston moves througha displacement amplitude ζ eq so as to sweep the volume displacement of the actualsource. The volume displacement amplitude of the simple piston equivalent to thecircular membrane vibrating in its fundamental mode isS eq ζ eq = 0.432πa 2 A 1The nodal vibrations of actual membranes cannot be sustained with constantamplitudes because of damping forces occasioned by internal friction and externalforces associated with the radiation of acoustic energy. The amplitude of eachmode tends to decay exponentially with time as e −β nt , where β n represents thedamping constant for mode n. This damping constant generally increases withfrequency, with the result that higher frequencies damp out more quickly thandoes the fundamental.6.6 Application of Membrane Theory to the KettledrumIn addition to the damping forces mentioned above, other forces may act on amembrane and affect its vibration. The kettledrum is an example of the case ofa membrane that covers a closed space in which changes of pressures occur asthe entrapped volume of air changes in pressure incurred by the vibration of thedrumhead. A similar situation occurs with the air entrapped behind the diaphragmof a condenser microphone.The kettledrum consists of a membrane stretched taut over the open end ofa hemispherical shell. As this membrane (i.e., the drumhead) vibrates, the aircontained inside the shell undergoes alternative compressions and rarefactions.With the radial velocity of the transverse waves being considerably less than thespeed of sound in air, the pressure arising from the alternative compression anddecompression of the entrapped air is fairly uniform across the entire drumheadand depends only on the average displacement 〈z〉. With the radius of the drumheaddesignated by a, the incremental or displaced volume of the enclosed air is givenby dV = πa 2 〈z〉. Let us denote by V 0 the equilibrium or quiescent volume of theair enclosed in the kettledrum. The corresponding unperturbed pressure is P 0 . Thevibration of the air enclosed in the kettledrum is essentially an adiabatic process,with the result that the instantaneous pressure P and volume V are related to thequiescent values byPV γ = P 0 V γ0= constant (6.26)Here γ is the ratio of c p , the specific heat of the contained air at constant pressure,to c v , the specific heat at constant volume. Differentiating Equation (6.26) yields

the pressure deviation dP:6.6 Application of Membrane Theory to the Kettledrum 121dP =− γ P 0dV =− γ P 0πa 2 〈z〉 (6.27)V 0 V 0This gives rise to an incremental force dP dx dy over incremental area dx dy ofthe membrane. Modifying Equation (6.3) to include this incremental force weobtain∇ 2 z − 1 γ P 0πa 2 〈z〉 = 1 ∂ 2 z(6.28)c 2 ρ 0 V 0 c 2 ∂t 2The term 〈z〉 is an integral function of all the permitted modes of vibration, whichmust also include the influences of their relative amplitudes and phases. We cangreatly simplify the solution of Equation (6.28) by assuming only one mode ofvibration and disregarding all of the other modes which constitute the generalsolution.The average displacement is zero for all normal modes dependent upon θ;therefore, none of these modes are affected by the pressure fluctuation of the airinside the drum. We need only to consider the symmetric modes entailing theBessel function J 0 . The solution with only one frequency present is of the formdepending only on the coordinate rz = Ψe iωtInserting the above into Equation (6.28) yieldsd 2 Ψdr + 1 dΨ2 r dr + k2 Ψ = γ P ∫ a02πrΨ dr (6.29)TV 0 0In order to establish the solution to the differential equation, we examine the salientfeatures of Equation (6.29). If the right-hand integral term were not present, thesolution would entail J 0 (kr). But the presence of this integral term involvingthe radius a suggests an additional term to the solution, one that is a functionof a, namely J 0 (ka). Moreover, the assumed solution should meet the boundarycondition that Ψ = 0atr = a, regardless of the value of k.We now integrate the right-hand side of Equation (6.29) as follows[2πγ P 0 rJ1 (kr)A − r 2 a0(ka)]TV 0 k 2 J 0= γ P 0TV 0πa 2 A[ ]2J1 (ka)− J 0 (ka)ka= γ P 0TV 0πa 2 AJ 2 (ka) (6.30)Insertion of Equation (6.30) into Equation (6.29) provides the condition for theviability of the assumed solution−k 2 J 0 (ka) = γ P 0TV 0πa 2 J 2 (ka)

122 6. Membrane and PlatesTable 6.2. Allowed Frequencies of a Kettledrum.B 0 1 2 5 10k 1 a 2.405 2.545 2.68 3.02 3.485k 2 a 5.520 5.54 5.55 5.59 5.87k 3 a 8.654 8.657 8.660 8.67 8.69orwhereJ 0 (ka) =− BJ 2 (ka)(ka) 2B = γ P 0TV 0πa 2The parameter B is a dimensionless constant that compares the relative magnitudesof the restoring forces arising from the compression effects of the air trappedinside the drum and the tension applied to the drumhead. The value of B is smallif either the volume or the tension in the membrane is quite large compared withthe compressive pressure acting over the area of the membrane. In the limit whereB approaches zero, the allowed frequencies become those corresponding to thefreely vibrating circular membrane which was described earlier in this chapter.The allowed values of ka are listed in Table 6.2 for selected values of B rangingfrom 0 (which corresponds to an unimpeded vibrating circular membrane) to10 (indicative of low drum volume or light drumhead tension). The effect of theadditional term in Equation (6.28), which is proportional to the displacement andtherefore is indicative of membrane stiffness, is to elevate the allowed frequencies.The effect on the fundamental frequency is much more considerable than it is onthe higher modes of vibration. This stems from the fact that the average displacementamplitude becomes smaller with increasingly higher modes of vibration witha consequently larger number of oppositely phased segments. It is also apparentthat since pressure fluctuations inside the drum affect only the basic frequencymodes z 0n , the tonal qualities of a kettledrum can be varied by parametric changesof the drum volume V 0 and the area πr 2 of the drumhead.6.7 Forced Vibrations of a MembraneConsider a circular membrane that is acted only on one side by a evenly distributedsinusoidal driving pressure p = P cos ωt. In complex notation the pressure isgiven byp = Pe iωtand the equation of motion (6.3) becomes modified as follows:∂ 2 z∂t = 2 c2 ∇ 2 z + P e iωt (6.31)ρ s

6.7 Forced Vibrations of a Membrane 123Assume a steady-state solutionz = Ψe iωt (6.32)which is then inserted into Equation (6.31), resulting in∇ 2 Ψ + k 2 Ψ =Pρ s c =−P (6.33)2 Twhere k = ω/c. In this situation of a driven membrane the angular frequency ωmay have any value, and the wave number k is thus not limited to discrete sets ofvalues which prevail in freely vibrating membranes.The solution to Equation (6.33) consists of two parts, one being a general solutionof the homogeneous equation 2 Ψ h + k 2 Ψ h = 0 and the second being theparticular solution Ψ p =−P/(k 2 T ). Then the complete solution can be written asΨ = AJ 0 (kr) −Pk 2 TThe immobility of the membrane at the rim r = a provides the boundary conditionΨ(a) = 0, andA = 1J 0 (ka) · Pk 2 TThe displacement of the membrane becomesz(r, t) =Pk 2 T[J0 (kr)J 0 (ka) − 1 ]e iωt (6.34)with the corresponding amplitude of the displacement at any position in the membranegiven byΨ(r) = P [ ]J0 (kr) − J 0 (ka)(6.35)T k 2 J 0 (ka)From Equation (6.35) it is seen that the amplitude of the displacement is directlyproportional to the driving force P and inversely proportional to the tensionT . The vibrational amplitude at any location on the membrane depends on thetranscendental terms enclosed by the square bracket in Equation (6.35). But ifthe driving frequency ω corresponds to any of the free-oscillation frequencies ofEquation (6.22), the overtones, the Bessel function J 0 (ka) assumes zero values,presaging infinite amplitudes. But damping forces occur in real cases, and thesemay be represented in Equation (6.31) by a damping factor –(R/ρ s )(∂z/∂t) thatlimits the amplitudes to finite maximum values.The average displacement 〈z〉 s of the driven membrane is found by averagingover the surface area of the membrane:∫ a[ ]2πe iωt P J0 (kr)0 k〈z〉 s =2 T J 0 (ka) − 1 rdrπa 2= P J 2 (ka)k 2 T J 0 (ka) eiωt (6.36)

124 6. Membrane and PlatesAt low frequencies ka assumes a value less than unity and the following approximationsfor Bessel functions hold true:J 0 (ka) ≈ 1 − 3(ka) 2J 2 (ka) = k2 a 2 [1 − k2 a 2 ]8 12Introducing the above approximations into Equation (6.36) yields the followingexpression for the average displacement at low frequencies,〈z〉 s ≈(1 Pa2 + k2 a 2 )(6.37)8T 6If we apply the situation as represented by Equation (6.37) to the design of a condensermicrophone, it is apparent that as long the driving frequency is sufficientlylow, i.e., ka ≪ 1, the output of the microphone will be virtually independent of thefrequency. No resonances should occur in that frequency range. The first resonanceoccurs at ka = 2.405. Becausek = 2π f ( ρS) −1/2= 2π fc Tand if we set the limiting frequency of the uniform microphone response to ka < 1,then√f < 1 T2πa ρ sThe upper frequency limit of the microphone can be elevated by either increasingthe tension T or decreasing the radius a, all other factors being equal. But thisalso has the effect of lessening the amplitude of the average displacement 〈z〉 s and,consequently, the voltage output of the microphone.When a damping factor –(R/ρ s )(∂z/∂t) is included in Equation (6.31), theresulting solution does not change except that k is replaced by k, a complexexpression represented byk 2 = ω2c 2 − iωRTThe presence of the imaginary component −ωR/T causes the average displacementto assume a finite value at resonance. Figure 6.4 displays the average displacementresponse 〈〉 s of a freely vibrating dissipationless membrane, as computedthrough the use of Equation (6.36). The amplitude assumes a value of infinity atka = 2.405. Another curve that includes the effect of damping is also plotted, andthe corresponding amplitude assumes a finite value at ka = 2.405. Both of thesecurves indicate zero responses at ka = 5.136, for which J 2 (ka) = 0. If the frequencyis increased beyond the first resonance value to approximately 1.60 timesthe first resonant frequency, a circular nodal line will appear near the rim of themembrane. As the frequency is increased, the nodular line moves inward as acircle of decreasing radius. The displacement of the membrane’s center is out of

6.8 Vibrating Thin Plates 125Figure 6.4. Plot showing the average (normalized) displacement response as a functionof frequency. The effect of damping is also shown.phase with the driving force, while that of the membrane’s outer portion remains inphase. As the driving frequency increases and the nodal circle shrinks, the averagedisplacements of the two zones tend to cancel each other out. The cancellationbecomes complete at ka = 5.136, and no displacement occurs across the entiresurface of the membrane.6.8 Vibrating Thin PlatesThe principal difference between the vibration of a membrane and a thin plateis the restoring force in the former is due entirely to the tension acting on themembrane and in the latter there is no tension applied, and the restoring force isattributed entirely to the inherent stiffness of the plate.To keep matters simple, we consider only symmetrical vibrations of a uniformcircular diaphragm. The appropriate equation, essentially equivalent to Equation(6.36) but modified to include the effect of stiffness, is∂ 2 z∂t = κ 2 E2 ρ(1 − μ 2 ) ∇2 (∇ 2 z) (6.38)where ρ is the density of the material, μ is the Poisson’s ratio, E is Young’smodulus, and κ is the surface radius of gyration. For a circular plate of uniformthickness b, the radius of gyration is given byκ =b √12

126 6. Membrane and PlatesThe elastic resistance to flexing provides the restoring force that acts on thecircular plate. While there may be some temptation to consider the coefficient−κ 2 E/[ρ(1 − μ 2 )] 2 on the right-hand side of Equation (6.38) as being analogousto −κ 2 E/ρ in Equation (5.28) for the transverse vibration of a bar, this is notstrictly true because a sheet will curl up sideways as it is bend downward along itslength. This is the Poisson’s effect in which the curling occurs from the lateral expansionas the longitudinal compression ensues from the bending of the plate. Anincrease in the effective stiffness is thereby produced. The Poisson’s ratio μ givenin Equation (6.38) is the negative ratio of the lateral strain Mξ/Myto Mζ/Mx, i.e.,μ =− ∂ξ/∂y∂ς/∂xIn order to keep the Poisson’s ratio a positive number, it is necessary to introducethe minus sign to counteract the effect that a positive longitudinal strain gives riseto a negative lateral strain of compression. The value of μ, which is a property ofthe material, may be obtained from standard tables and is generally of the value0.3. In Equation (6.38), the factor (1 − μ 2 ) −1 embodies the effective increase inthe stiffness of the plate resulting from the curling.In solving Equation (6.38) it is assumed thatz = Ψ(r) iωtwhich is then substituted into that equation to givein which∇ 2 (∇ 2 Ψ) − K 4 Ψ = 0 (6.39)K 4 = ω2 ρ(1 − μ 2 )κ 2 EThe substitution of the Helmholtz equation 2 Ψ =−K 2 Ψ into Equation (6.39)indicates that if Ψ can satisfy the Helmholtz equation, it will also constitute asolution to Equation (6.39). The function Ψ in the relationship 2 Ψ = K 2 Ψ willalso satisfy Equation (6.39), so the complete solution of this equation must be thesum of four independent solutions to∇ 2 Ψ ± K 2 Ψ = 0 (6.40)Equation (6.40) with the positive sign is the Helmholtz equation with circular symmetry,which yields the solutions J 0 (Kr) and Y 0 (Kr). But the boundary conditionthat the displacement must be finite at r = 0 at the center of the plate requiresthat the latter solution must be scrapped. The solution of Equation (6.40) with thenegative sign yields J 0 (iKr) and Y 0 (iKr); the latter term is also discarded. Theterm J 0 (iKr) is a modified Bessel function of the first kind, generally written as 22 The modified Bessel functions I n (x) are solutions of the modified Bessel differential equationd 2 ydx 2 + 1 ( )dyx dx − 1 + n2x 2 y = 0

I 0 (Kr). The complete applicable solution of Equation (6.39) is6.8 Vibrating Thin Plates 127Ψ = AJ 0 (Kr) + BI 0 (Kr) (6.41)The function J 0 (Kr) is an oscillating function that damps out with increasing rwhile I 0 (Kr) increases continuously with r.The manner in which the plate is supported determines the conditions whichare used to evaluate the constants A and B. A common type of support is one inwhich the circular plate is rigidly clamped at its periphery r = a. The boundaryconditions therefore areThese yieldΨ = 0 and ∂Ψ/∂r = 0 atr = aAJ 0 (Ka) =−BI 0 (Ka), AJ 1 (Ka) =−BI 1 (Ka) (6.42)and through elimination of the constants A and B we obtain the transcendentalequation which gives the permissible values of Ka:J 0 (Ka)J 1 (Ka) =−I 0(Ka)(6.43)I 1 (Ka)Both I 0 and I 1 remain positive for all values of Ka, so solutions occur only when J 0and J 1 have opposite signs. The sequence of solutions satisfying Equation (6.43)isKa = 3.20, 6.30, 9.44, 12.57,...The above can be approximated by Ka = nπ, where n = 1, 2, 3,....This approximationimproves with increasing values of n.From the definition of K for Equation (6.39), it is apparent that the frequencycan be found from√f = ω2π = κ K 22πEρ(1 − μ 2 )By setting K = 3.20/a, the fundamental frequency f 1 is found to be√√f 1 = ω 12π = 3.22 b E√2πa 2 12 ρ(1 − μ 2 ) = 0.47 b Ea 2 ρ(1 − μ 2 )where b represents the thickness of the plate. The frequencies of the overtones aregiven by( ) 6.3 2f 2 = f 1 = 3.88 f 13.2f 3 = 8.70 f 1 , etc.These frequencies are spread out much further apart than those for the circularmembrane.

128 6. Membrane and PlatesFor the fundamental mode of vibration, the displacement of a thin circular plateis given by[ ( ) ( )]3.23.2z 1 = cos(ω 1 t + φ 1 ) A 1 J 0a r + B 1 I 0a rFrom the boundary condition relationships of Equation (6.42) the last expressionbecomes[ ( ) ( )]3.23.2z 1 = A 1 cos(ω 1 t + φ 1 ) J 0a r + 0.555I 0a rIt is interesting to observe that the amplitude at the center r = 0 is 1.0555A 1 ,not A 1 . If we compare the shape function represented by the bracketed terms onthe right-hand side of the last equation with the corresponding shape functionJ 0 (2.405r/a) for the fundamental mode of a similar-sized vibrating circular membrane,it will be found that the relative displacement of the plate near its edgeis considerably smaller than that of the membrane. Hence, the ratio of the averageamplitude to the amplitude at the center is less than that in the case for themembrane. The average displacement amplitude is given by〈Ψ 1 〉 s = 0.326A 1 = 0.309z 0where z 0 = 1.0555A 1 represents the amplitude at the center r = 0 of the plate.In the same manner we used to represent the membrane, the circular plate can bedepicted by an equivalent flat piston so thatS eq ζ eq = 0.309πa 2 z 0Plates can also undergo loaded and forced vibrations. The mathematical treatmentsof these cases are analogous to those for membranes, and the response curves aresimilar to those shown in Figure 6.4. Large amplitudes will also occur at resonancefrequencies unless there is appreciable damping.The most apparent use of the vibrating thin plate is that of the telephone diaphragms(both receiver and microphone). While these diaphragms do not providethe flatter frequency responses or frequency range of membranes in condenser microphones,they do provide adequate intelligibility, are generally far more ruggedin their construction and cheaper to manufacture. Sonar transducers used to generateunderwater sounds less than 1 kHz constitute another class of vibrating plates;the signals are produced by the variations of an electromagnetic field in an electromagnetpositioned closely to a thin circular steel plate.ReferencesFletcher, Neville H. and Rossing, Thomas D. 1998. The Physics of Musical Instruments,2nd ed. New York: Springer-Verlag, Chapter 3.Kinsler, Lawrence E., Frey, Austin R., Coppens, Alan B., and Sanders, James V. 1982.Fundamentals of Acoustics, 3rd ed. New York: John Wiley & Sons, Chapter 4.

Problems for Chapter 6 129Morse, Philip M. and Ingard, K. Uno. 1968. Theoretical Acoustics. New York: McGraw-Hill, Sections 5.2 and 5.3.Reynolds, Douglas R. 1981. Engineering Principles of Acoustics, Noise and VibrationControl. Boston: Allyn and Bacon, pp. 247–255.Wood, Alexander. 1966. Acoustics. New York: Dover Publications, pp. 429–436.Problems for Chapter 6All membranes described below may be assumed to be fixed at their perimetersunless otherwise indicated.1. Consider a square membrane, having dimensions b × b, vibrating at its fundamentalfrequency with amplitude δ at its center. Develop an expression thatgives the average displacement amplitude. Obtain a general expression forpoints having an amplitude of δ/2. Plot at least five points from this generalexpression. Do these points fall in a circle?2. A rectangular membrane has width b and length 3b. Find the ratio of the firstthree overtone frequencies relative to the fundamental frequency.3. Consider a circular membrane with a free rim. Develop the general expressionfor the normal modes and sketch the nodal patterns for the three normal modeswith the lowest natural frequencies. Express the frequencies of these normalmodes in terms of tension and surface density.4. A circular aluminum membrane of 2.5 cm radius and 0.012 cm thickness isstretched with a tension of 15,000 N/m. Find the first three frequencies of freevibration, and for each of the frequencies, determine any nodal circles.5. Prove that the total energy of a circular membrane vibrating in its fundamentalmode is equal to 0.135πρ s (aωA f ) 2 where ρ s is the area density, a the radiusof the membrane, and ω the angular frequency of the vibration, and A f thefundamental amplitude at the center.6. Steel has a tensile strength of 1.0 GPa (= 10 9 Pa) and aluminum, 0.2 GPa.Using these values as the maximum tensions, what will be the maximumfundamental frequency of a 2-cm-diameter circular membrane made up of eachof these materials? Note: for thin membranes these fundamental frequenciesare independent of the thicknesses.7. A damping force is applied uniformly over the surface of a circular membrane.This damping force per unit area =−I∂z/∂t should be introduced into theappropriate wave equation in a manner consistent with the dimensions of theterms of the equation. Solve the equation to demonstrate that the amplitudesof the free vibrations are damped exponentially as e −1/2It/ρ s.8. A kettledrum consists of a circular membrane of 50 cm diameter, with anarea density of 1.0 kg/m 2 . The membrane is stretched under a tension of10,000 N/m.(a) Determine the fundamental frequency of the membrane without a backingvessel.

130 6. Membrane and Plates(b) Determine the fundamental frequency for the membrane with a backingvessel that is a hemispherical bowl of 25 cm radius. The vessel isfilled with air at a pressure of 100 kPa, and γ (the ratio of specific heats)is 1.4.9. An undamped membrane of 4-cm radius has an area density of 1.6 kg/m 2and is stretched to a tension of 1200 N/m. It is driven by a uniform pressure7000 sin ωt Pa applied over the entire surface.(a) Determine and plot the amplitude of the displacement at the center as afunction of frequency ranging 0–2 kHz.(b) Compute and plot the shape of the membrane when driven by the appliedfrequency of 600 Hz and the applied frequency of 1000 Hz.10. A condenser microphone contains a circular aluminum diaphragm of 30-mmdiameter and 0.02-mm thickness. Aluminum has a maximum tensile strengthof 0.2 GPa.(a) What is the allowable maximum tension in N/m in the diaphragm?(b) What will be the fundamental frequency under these conditions?(c) What will be the displacement of the diaphragm at its center under theimpetus of a 500-Hz sound wave having a pressure amplitude of 1.5 Pa?(d) What will be the average displacement under the conditions of (c)?11. If the volume of air trapped behind the diaphragm of the condenser microphoneof the preceding problem is 2.5 × 10 −7 m 3 , by how much will the fundamentalfrequency be raised? Assume the normal air pressure to be equal to 100 kPaand γ = 1.4.12. Use integration over the surface of a circular thin plate vibrating in its fundamentalmode to show that the average displacement amplitude is 0.327A,where A denotes the displacement in the center of the plate.13. The diaphragm of a typical telephone receiver comes in the form of a circularsheet of steel, 4 cm in diameter and 0.18 mm thick.(a) What is the fundamental frequency, if the diaphragm is rigidly clamped atits rim?(b) How will this fundamental frequency change if the diaphragm thicknessis doubled?(c) What would happen to the fundamental frequency if the diameter of thediaphragm was increased by 50%?14. Find the fundamental frequency of a vibrating circular steel plate which isclamped at its rim and is of 25 cm diameter and 0.55 mm thickness.

7Pipes, Waveguides, and Resonators7.1 IntroductionIn dealing with strings, bars, and membranes in Chapters 4–6, we considered relativelysimple geometric conditions. The situation becomes more complex whenthe sound waves are confined in a restricted amount of space. For example, whensound propagates inside a rigid-walled pipe with a wavelength that exceeds theradius of a rigid-wall pipe, the acoustic propagation inside the pipe becomes fairlyplanar. The resonance properties of the pipes driven at one end and closed off atother end constitute the basis for measuring acoustical impedances and absorptionproperties of materials. In our study of pipes we establish the models for physicalanalyses of wind musical instruments, organ pipes, and ventilation ducts (Fletcherand Rossing, 1991). In larger spaces, where the dimensions may exceed wavelengths,two- and three-dimensional standing waves can occur. We shall treat thesimple case of a waveguide with a uniform cross section, establish the conceptof group speed and phase speed which occurs with a wave propagating inside awaveguide. The acoustic waveguide is very much analogous to the electromagneticwaveguides, and it finds applications in surface-wave delay lines and in thepropagation of sound in ocean and atmospheric layers. We shall also consider thephysics of a Helmholtz resonator.7.2 The Simplest Enclosed System: InfiniteCylindrical PipeThe simplest enclosed system inside which sound propagation occurs is an infinitecylindrical pipe with its axis parallel to the direction of the propagation of the planewave in the enclosed medium. The pipe wall is assumed to be rigid, perfectlysmooth, and adiabatic (i.e., no heat transfer occurs through the wall). The pipethus has no effect on the wave propagation. A pressure wave generated by a pistonmoving in the x-direction can be expressed asp(x,t) = p 0 e i(ωt−kx) (7.1)131

132 7. Pipes, Waveguides, and ResonatorsHere p 0 is the maximum amplitude of the pressure wave. The volume flow is givenbyU(x, t) = p 0Sρc ei(ωt−kx)where ω is the angular frequency, k = 2π/λ = ω/c the wave number, S the crosssectional area of the pipe, ρ the density of the fluid inside the pipe, and c thepropagation velocity.7.3 Resonances in a Close-Ended PipeAs shown in Figure 7.1 consider a pipe of length L and cross-sectional area S,filled with a fluid, and sealed off at one end, x = L. Let the fluid inside the pipebe driven by a piston at x = 0. The pipe has a mechanical impedance Z nL. Thepiston vibrates harmonically at a sufficiently low frequency so that only planewaves are considered to exist inside the pipe. The wave inside the pipe can bedescribed byp = Ae i[ωt+k(L−x)] + Be i[ωt−k(L−x)] (7.2)where A and B are established by the boundary conditions at x = 0 and x = L.At x = L the mechanical impedance of the wave must equal the mechanicalimpedance Z nL at the termination so as to sustain the continuities of force andparticles. The force of the fluid acting at the end of the pipe is p(L, t)S, and thecorresponding particle speed u(L, t) derives from the integrated Equation (2.23)∫u =−1ρ δ ( ∂p∂x)dtFigure 7.1. A pipe close-ended at x = L.

Mechanical impedance Z n , expressed as7.3 Resonances in a Close-Ended Pipe 133Z n = f u(7.3)represents a complex value that is the ratio of the complex driving force f to thecomplex speed u at the point where the force is applied. In the case of the finitepipe, the mechanical impedance at x = L is given byZ nL = ρ 0 cS A + B(7.4)A − BThe value of the input mechanical impedance at x = 0 is expressed asZ n0 = ρ 0 cS AeikL + Be −ikL1 + iZ nLρ 0 cS tan kL (7.5)Eliminating A and B by combining Equations (7.4) and (7.5) yieldsZ n0ρ 0 cS =Z nL+ i tan kLρ 0 cS1 + Z (7.6)nLρ 0 cS tan kLThe term ρ 0 cS is the characteristic mechanical impedance of the fluid. The complexquantity Z n0 can be recast in terms of real and imaginary components, r andψ, respectively,Z nL= r + iψ (7.7)ρ 0 cSThe ratio on the left-hand side of Equation (7.7) constitutes a normalizedimpedance. Inserting Equation (7.7) into Equation (7.6) yieldsZ n0ρ 0 cS=(r + iψ) + i tan kL1 + i(r + iψ) tan kL= r(tan2 kL + 1) − i[ψ tan 2 kL + (r 2 + ψ 2 − 1) tan kL − ψ](ψ 2 + r 2 ) tan 2 (7.8)kL − 2ψ tan kL + 1When r = 0, the input impedance Z n0 vanishes when the reactance vanishes, i.e.,−i[ψ tan 2 kL + (r 2 + ψ 2 − 1) tan kL − ψ](ψ 2 + r 2 ) tan 2 kL − 2ψ tan kL + 1and this results inthat is,= 0 (7.9)ψ tan 2 kL + (ψ 2 − 1) tan kL − ψ = 0 (7.10)ψ =−tan kL (7.11)

134 7. Pipes, Waveguides, and ResonatorsThe impedance becomes infinite whenorψtan 2 kL + (ψ 2 − 1) tan kL − ψ = 0ψ = cot kLLet us briefly examine the situation for a constant driving force at x = 0. Thevanishing of Z n0 = f/u 0 connotes that the speed amplitude at the point of forceapplication (x = 0) is infinite, the condition for mechanical resonance. Converselythe input impedance reaching infinity means that the speed amplitude approacheszero, which describes the condition of antiresonance.To obtain the condition of resonance in a pipe driven at x = 0 and sealed with arigid cap at x = L,welet|Z nL /ρ 0 cS| approach infinity in Equation (7.4), givingZ n0=−icot kLρ 0 cSThe reactance becomes zero when cot kL = 0,k n L − (2n − 1)π/2 n = 1, 2, 3,...and so we obtain the set of resonant frequencies as for the forced-fixed string:f n = 2n − 14With the odd harmonics of the fundamental constituting the resonance frequencies,the driven closed pipe contains a pressure antinode at x = L and a pressure node atx = 0. This means that the driver must present a vanishing mechanical impedanceto the tube.cL7.4 The Open-Ended PipeConsider a pipe driven at x = 0 but open-ended at the other end x = L. The assumptionthat Z nL = 0atx = L (which would lead to resonances at f n = 1 / 2 nc/L)is not valid, because the open end of the pipe radiates into the surrounding air. Theappropriate condition is Z nL = Z r , where Z r is the radiation impedance at theopen end of the tube. Also, the presence of a flange at the open end affects theexit impedance. Consider the case of a flange at the open end of a circular pipe ofradius a. The flange is large with respect to the wavelength of the sound, which, inturn, is considerably larger than the tube radius (λ ≫ a). This situation resemblesa baffled piston in the low-frequency limit. From theory (Kinsler and Frey, 1962)Z nLρ 0 cS = (ka)2 + i 8 ka2 3 π(7.12)where the real component r = (ka) 2 /2 and the imaginary component ψ = 8ka/3πare both much less than unity and r ≪ ψ. Under these conditions the solution to

7.5 Radiation of Power from Open-Ended Pipes 135Equation (7.9) yields tan kL =−ψ in order for resonance frequencies to occur.With the assumption ψ ≪ 1, we obtaintan(nπ − k n L) = 8ka 8ka≈ tan (7.13)3π 3πwhere n = 1, 2, 3,.... Hencenπ = k n L + 8k na(7.14)3πThe resonance frequencies therefore aref n = n c2L + 8(7.15)3π aAll of the resonance frequencies are harmonics of the fundamentals. We also notethat the denominator L + 8a/(3π) constitutes the effective length of the pipe ratherthan actual length L.In the case of the unflanged pipe, the radiation impedance, indicated by boththeory and experiment is given approximately byZ nLρ 0 cS = (ka)2 + i(0.6ka) (7.16)4Here the end correction for the unflanged pipe equals 0.6a, with the effectivelength being L + 0.6a. We also note that the end corrections do not depend onthe frequency. Providing that λ n ≫ a, the resonance frequencies of flanged andunflanged pipes constitute harmonics of the fundamental. Hence, the driving frequencyof an open-ended organ pipe yields resonances that are harmonics of thedriving frequency. The above exposition so far has dealt with pipes of constantcross sections. If a pipe is flared at the open end, as is the case with many windinstruments such as the clarinet and the oboe, the results are modified, and theresonances may not necessarily be harmonics of the fundamental. Variations inthe flare design will emphasize or lessen certain harmonics present in the forcingfunction, thereby affecting the quality or timbre of the sound emanated by the pipe.7.5 Radiation of Power from Open-Ended PipesEquation (7.4) may be revised to readBA =Z nLρ 0 cS − 1Z nLρ 0 cS + 1 (7.17)When the termination impedance Z nL is known the power transmission coefficientT n can be established fromT n = 1 −B2∣A∣(7.18)

136 7. Pipes, Waveguides, and ResonatorsThrough the use of Equation (7.16), Equation (7.17) applied to the case of anopen-ended pipe becomes[ ]1 − (ka)2 − i 8kaBA = 2 3π][1 + (ka)2 + i 8ka(7.19)2 3πwhich is then inserted into Equation (7.18), which now becomes2(ka) 2T n =] 2 ( ) [1 + (ka)28 2(7.20)+ (ka)22 3πNormally ka ≪ 1, so the power transmission coefficient is quite small and it canbe simplified toT n ≈ 2(ka) 2 (for flanged pipe) (7.21)From Equation (7.18) it can be ascertained that B/A is almost equal to –1. Thepressure amplitude of the reflected wave is barely less than that of the incident wave.At x = L, its pressure differs by nearly 180 ◦ out of phase. On the other hand, theincident and reflected particle speeds remain nearly in phase at the opening of thepipe, so that the location is (nearly) the antinode of the particle speed. Even thoughthe amplitude of the particle speed at the opening is nearly twice that of the incidentwave, only a small percentage of the incident power transmits out of the flangedpipe. Thus, sources having dimensions small compared with the wavelength of thesound behave as inefficient radiators of sonic energy.Inserting Equation (7.16) for the unflanged pipe into Equation (7.17), the transmissioncoefficient becomes(ka) 2T n =] 2≈ (ka) [1 2 (7.22)+ (ka)2 + (0.6ka) 24By comparing Equations (7.20) and (7.22) we perceive that adding a flange at theend of the pipe essentially doubles the radiation of sound at low frequencies. Agradual flare at the open terminal of the pipe will increase the low-frequency powertransmission even more.7.6 Standing Waves in PipesThe existence of phase interference between transmitted and reflected waves insidea terminated pipe gives rise to a standing wave pattern. The properties of thestanding waves can be applied to gauge the load impedance. In Equation (7.2) letus setA = A, B = Be iθ (7.23)

7.6 Standing Waves in Pipes 137where A and B are real, positive numbers. Combining Equations (7.4) and (7.23)results inZ1 + BnLρ 0 cS = A eiθ1 − B (7.24)A eiθInserting Equation (7.23) into Equation (7.2) we obtain for the pressure amplitudep =|p| of the wave√p = (A + B) 2 cos 2 [k(L − x) − θ/2] + (A − B) 2 sin 2 [k(L − x) − θ/2](7.25)At a pressure antinode, the pressure amplitude is A + B, while the amplitudepressure at the pressure node is A − B. The standing wave ratio (SWR) occurs asthe ratio of these two pressure amplitudes, respectively,SWR = A + BA − B or BA = SWR − 1(7.26)SWR + 1SWR is measured by probing the sound field along the pipe (also known as animpedance tube) with a tiny microphone to obtain the value of B/A. The phaseangle θ can be found by measuring the distance of the first node from the end atx = L. According to Equation (7.25) the nodes are located atk(L − x n ) − θ (2 = n − 1 )π (7.27)2For the first node (n = 1),θ = 2k(L − x 1 ) − π (7.28)Example Problem 1An impedance tube is found to have a SWR of 3 and the first node is 3/8 of thewavelength from the end. Find the normalized mechanical impedance at x = L.SolutionL − x = 3λ/8Hence from Equation (7.28) θ = 2(2π/λ)(3λ/8) − π = π/2. From Equation(7.26)BA = 3 − 13 + 1 = 1 2Equation (7.24) now readsZ nLρ 0 cS = 1 + 1 2 eiπ/21 − 1 2 eiπ/2 = 0.60 + i0.80

138 7. Pipes, Waveguides, and ResonatorsMechanical impedances at terminations occur as complicated functions of thefrequencies, so it may be necessary to conduct measurements over the range offrequencies under consideration. Smith nomographs (Beranek, 1949) are usefultools to expedite computations for r and ψ, the real and imaginary components ofthe impedances, from the measurements of the standing wave ratio and the positionof the node most adjacent to the end.The impedance tube is used to measure the reflective and absorptive propertiesof small sections of materials, such as acoustic tiles and sound control absorbers,mounted at the end of the tube.7.7 The Rectangular CavityIn Figure 7.2 a rectangular cavity is shown having dimensions L x , L y , L z in thex-, y-, and z-directions, respectively. This parallelepiped can represent a simpleauditorium or any other rectangular space that contains rigid walls, few windows˘and other openings. We assume the walls of the cavity to be perfectly rigid thatn · ⃗u = 0 at all of the boundaries (i.e., the walls will not move in the directions oftheir normals). This also means that n ·∇p = 0, i.e.,( ) ( ) ⎫∂p ∂p== 0∂x x=0 ∂x x=L x( ) ( )∂p ∂p⎪⎬=(7.29)∂y)( ∂p∂zy=0z=0˘== 0∂y y=L y)= 0⎪⎭z=L z( ∂p∂zFigure 7.2. The rectangular cavity.

7.7 The Rectangular Cavity 139Because acoustic energy cannot escape from a completely closed cavity withrigid walls, standing waves constitute the only appropriate solutions of the waveequation. Insertinginto the wave equationp(x, y, z, t) = X(x)Y (y)Z(z)e iωt (7.30)∇ 2 p = 1 ∂p(7.31)c 2 ∂tand separating variables results in the following set of equations:( d2) ( d2) ( d2)dx + 2 k2 x X = 0,dy + 2 k2 y Y = 0,dz + 2 k2 z Z = 0 (7.32)Here the separation constants are related as follows:k 2 = ω2c 2 = k2 x + k2 y + k2 z (7.33)The boundary conditions of Equation (7.29) stipulate cosine solutions, and Equation(7.30) revises towhere the components of k arep lmn = A lmn cos k xl x cos k ym y cos k zn xe iω lmnt(7.34)k xl = lπ ,L xl = 0, 1, 2,..., k ym = mπ ,L ym = 0, 1, 2,...k zn = nπ ,L zn = 0, 1, 2,..., (7.35)This leads to the quantization of allowable frequencies of vibration√l 2 π 2ω lmnc=L 2 x+ m2 π 2L 2 y+ n2 π 2L 2 z(7.36)The above gives rise to eigenfunctions of Equation (7.34). Each eigenfunctionis characterized by its own eigenfrequency (7.36) specified by the ordered integers(l,m,n). Equation (7.34), which is the solution to the wave equation (7.31), yieldsthree-dimensional standing waves in the cavity with nodal planes parallel to thewalls. The pressure varies sinusoidally between these nodal planes. In the samemanner that a standing wave on a string could be resolved into a pair of waves travelingin opposite directions, we can separate the eigenfunctions in the rectangularcavity into traveling plane waves. This is done by casting the solutions (7.34) intocomplex exponential form and expanding it as a sum of products:p lmn = A ∑lmn eiω lmn t(±k x x±k x y±k z z)(7.37)8where the summation is taken over all permutations of plus and minus signs.There are eight terms in all, each representing a plane wave traveling along the

140 7. Pipes, Waveguides, and Resonatorsdirection of its propagation vector ˆk which has projections ±k xl ′, ±k ym ′, ±k zn ′ onthe coordinate axes. The standing wave solution results from the superpositionof eight traveling waves (one into each quadrant) whose directions of travel areconstrained by the boundary conditions. The methodology involving separation ofvariables can also be used to treat standing waves in other simple cavities, suchas cylindrical and spherical cavities, with the eigenfunctions entailing Bessel andLegendre functions.7.8 A Waveguide with Constant Cross SectionIn Figure 7.3 a waveguide of rectangular cross section is assumed to have rigidside walls and a source of acoustic energy located at its boundary z = 0. Thereis no other boundary on the z-axis, which permits energy to propagate down thewaveguide. This results in a situation where the wave pattern consists of standingwaves in the transverse directions x and y and a traveling wave in the z-direction.The mathematical solution which would contain applicable eigenfunctionsisp lmn = A lmn cos k xl x cos k ym ye i(ωt−k zz)(7.38)Upon substituting Equation (7.38) into the wave equation (7.31) we obtain thefollowing relationship:ω 2c 2 = k2 = k 2 xl + k2 ym + k2 z (7.39)Figure 7.3. A waveguide having a rectangular cross section. The travel is along the z-axis.

7.8 A Waveguide with Constant Cross Section 141with permitted values of k xl and k ym , resulting from the boundary conditions ofrigidity, these beingk xl = lπ ,L xl = 0, 1, 2,...k ym = mπ ,L ym = 0, 1, 2,... (7.40)We rearrange Equation (7.39) to find k z√ω2k z =c − 2 k2 xl − k2 ym (7.41)Because ω may have any value, Equation (7.38) comprises a solution for all valuesof ω, in contrast to the totally enclosed cavity which allows for only quantizedfrequencies. Setting√k lm = kxl 2 + k2 ym (7.42)we can shorten Equation (7.41) to√ω2k z =c − 2 k2 lm(7.43)The value k z is real when ω/c > k lm . We then obtain a propagating mode,asthewave moves in the +z-direction. The cutoff frequency occurs when ω/c = k lm ,sodefining the limit for which k z remains real; is given byω lm = ck lm (7.44)for the (l, m) mode. A frequency below the threshold value of ω lm results in apurely imaginary value of k z ,√k z ≡±i klm 2 − ω2c 2 (7.45)We need to include the negative sign in Equation (7.45) so that p → 0asz →∞and the eigenfunctions assume the form(√p lm = A lm cos k xl x cos k ym ye − klm )z 2 − ω2c 2 eiωt(7.46)Equation (7.46) represents a standing wave that decays exponentially with z. Thisform of eigenfunction is termed an evanescent mode—i.e., no energy is propagatedalong the waveguide. If the frequency exciting the waveguide is just below thecutoff value of some particular mode, then this and higher modes are evanescentand are of little consequence at appreciable distances from the source. The lowermodes may propagate energy and can be detected at large distances from thesource.Only plane waves can propagate in a rigid-walled waveguide if the frequencyof the sound is sufficiently low. This frequency is less than c/(2L), where L is thelarger dimension of the rectangular cross section.

142 7. Pipes, Waveguides, and ResonatorsFrom Equation (7.38), the phase speed for mode lm is not c butc p = ω/k z = √1 −c(klmk)= √21 −c( ωlmω) 2(7.47)A physically meaningful solution can be written in the complex exponential formp lm = 1 4 A ∑lm e i(ωt±k xlx±k ym y−k z z)(7.48)±The absence of the boundary at a location on the +z-axis necessitates only anegative sign for k z , since there will be no reflected waves propagating in the −zdirection.The propagation vector k for each of the four traveling waves forms anangle θ with the z-axis, according tocos θ = k z√k = 1 − ( ω lm)(7.49)2ωand the corresponding phase speed c p is˘c p =c(7.50)cos θThe surfaces of constant phase for two component waves representing the (0,1)mode are shown in Figure 7.4 for a rigid waveguide. The waves cancel eachother precisely for y = 1/2L y , with the result there exists a nodal plane halfwayFigure 7.4. The surfaces of constant phase for two-component waves in the (0,1) mode.

7.9 A Waveguide with Constant Cross Section 143between the walls. The waves remain in phase at the upper and the lower walls,so the pressure amplitude is maximized at these rigid boundaries. The apparentwavelength λ z in the z-direction is given byλ z =λ(7.51)cos θIn the lowest mode (0,0), k z = k and the four component waves form a singleplane wave that travels down the axis of the waveguide with speed c. For allthe other modes, the propagation vectors of the component waves generally formangles with the z-axis, with one aimed into each of the four forward quadrants.According to Equation (7.47), at frequencies much greater than the cutoff of the(l, m) mode, i.e., ω ≫ ω lm , the angle θ approaches zero and the waves are travelingfairly straight down the waveguide. As the input frequency approaches the cutoffvalue, θ increases with the result that the component waves move in increasinglytransverse directions. In fact, when the frequency reaches the stage that ω = ω lm ,the component waves are traveling transversely to the axis of the waveguide.Each component wave carries energy along the waveguide through the process ofcontinual reflection from the rigid walls (in the same manner a signal is transmittedthrough a fiberoptic line, bouncing from one wall back to the opposite wall down˘along the line). With the energy of the wave propagating at a speed c in the directionk, the corresponding speed c p (the group speed) of the energy in the z-direction isgiven by the component of the plane-wave velocity c along the waveguide axis:√( ωlm) 2c g = c cos θ = c 1 −(7.52)ωGiven the driving frequency ω, each normal mode, in which ω lm

144 7. Pipes, Waveguides, and Resonators7.9 Boundary Condition at the Driving Endof the WaveguideAn impedance match must be made at the waveguide entry z = 0 so that thepermissible mode solutions comply with the acoustic behavior of the active surface.If we know the pressure or velocity distribution of the driving source, these can becorrelated with the behavior of p(x, y, 0,t) for the pressure or z ·⃗u(x, y, 0, t) forthe velocity. If the pressure distribution of the source is given, then the boundarycondition becomesp(x, y, z, t) = P(x, y)e iωt (7.53)However, the left-hand side of Equation (7.53) can also be expressed as a superpositionof the normal modes of the waveguide, i.e.,p(x, y, z, t) = ∑ A lm cos k xl x cos k ym ye i(ωt−k zt)(7.54)l,mSetting z = 0, Equation (7.53) becomesP(x, y) = e iωt ∑ A lm cos k xl x cos k ym y (7.55)With Equation (7.55) we can establish the values of A lm by applying Fourier’stheorem, first for the x direction and then for the y direction.˘7.10 Rigid-Walled Circular WaveguideThe treatment of the rigid-walled circular waveguide of radius r = a is fairlystraightforward, beginning withP ml = A ml J m (k ml r) cos(mθ) e (iωt−k zt)where (r, θ, z) are the customary cylindrical coordinates, J m is the mth-orderBessel function, and√ω2k z =c − 2 k2 lmwhere k ml is found from the boundary conditions. Because r ·∇p = 0atthewallr = a,k ml = j ml′awhere jlm ′ are the zeros of dJ m(z). When the values of kdzml are determined, theapplicable results developed for the rectangular waveguide may be applied bysubstituting the values of k ml for the circular waveguide. As with the rectangularunit, the (0,0) mode of the circular waveguide is a plane wave that propagates withphase velocity c p = c for all ω>0. For frequencies below f 1,1 , only plane wavescan propagate in a rigid-walled circular waveguide.

7.11 The Helmholtz Resonator 1457.11 The Helmholtz ResonatorMany analyses of acoustic devices become simplified with the assumption that thewavelength in the propagation fluid is much greater than the dimensions of thedevices. If the wavelength is indeed much larger in all dimensions, the acousticvariable may be time varying, but it is virtually independent of the distance withinthe confines of the device. In such a case, the device can be viewed as a harmonicoscillator with one degree of freedom; and in the long-wavelength limit, such adevice can be considered a lumped acoustic element. An example of this typeof device is the Helmholtz resonator, illustrated in Figure 7.6 (three types areshown). The resonator is a rigid-walled cavity of volume V , with a neck of area Aand length L.According to theory, if λ ≫ L, the fluid in the neck behaves somewhat as asolid plug which vibrates back and forth. As the fluid (usually air) moves back andforth, the acoustic energy converts to heat as a result of friction along the neck.These losses can be increased by placing a light porous material across the neckor by placing the material inside the volume. Maximum sound absorption occursat the resonant frequency of the mass of air in the neck with the “spring” suppliedby the air resistance inside the enclosed volume. Very little sound is absorbed atother frequencies. However, for necks greater than 1 cm in diameter, the viscouslosses are considerably less than those associated with radiation. For the purposeof this analysis we can ignore the effects of viscosity in analyzing the Helmholtzresonator viewed as an analogous spring-mass system. The air in the neck has atotal effective massm = ρ 0 AL ′ (7.56)where the effective neck length L ′ is longer than the physical length L becauseof its radiation mass loading. We have seen earlier in this chapter that at lowfrequencies a circular opening of radius a is loaded with a radiation mass equalto that of the fluid occupying area πa 2 ; and effective length 8a/(3π) = 0.85a ifterminated in a wide flange and 0.6a for an unflanged terminal. We assume themass loading at the inner end of the neck is equivalent to a flanged termination.Figure 7.6. Three simple Helmholtz resonators.

146 7. Pipes, Waveguides, and ResonatorsThen,L ′ = L + 2(0.85a) = L + 1.7a (flanged out end)L ′ = L + (0.85 + 0.6)a = L + 1.5a (unflanged outer end) (7.57)Now consider the neck of the resonator to be fitted with an air-tight piston. Whenthe piston travels a distance δ, the volume of the cavity changes by V =−δ A.Then,ρρ=−V V= AδVand from the thermodynamic relationp = ρ 0 c 2 ppCombining the last two equations gives the following:p = ρ 0 c 2 Aδ(7.58)VThe force f = pAnecessary to execute the displacement is ρ 0 c 2 A2δ. The effectiveVstiffness S (from the spring formula f = Sδ)isS = ρ 0 c 2 A2(7.59)VThe fluid moving in the neck radiates sound into the surrounding medium in thesame manner as an open-ended pipe. For wavelengths much larger than the radiusof the neck, the radiation resistance [cf. Equations (7.12) and (7.16)] isR r = ρ 0 c k2 A 2(flanged)(7.60a)2πorR r = ρ 0 c k2 A 2(unflanged)(7.60b)4πThe sound wave impinging on the neck opening is represented as an instantaneousdriving force with a pressure amplitude P:f = APe iωt (7.61)We then can write the differential equation for the inward displacement δ of thefluid “plug” in the neck asm d2 δdt + R dδ2 rdr + Sδ = APeiωt (7.62)This last equation is analogous to that of a sinusoidally driven oscillator which hasanalogous solutions. Solution of Equation (7.62) gives a complex displacement δ.The real part of the driving force represents the actual driving force AP cos ωt,and the real part of the complex displacement represents the actual displacement.

7.11 The Helmholtz Resonator 147Because f is periodic with angular frequency ω, δ = Be iωt must be the solutionwhere B exists as a complex constant. Then inserting the solution into Equation(7.62) results in(−Bω 2 m + iBωR r + BS)e iωt = APe iωtSolving for B provides the following complex displacement:δ =[iω R r + iAPe iωt(mω − S ω)] (7.63)Differentiating Equation (7.63) yields the complex speed of the fluid in the resonator’sneck,⃗u =R r + iAPe iωt(mω − S ω) (7.64)The impedance found from the relation Z m = f/⃗u, and making use of Equations(7.61) and (7.64) yields(Z m = R r + i mω − S )(7.65)ωwhich means that the mechanical reactance isX m = mω − S ωResonance occurs when the reactance becomes zero, i.e.,√ √√Sω 0 =m = ρ 0 c 2 A 2 /V A= c(7.66)ρ 0 L ′ A L ′ VWe note that the resonance frequency depends on the volume of the cavity, noton the shape. Experimentation with differently shaped resonators having the sameS/L ′ V ratios indicate identical resonant frequencies. This holds true as long asall dimensions of the cavity are appreciably less than a single wavelength and theopening is quite small. There are additional frequencies in Helmholtz resonatorshigher than that given by Equation (7.66), which arise from standing waves in thecavity rather than from the oscillatory motion of the mass of air in the neck. Theseovertone frequencies are not harmonics of the driving fundamental, and the firstovertone may be several times the frequency of the fundamental.The sharpness (or narrowness of spread) of the resonance of a Helmholtz resonatoris define by the quality factor Q,√Q = mω ( )0L′ 3= 2π V(7.67)R rAAssumptions have been made, there are no losses other than those arising fromacoustic radiation and the termination of the neck is flanged.

148 7. Pipes, Waveguides, and ResonatorsIn conducting his experiments of complex musical tones, Helmholtz used aseries of resonators of the type illustrated in Figure 7.6(c), with small nipplesopposite the necks of the resonators. He used a graduated series of resonators, withdiffering volumes and neck sizes to achieve a wide range of resonant frequencies.When an incident sound wave contains a frequency component that correspondsto the resonant frequency of a Helmholtz device, a greatly amplified signal willbe generated within the cavity of the resonator, and it can be aurally detected byconnecting a short, flexible tube from the nipple to the ear. These phenomenaled to the definition of pressure amplification, which is the ratio of the acousticpressure amplitude P c inside the cavity to the external driving pressure amplitudeP. Equation (7.58) provides the pressure amplitude P c . At resonance|δ| =PAω 0 R rThen applying Equations (7.60 ) and (7.66), we find for the resonator with a flangedneck that√( )P cLP = 2π ′ 3V = QAThe gain, therefore, is the same as the value of the quality factor defined inEquation (7.67), and we see the Helmholtz resonator behaving as an amplifierof gain Q at resonance, a fact which explains why a loudspeaker mounted in aclosed chamber can be regarded as a Helmholtz resonator in which the air’s reactanceand the speaker cone mass constitute the effective mass of the system. The airresistance within the box and the cone’s stiffness combine to provide the effectivestiffness, while the effective resistance results from the sum of that attributable tothe radiation of acoustic energy and that arising from the mechanical resistance ofthe speaker cone.Example Problem 2A Helmholtz resonator is a sphere of 8 cm internal diameter. If it is to resonate at450 Hz in air, what is the hole diameter that should be drilled into the sphere?SolutionApplying Equation (7.66) we have√ √Aω 0 = 450 × 2π rad s −1 = cL ′ V = 341 m πa s−1 =21.5a × 4 π(0.04 m)33which results in a = 0.00865 m = 0.865 cm, or 1.73 cm diameter. Here we assumedan unflanged “pipe” with an effective value of L ′ = 1.5a.

Problems for Chapter 7 149ReferencesBeranek, Leo L. 1986. Acoustics. New York: American Institute of Physics, Chapters 5and 6.Beranek, Leo L. 1949. Acoustic Measurements. New York: John Wiley & Sons, pp. 317–321.Fletcher, Neville H. and Rossing, Thomas D. 1991. The Physics of Musical Instruments,1 st ed. New York: Spring-Verlag, pp. 150–155.Kinsler, Lawrence E. and Frey, Austin R. 1962. Fundamentals of Acoustics, 2nd ed. NewYork: John Wiley & Sons, Chapter 8. (Many users consider this edition to be the bestof the four issued to date.)Kinsler, Lawrence E., Frey, Austin R., Coppens, Alan B., and Sanders, James V. 1982.Fundamentals of Acoustics, 3rd ed. New York: John Wiley & Sons, Chapters 9 and 10.Olson, Harry F. 1967. Music, Physics and Engineering, 2nd ed. New York: Dover Publications,Chapter 4.Wood, Alexander. 1960. Acoustics. New York: Dover Publications, pp. 103–110.Problems for Chapter 71. Determine the minimum length of a pipe in which the input mechanical reactanceis equal to the input mechanical resistance for the frequency of 700 Hz.Assume the loading impedance is four times the pipe’s characteristic mechanicalimpedance ρ 0 cA.2. A condenser microphone is constructed by stretching a diaphragm across oneend of a hollow cylinder with a diameter of 2 cm and length of 0.75 cm. Theother end of the cylinder is open. Determine the ratio of the pressure at thediaphragm (considered to be a rigid plane) to the pressure at the open end asa function of frequency from 64 Hz to 2 kHz.3. A pipe is 1.2 m long and has a radius of 6 cm. It is being driven at one endby a piston (negligible mass). The other open end of the pipe terminates in aflange.(a) Find the fundamental resonance of the system.(b) If the piston has a displacement of 1 cm when being driven at 500 Hz,what is the amount of acoustic power being transmitted by plane wavestraveling to the open end of the pipe?(c) What is the acoustic power (in watts) being radiated out at the open endof the pipe?4. Consider a pipe that is 1.2 m long with a radius of 6 cm. It is being driven atone end by a piston of 0.02 kg mass and the same radius as the pipe. The otherend of the pipe terminates in an infinite baffle.(a) For 200 Hz frequency, find the mechanical impedance of the piston of thepipe, inclusive of the loading effect of the air inside the pipe.(b) For the above frequency, determine the amplitude of the force necessaryto drive the piston with a displacement amplitude of 0.50 cm.

150 7. Pipes, Waveguides, and Resonators(c) What will be the amount of acoustic wattage that emanates out throughthe open end of the pipe?5. A 70 dB (re 20 μPa) 1 kHz signal is incident normally to a boundary betweenthe air and another medium having a characteristic impedance of 780 Pa s/m.(a) Determine the effective root-mean-square pressure amplitude of the reflectedwaves.(b) Determine the effective root-mean-square pressure of the transmittedwaves.(c) How far from the boundary is the location where pressure amplitude in thepattern of standing waves equals the pressure amplitude of the incidentwave?6. The speed of sound in water is 1480 m/s. Consider a series of 2960-Hz planewaves in water normally incident to a concrete wall. It can be assumed thatall of the sound energy is completely absorbed by the wall. The pattern ofstanding waves results in a peak pressure amplitude of 30 Pa at the wall and apressure amplitude of 10 Pa at the nearest pressure node at a distance of 50 cmfrom the wall.(a) What is the ratio of the intensity of the reflected wave to that of the incidentwave?(b) Find the specific acoustic impedance of the wall.7. An acoustic signal consisting of 400-Hz plane wave is normally incident to anacoustical tile surface having a complex impedance of 1500 – i3000.(a) Find the standing wave ratio in the resultant pattern of standing waves.(b) Determine the locations of the first four nodes.8. A loudspeaker is fitted to one end of an air-filled pipe of 12 cm radius togenerate plane waves inside the pipe. The far end of the pipe is closed off by arigid cap. A 5-kHz signal is fed into the loudspeaker. The measured standingwave ratio of pressure at one location in the pipe is 7. At another location insidethe pipe 50 cm downstream of the pipe, the measured standing wave ratio is9. Derive an equation that includes these ratios and the distance betweenthem which can be used to establish the absorption constant for the signalbeing propagated inside the pipe. To simplify matters, assume that absorptioncoefficient α ≪ 1.9. Prove analytically (assuming ka ≪ 1) that the frequencies of the resonance andthose near antiresonance of the forced-open tube with damping correspond tothe frequencies of maximum and minimum power dissipation, respectively.10. Determine the lowest normal mode frequency of a fluid-filled cubic cavity (oflength L to a side) that consists of five rigid walls and one pressure releaseside. Plot the pressure distribution associated with that node.11. Given a rigid-walled rectangular room with dimensions 6.50 m × 5.65 m ×3 m, calculate the first 10 eigenfunctions. Assume the sound propagation speedc = 344 m/s.12. A concrete water sluice measures measure 25 m wide and 8 m deep. It iscompletely filled with water. Find the cutoff frequency of the lower mode ofpropagation, assuming the concrete to be totally rigid.

8Acoustic Analogs, Ducts, and Filters8.1 IntroductionIn this chapter we deal with lumped and distributed acoustic elements, applyingelectrical and mechanical analogs to acoustic behavior in order to treat differentduct geometries, acoustic filters, and networks. The reflection and transmissionof sound waves at piping interfaces, where the acoustic impedance changes, areanalogous to the behavior of current waves in a transmission line at locations wherethe electrical impedance undergoes a change.A simple mechanical system can often be converted into analogous electricalsystems and solved in analogous terms. The motion of a fluid is compared to thebehavior of current in an electrical circuit, with a pressure gradient between twopoints playing the role of voltage across the corresponding parts of the circuit.In terms of electricity the impedance is voltage divided by the current, whichcorresponds to the effect of lumped elements of inductance, capacitance, andresistance. In acoustics the acoustic impedance Z of a fluid acting with acousticpressure p on a surface of area A is given byZ = p U(8.1)where U represents the volume velocity of the fluid in the acoustic element ofinterest. U is not really a vector; it is a speed representative of a scalar quantity,unlike velocity which is a magnitude with an indicated direction. The acousticimpedance Z defined by Equation (8.1) is a complex quantity.The specific acoustic impedance z is given byz = p u(8.2)where u is the particle velocity, not the volume velocity. The specific acousticimpedance, which is useful for treating the transmission of acoustic waves fromone medium to another, is a characteristic of the propagation medium and the typeof propagating wave. The acoustic impedance, defined by Equation (8.1) as the151

152 8. Acoustic Analogs, Ducts, and Filtersratio of pressure to volume velocity, is applied to treat acoustic radiation fromvibrating surfaces and the transmission of this radiation across lumped acousticelements and through ducts and horns. These two impedances are related to eachother byZ = z A(8.3)A is the area of a vibrating surface. If the vibrating surface is driven with a velocityu, with a force f acting on the fluid, the radiation impedance Z r is given by theratio of the force to the speed:Z r = f u(8.4)This type of impedance constitutes a part of the mechanical impedance Z m of thevibrating system. Radiation impedance relates to specific impedance at a surfaceas follows:Z r = z r(8.5)AIt is useful for dealing with coupling between acoustic waves and a driving surfaceor a driven load.8.2 Lumped Acoustic ImpedanceIn the use of lumped parameters, the advantage is taken of the assumption thatthe signal wavelength is larger than all principle dimensions, which allows forfurther simplification. Each acoustic parameter may be time varying, but it becomesvirtually independent of distance over the spatial extent of the device. When weconsider lumped or concentrated impedances rather than distributed impedances,we define that impedance in a segment of the acoustic system as the (complex) ratioof the pressure difference p (which propels that segment) to the resultant volumevelocity U. The unit of acoustic impedance is Pa s/m 3 , and it is often referred toas an acoustic ohm.Example Problem 1Consider a Helmholtz resonator described by differential Equation (7.62) whichis repeated below:m d2 δdt + R dδ2 rdr + Sδ = APeiωt (8.6)and recast the system as a lumped acoustic impedance with an electric analog.

Solution8.2 Lumped Acoustic Impedance 153Divide Equation (8.6) by surface area A and utilize the fact that U = dδ A, whichwill result indtZ = p (U = R + i Mω − 1 )(8.7)ωCwhereR = R rA 2 = ρ 0ck 22π , M ≡ m A 2 = ρ 0L ′A ,and the stiffness is given byC ≡ A2S = Vρ 0 c 2S = ρ 0 c 2 A2Vaccording to Equation (7.59). We also assume here a flanged resonator. In electricalterms this constitutes a RLC series circuitry, where the inductance L is the electricalanalog to M. This electrical analog to the Helmholtz resonator is illustrated inFigure 8.1. As a further aid to analytical treatment of lumped parameters, Figure 8.2summarizes the fundamental analogous elements of acoustic, mechanical, andelectrical systems. The inertance M in the acoustic system is represented by a“plug” of fluid that is sufficiently short so that all particles in the fluid can bedepicted as moving in phase under the impetus of sound pressure. The complianceC of the acoustic system is represented by an enclosed volume incorporating aFigure 8.1. The electrical analog to the Helmholtz resonator.

154 8. Acoustic Analogs, Ducts, and FiltersFigure 8.2. Fundamental mechanical, acoustical, and electrical analogs.stiffness. A number of difference situations can contribute to resistance; we canrepresent acoustical resistance in the conventional manner by a narrow slit insidea pipe segment.8.3 Distributed Acoustic ImpedanceWhat if one or more of the principal dimensions of an acoustic system is of thesame order of magnitude as a wavelength? In this case it may not be possible totreat the system as one possessing lumped parameters. The alternative is to analyzeit as having distributed physical constants. Consider a very simple case of planewaves propagating through a pipe in the positive x-direction. The characteristicimpedance of the pipe is given by the ratio of acoustic pressure to particle speed;and the acoustic impedance at any cross section A of the pipe isZ = p U = 1 pA u = ρ 0c(8.8)AThe case of such propagation in a pipe is electrically equivalent to high-frequencycurrents traveling along a transmission line that has an inductance per unit length L sand a capacitance per unit length C s . The corresponding input electrical impedanceis √ L s /C s . In compliance with the electrical analogy, we may consider the fluidin the pipe to have a distributed inertance M s per unit length and a distributed

8.4 Waves in Pipes: Junctions and Branches 155compliance C s per unit length. It also follows that the distribution of mass per unitlength of the pipe can be represented by m s = ρ 0 A. The acoustic inductance perunit length becomes M s = m s /A 2 = ρ 0 /A.We shall now find the mechanical stiffness per unit length. When the fluid is compressedadiabatically by a small linear displacement δ , then p = ρ 0 c 2 (δ / ), andthe force providing this impetus is pA. Hence the stiffness becomes S = pA/δ ;and the stiffness per unit length is S s = ρ 0 c 2 A. The mechanical compliance C mrelates to acoustic compliance C by C = A 2 C m , and on a per unit length basis, it followsthat C s = A/(ρ 0 c 2 ). By analogy the acoustic impedance of the pipe is given byZ =which corresponds to Equation (8.8).√M sC s= ρ 0cA8.4 Waves in Pipes: Junctions and BranchesConsider a sound wave traveling in the positive x-direction, represented byp i = ae i(ωt−kx) , (8.9)is incident upon a point x = 0, where the acoustic impedance changes from ρ 0 c/Ato some complex value Z 0 . At this point a reflected wavep r = be i(ωt+kx) (8.10)will be produced and travel in the negative x-direction. It is our task to findthe power reflection and transmission coefficients for this point. The acousticimpedance at any point in the pipe is given byZ = p i + p r= ρ 0c ae −ikx + be +ikx(8.11)U i + U r A ae −ikx − be +ikxwhich, at x = 0, reduces toZ 0 = ρ 0cAa + ba − b(8.12)Equation (8.12) can be rearranged tobZ 0 − ρ 0ca = AZ 0 + ρ (8.13)0cAThe sound power reflection coefficient R p , which yields the fraction of the incidentpower that is reflected, is given by(R p =b2 R 0 − ρ )0c 2+ X2 ∣ a ∣ =A 0(R 0 + ρ )0c 2(8.14)+ X2A 0

156 8. Acoustic Analogs, Ducts, and FiltersFigure 8.3. Transmission and reflection of a plane wave at the junction x = 0 betweentwo pipes of different cross sections.In Equation (8.14) we have set Z 0 = R 0 + iX 0 . The sound power transmissionT p = 1 − R p represents the portion of the incident sound power that travels pastx = 0. We obtainExample Problem 2T p = (4R 0 ρ 0 c/A) 2(8.15)+ X20R 0 + ρ 0cAApply the above equations to plane waves in a pipe of cross-section area A 1 thatis mated to a pipe of cross-section area A 2 , as shown in Figure 8.3. The secondpipe is of infinite extent or the proper length so that no reflected wave is returnedfrom its far terminus. Assume that the wavelength is considerably larger than thediameter of either pipe so that the region of complicated flow at the junction, wherethe wave adjusts from cross-sectional area to another, is considerably smaller thanthe wavelength itself.SolutionUnder the conditions stipulated above, the acoustic impedance seen by the incidentwave at the junction is Z 0 = ρ 0 c/A 2 . Inserting this value of Z 0 into Equations (8.14)and (8.15), we have( )A1 − A 22R p =and T p = 4A 1 A 2A 1 + A 2 (A 1 + A 2 ) 2 (8.16)Note that if the above pipe is closed at x = 0, A 2 = 0, then Z 0 becomes infinity,which results in a reflection coefficient of unity. On the other hand, if the pipe isopen at x = 0, the impedance at the junction is not zero but corresponds to theimpedance given by Equation (7.12) for an unflanged pipe.

8.4 Waves in Pipes: Junctions and Branches 157Figure 8.4. A three-way junction.In Figure 8.4 we have the more complex case of a pipe that branches into twopipes, each with its own input impedance. Let the junction be located at the origin.The pressures produced by the waves in the three pipes at x = 0 are representedbyp i = ae iωt , p r = be iωtp 1 = Z 1 U 1 e iωt , p 2 = Z 2 U 2 e iωt (8.17)Here a and b denote the amplitudes of the incident and reflected waves, respectively;and Z 1 , Z 2 and U 1 , U 2 the input impedances and volume velocity complexamplitudes in branches 1 and 2. Again, under the assumption of a large wavelengthso that the impact of branching remains confined to a small region at the juncture,we apply the condition of continuity of pressure so thatp i + p r = p 1 = p 2 (8.18)Likewise, the continuity of the volume velocity demands thatU i + U r = U 1 + U 2 (8.19)which is analogous to Kirchhoff’s law of electric currents. Dividing Equation(8.19) by Equation (8.18) yields the impedance relationship1= 1 + 1 (8.20)Z 0 Z 1 Z 2The reciprocal of an impedance, Z –1 , is called the admittance. Equation (8.20)indicates that the combined admittance 1/Z 0 equals the sum of the admittances ofthe two branches 1 and 2.Example Problem 3An infinitely long pipe of cross-sectional area A has a branch at x = 0 that presentsa given impedance of Z g . Find the appropriate power reflection and transmissioncoefficients.

158 8. Acoustic Analogs, Ducts, and FiltersSolutionIn this case we consider this pipe to have two branches, one corresponding tothe given impedance and the other to an infinite pipe which does not provide anyreflections but does present an impedance of ρ 0 c/A. Applying Equation (8.20) weobtainba =ρ 0 c2Aρ 0 c2A + Z g(8.21)The ratio of pressure amplitude a t of the wave transmitted beyond x = 0 along theinfinite pipe to the pressure amplitude of the incident wave is found by insertingEquation (8.21) into Equation (8.18), yieldinga ta =Z gZ g + ρ 0c2A(8.22)We resolve the acoustic impedance Z g of the branch into real and imaginary components,i.e., Z g = R g + iX g . The reflection and transmission coefficients become( ρ0 c) 2R p =b22A∣ a ∣ ( ρ0 c) 2(8.23)2A + R g + X2 g T p = ∣ a t∣ 2 Rg 2 =+ X g2(a ρ0 c) 2(8.24)2A + R g + X2 gThe portion T pg of the power that is transmitted into the branch is T pg = 1 − R p −T p ,orR g ρ 0 c/AT pg = ( ρ0 c) 2(8.25)2A + R g + X2 gIf R g has a positive finite value, some acoustic energy is dissipated at the branchand some is transmitted beyond the junction, no matter what the value of X g is.When either R g or X g greatly exceeds ρ 0 c/A, all of the incident power is transmittedpast the branch. At the other extreme, when R g X g =∞, which correspondsto no branch, the power transmission is unity.8.5 Acoustic FiltersAdvantage can be taken of the fact that a side branch can attenuate sound energytransmitted in a pipe. The input impedance of the side branch determines whetherthe system can behave as a low-pass, high-pass, or band-pass filter. We shall nowconsider each of these filters.

8.5 Acoustic Filters 159Figure 8.5. (a) A simple low-pass acoustic filter, (b) its analogous electrical filter, and (c)the corresponding power–transmission curves for the acoustic filter of (a).1. Low-Pass Filters. Figure 8.5 illustrates the construction of a simple low-passfilter, essentially consisting of an enlarged segment of a pipe of cross-sectionalarea A 1 and length L in a pipe of cross section A. At sufficiently low frequencies(kL ≪ 1), this filter may be viewed as a side branch with acoustic complianceC = V/(ρ 0 c 2 ), where V = A 1 L represents the volume of the expansion chamber.The acoustic impedance of this type of branch is pure reactance, henceR g = 0, X g =− 1Cω = ρ 0c 2A 1 LωInserting the above into the expression for the transmission coefficient, Equation(8.23) yields1T p = ( ) 2 A12A kL + 1(8.26)Equation (8.26), plotted in Figure 8.5, indicates that as the frequency approacheszero, the transmission coefficient approaches unity (100% transmission), but asthe frequency becomes higher, this coefficient tends toward zero. Curve 1 is foran expansion chamber 5 cm in length and a cross-sectional ratio of A 1 /A = 4.However, Equation (8.26) does not apply to kL > 1.In order to treat the case of the above-mentioned acoustic filter for kL > 1, theincident, reflected and transmitted waves in the three regions of the pipe must berelated to one another by the fact that the continuity of pressure and volume velocity

160 8. Acoustic Analogs, Ducts, and Filtersmust occur at the two junctions of the pipe. This results in a power-transmissioncoefficient expressed as follows:4T p =(A14 cos 2 kL +A + A ) 2sin 2 kLA 1(8.27)In Figure 8.5, Curve 2 constitutes the plot of Equation (8.27) for the same filtersystem used to obtain Curve 1. At low values of frequencies, i.e., kL ≪ 1, thetwo curves basically coincide. Equation (8.27), which is physically more valid,exhibits a minimum transmission coefficient of( ) 2A1 A 2T p (minimum) =S1 2 + (kL = π/2) (8.28)S2for the case where the length of the filter segment equals a quarter wavelength.Beyond this saddle point, T p gradually rises with increasing frequency until itreaches 1.0 (100%) at kL = π. At even higher frequencies the transmission coefficientvacillates through a series of maxima and minima until ka (a is the radiusof the through pipe) becomes somewhat larger than unity. From this point on,the transmission coefficient remains at unity. This trait of the transmission coefficientreaching a plateau of unity is also shared by high-pass and band-passfilters.Equation (8.28) may also be used to treat the constriction-type low-pass filterillustrated in Figure 8.6, since it does not matter if A 1 is larger or smaller thanA. The decrease in the area can be viewed as introducing an inertance in serieswith the pipe, but the validity of this analog also extends over a limited range offrequencies, as with the case of the expanded-area low-pass filter of Figure 8.5.In the real world of filter design (of mufflers, sound-absorption plenum chambersfor ventilating systems, etc.) the filter cross section cannot be radically differentin value from the cross-section area of the pipe. As is demonstrated in the curvesof Figure 8.5, a limited range of frequencies exists for the practical operation ofthe filters.Figure 8.6. A pipe with a constriction and its electrical analog.

8.5 Acoustic Filters 1612. High-Pass Filters. A high-pass filter can be constructed by attaching a shortlength of pipe as a branch to a main pipe, in effect creating an orificed Helmholtzresonator. Both the radius a and length L of this appendage are small comparedto a wavelength. Equations (7.60b) and (7.57) for the unflanged resonator applyto the branch impedance of this orifice given byZ g = kρ 0c 24π + i ρ 0ωL ′(8.29)πa 2where L ′ = L + 1.5a. The first term in the right-hand side of Equation (8.29)stems from the radiation of sound through the orifice into the external medium,and the second is attributable to the inertance of the gas in the orifice. The ratioof the branch acoustic resistance to its acoustic reactance is R g / X g = 1/4ka 2 /L ′ .Because it has been assumed that ka ≪ 1, we can neglect the acoustic resistancein comparison with the acoustic reactance in the use of Equation (8.24) to find thepower-transmission coefficient T p , with the result:1T p = ( πa2) 2(8.30)+ 12AkL ′We observe that this transmission coefficient is virtually zero for low frequenciesand it rises to nearly unity at higher frequencies, as shown in Figure 8.7. Thehalfway point at which the transmission coefficient is 50% is reached whenk = πa22AL ′Figure 8.7. Attenuation produced by an orifice-type branch, resulting in high-pass transmission.

162 8. Acoustic Analogs, Ducts, and FiltersThe presence of one orifice turns a pipe into a high-pass filter. If the radiusof the orifice is increased, the attenuation of the low-frequency components alsoincreases. If a pipe contains several orifices separated by only a fraction of a wavelength,these orifices can be treated as a group acting with their equivalent parallelimpedance. But if the distances between the orifices constitute an appreciable portionof the wavelength, the system becomes analogous to an electrical network offilters or to a transmission line which has a number of impedances shunted acrossit, spaced apart at wide intervals. Waves reflected from these different orificesare then out of phase with respect to each other, and Equation (8.30) no longerremains valid. Electrical filter theory must then be utilized to compute the transmissioncoefficient. As a rule, a number of orifices strategically placed apart canattenuate at low frequencies more effectively than a single orifice of equal totalarea.The sound power transmission coefficient T pg into a single orifice is approximatedbyT pg=2k 2 A[ (2AkL′) ] 2(8.31)π+ 1πa 2The filtering action of an orifice is principally that of the reflection of energy backtoward the source, not so much the loss of acoustic energy out of the pipe throughthe orifice into the ambient medium.A common example of the application of orifices is the control of the behaviorof a wind instrument such as a flute or a saxophone. When such an instrument isplayed in its fundamental register, all or nearly all of the orifices some distancebeyond the mouthpiece are kept open by the player. The diameters of these orificesnearly equal the bore of the tube, essentially shortening the effective length of theinstrument. The acoustic energy reflected from the first open orifice generates apattern of standing waves between the first open orifice and the mouthpiece. Theflute behaves like an open pipe, with the wavelength approximately equal to twicethe distance between the first orifice and the opening of the mouthpiece. A clarinetor a saxophone contains a vibrating reed at the mouthpiece, which approximatesthe conditions of the closed end of a tube. In this case the wavelength will equalnearly four times the distance from the reed to the first open orifice.Both the reed-type (clarinet, saxophone, coronet, etc.) and tubular instruments(flute, recorder, piccolo, etc.) contain a number of harmonics, those of the reedinstruments being primarily odd harmonics (characteristic of closed pipes). Whenhigher notes are played on either type of instruments, the fingering of these holesbecome more complicated, with some orifices beyond the first orifice closed andsome others opened. The fingering of these orifices control the standing wavespatterns which correspond to specific notes.3. Band-Pass Filters. A side branch in the form of a long pipe rigidly capped atits far end or a fully enclosed Helmholtz resonator (shown in Figure 8.8) contains

8.5 Acoustic Filters 163Figure 8.8. The effect of a Helmholtz resonator branch, resulting in band-pass transmission.both inertance and compliance, so it will behave as a band-pass filter. Apart fromalmost negligible viscosity losses, no net dissipation of acoustic energy occursfrom the pipe into the resonator. All energy absorbed by the resonator during somephase of the acoustic cycle is returned to pipe during other phases of the cycle soR g = 0. Denoting the opening area by A g = πa 2 , the neck length by L and thevolume of the resonator by V , the branch reactance X g is expressed asX g = ρ 0( ωL′A g− c2ωVwhich is then inserted into Equation (8.24) to yield the following transmissioncoefficient:⎡⎤−1T p = ⎢⎣ 1 + c 2( ωL′) ⎥(8.32)4A 2 − c2 ⎦A g ωVThe resonant frequency occurs when the transmission coefficient becomes zero,i.e.,√Agω 0 = cL ′ Vwhich corresponds to the resonant frequency of a Helmholtz resonator. When thisfrequency occurs, large volume velocities prevail in the neck of the resonator, and)

164 8. Acoustic Analogs, Ducts, and FiltersFigure 8.9. A ladder-type network used as a filter.all acoustic energy that transmits into the resonator returns to the main pipe in such amanner as to be reflected back from the source. The plot of the power-transmissioncoefficient in Figure 8.8 is fairly typical for a bandpass resonator.Equation (8.32) serves well for a resonator that has a relatively large neck radius.Narrower and longer constrictions will cause the transmission coefficient to deviatefrom the prediction of Equation (8.32), unless consideration is taken of the viscousdissipation that manifest itself more with such geometries.4. Filter Networks. The design procedure for acoustic networks, which incorporatesresonators, orifices, and divergence and convergence of pipe areas, is renderedeasier by analogy with the deployment of electronic filters. The sharpness of cutoffof an electrical filter system, for example, can be enhanced by using the laddertypenetwork of Figure 8.9. This network is constructed by using the reactancesof one type of impedance Z 1 in series with the line and the reactance of anothertype of impedance Z 2 shunted across the line. The standard theory of wave filtersstates that a nondissipative repeating structure such as that illustrated in Figure 8.9causes a marked attenuation of all frequencies except those for which the ratioZ 1 /Z 2 meets the condition0 > Z 1 /Z 2 > −4 (8.33)Several examples of acoustic ladder-type filters are displayed with the electricalanalogs in Figure 8.10. The condition of Equation (8.33) provides the followingcutoff frequencies:1f =4π √ MCfor the high-pass filter as shown in Figure 8.10(a), and1f =π √ MCfor the low-pass filter as shown in Figure 8.10(b). The behavior projected by thefilters as shown in Figure 8.10 applies only to wavelengths that are large comparedto the dimensions of the filter. At higher frequencies, deviation of the behaviorpredicted by electrical analogs becomes more prominent, because the filter begins

8.6 Ducted Source Systems—Acoustic Modeling 165Figure 8.10. Examples of ladder-type acoustic filters.to manifest the properties of distributed parameters rather than those of lumpedparameters.8.6 Ducted Source Systems—Acoustic ModelingA ducted source system is one in which the source is the active component andthe load is the path, consisting of elements such as mufflers, ducts, and end terminators.Many mechanical systems such as engines and mufflers and air-movingdevices (flow ducts, fluid pumping) are very common examples of ducted sources.

166 8. Acoustic Analogs, Ducts, and FiltersSOURCELOADZ SP S P L -Z LV S Z SP L -Z L(a)(b)Figure 8.11. Electrical analog of a ducted-source-load system: (a) pressure source; and(b) volume velocity source.The source-load interactions generally determine the acoustic performance of thesystem. This section provides a brief overview of characterization of acousticperformance of ducted sources. Acoustic performance of a system incorporatinga muffler as a path element is usually described in terms of insertion loss andradiated sound pressure.Figure 8.11 illustrates a basic duct system. Let P S and V S denote the sourcepressure and volume velocity, respectively. P L and V L represent the pressureandvolume-velocity response of the source-load system, respectively. Z S and Z Lare the complex source and load impedances, respectively. The equations for thecourse-load system in terms of pressure and velocity complex source representationsare given byP L =P SZ LZ S + Z LandV L =V SZ SZ S + Z LThe source at one end of a duct system constitutes a boundary condition. Thesource is generally more difficult to characterize than the termination because ofthe dynamic nature of the source (Prasad, 1991).Prior to the development of direct and indirect methods for measuring sourceimpedance, characteristic or infinite impedances were assumed for the source,but these assumptions do not normally yield valid values of impedances. Bothdirect and indirect methods for the measurement of source impedance are basedon frequency-domain analysis. Analytical modeling efforts have been carried our

8.6 Ducted Source Systems—Acoustic Modeling 167Z sABCDZ r(a)Z sZ Lp r V r(b)Figure 8.12. Duct system models: (a) source-path-termination model and (b) source-loadmodel.principally in the time domain based on the method of characteristics. A number ofstudies have been conducted on the basis of modeling of the geometry of sources(Prasad, 1991).A duct system can be modeled as a source-path-termination model with asource load, as shown in Figure 8.12. These models are interrelated when thepath-termination is treated as a load. This type of model is commonly used, say,for engine-exhaust-pipe-tailpipe-radiation system. Referring to Figure 8.13, thethree most commonly used descriptors, namely, insertion loss (IL), transmissionloss (TL), and noise reduction (NR), are given by the following three equations:∣ ∣ ∣∣∣ AZ r + B + CZ S Z r + DZ s ∣∣∣IL = 20 log 10A ′ Z r + B ′ + C ′ Z S Z r + D ′ dB (8.34)Z S∣ ( ∣∣∣ 1TL = 20 log 10 A + BS2 ρc + Cρc )∣ ∣∣∣S + D dB (8.35)NR = 20 log 10∣ ∣∣∣(A + B Z r)∣ ∣∣∣dB (8.36)Here Z S and Z r are the source and radiation impedances, respectively, and A, B,C, and D are the four-pole parameters of the muffler including its upstream anddownstream ducts. The insertion loss IL is the most useful of the three descriptorsgiven in Equations (8.34)–(8.36). As the terminology implies, IL describes thereduction in the acoustic output when a muffler is inserted into an otherwise unattenuatedsystem. Radiated sound pressure level L p is also quite useful as it givesthe system output which can be used to determine IL. The other two descriptors,namely, TL and NR, do not require knowledge of the acoustic characteristics ofthe source. By their very definition in Equations (8.35) and (8.36), respectively,both TL and NR are independent of the influence of the source-load interaction.

168 8. Acoustic Analogs, Ducts, and FiltersL p1Insertion loss IL = L - Lp2 p1L p2L p2Area S ZIntensityArea S tI iI rI tTransmission Loss TL = 10 log10[( S I)/( S )]i it I tL p1Noise reduction NR = L- Lp2 p1Figure 8.13. Muffler system performance descriptors.While IL is fairly easy to measure, it is extremely difficult to predict because ofits dependence on the source impedance Z S (Davis, 1957; Munjai, 1987; Prasadand Crocker, 1998).ReferencesBeranek, Leo J. 1886. Acoustics. New York: The American Institute of Physics, Chapter 3.Davis, Jr., D. D. 1957. Acoustic filters and mufflers. In: Harris, Cyril M., ed. Handbook ofNoise Control. New York: McGraw-Hill, Chapter 21.

Problems for Chapter 8 169Kinsler, Lawrence E. Frey, Austin R., Coppens, Alan B., and Saunders, James V. 1982.Fundamentals of Acoustics, 3rd ed. New York: John Wiley & Sons, Chapters 9and 10.Munjai, M. L. 1987. Acoustics of Ducts and Mufflers. New York: John Wiley & Sons.Prasad, M. G. 1991. Characterization of acoustical sources in duct systems—progress andfuture trends. Proceedings of Noise-Con, Tarrytown, New York. Ames, Iowa: Instituteof Noise Control Engineering.Prasad, M. G. and Crocker, M. J. 1998. Acoustic modeling (ducted source systems).In: Crocker, Malcolm J., ed. Handbook of Acoustics. New York: John Wiley & Sons,Chapter 14.Problems for Chapter 8In the following problems, consider the fluid medium to be air at the standardconditions of 1 atmosphere and 20 ◦ C, unless otherwise stated.1. It is desired to make a Helmholtz resonator out of a sphere with a diameter 12cm for 350 Hz.(a) What should be the diameter of the hole in the sphere?(b) What should be the pressure amplitude of the incident acoustic plane waveat 350 Hz if it is to produce an excess internal pressure of 25 μbar?(c) If the hole is doubled in area, what will be the resonant frequency?(d) What will be the resonance frequency if two independent separate holes,each of the diameter found in part (a) are drilled in the sphere?2. Consider a loudspeaker system that is rigid-walled and back enclosed. Itsinside dimensions are 40 cm × 55 cm × 44 cm. The front panel of cabinetis 4 cm thick and it has a 22-cm diameter hole cut out to accommodate aloudspeaker.(a) What is the fundamental frequency of this cabinet considered as aHelmholtz resonator?(b) A direct-radiating loudspeaker having a cone of 22 cm diameter and 0.008kg mass and a suspension system of 1100 N/m stiffness is mounted inthe cabinet. Find the resonance frequency of the cone. It may be assumedthat the effective mass of the system is that of the cone and that of the airmoving in the opening of the cabinet. The effective stiffness is the sum ofthe stiffness of the cone and that of the cabinet.(c) What would be the resonant frequency of the loudspeaker if it were notmounted in the cabinet and it has no air loading?(d) Find the acoustic power emitted if the cone is driven with an amplitude of2.5 mm at the frequency established in part (b).(e) Under the conditions same as that of (d) what is the amplitude of the forceacting on one of the 55 cm × 44 cm panels?3. A rectangular room has internal dimensions of 3.0 m × 5.0 m × 2.6 m andwalls of 12 cm thickness. A door that opens into the room has dimensions of2.2 m × 0.8 m. Assume that the inertance of the door opening is equivalent ofa circular opening of equal area.

170 8. Acoustic Analogs, Ducts, and Filters(a) Find the resonance frequency of the room considered as a Helmholtzresonator.(b) Find the acoustic compliance of the room.(c) Find the inertance of the door opening.(d) What is the acoustic impedance presented to the sound source at 30 Hz insidethe room? Consider only the compliance of the room and the inertanceof the door opening.4. Pipe 1 having cross-section A 1 connects to pipe 2 with cross section A 2 .(a) Obtain an expression for the ratio of intensity of the waves transmitted intothe second pipe to that of the incident waves.(b) Under what conditions will the transmitted energy exceed the incidentenergy?(c) Develop a general expression for the standing wave ratio (SWR) producedin pipe 1 in terms of relative areas A 1 and A 2 .5. Consider two pipes that are connected to each other but separated by a thinrubber diaphragm. Pipe 1 is filled with a fluid with characteristic impedanceρ 1 c 1 and pipe 2 contains a fluid with characteristic impedance ρ 2 c 2 . A planewave travels in pipe 1 in the positive x direction toward pipe 2.(a) Obtain an expression for the power-transmission ratio from pipe 1 intopipe 2.(b) Under what conditions does 100% power transmission occur?6. An infinitely long pipe of cross-sectional area A has a side branch, also infinitelylong, of cross-sectional area A b . The main pipe is transmitting planewaves with frequencies such that their wavelengths are much larger than thediameter of either pipe.(a) Develop an equation for the transmission coefficient in the main pipe.(b) Do the same thing for the branch pipe.(c) Let the area of the main pipe be twice as much as that of the branch pipe.Obtain numerical values for the transmission coefficient into each pipe. Dothe sum of the two coefficients equal unity? If not, where did the remainingpower go?7. A ventilating duct in a basement has a square cross-sectional area, 35 cm tothe side. In order to quiet the duct in part, a Helmholtz resonator-type filter isconstructed about the duct by drilling a hole of 9 cm radius in one wall of theduct, leading into a surrounding closed chamber of volume V .(a) What is the volume V necessary to most effectively filter sounds at afrequency of 30 Hz?(b) What will be the sound power transmission coefficient of the filter at 45Hz and 60 Hz?8. Demonstrate that the radius r of the hole drilled into a pipe of radius r 0 as toresult in a 50% sound power transmission coefficient at a frequency f is givenbyr = 64 r0 2 f3 c

Problems for Chapter 8 1719. A 400-Hz plane wave of 0.2 W power propagates down an infinitely longpipe of 5 cm diameter. Find the power reflected, the power transmitted alongthe pipe, and the power transmitted out through a simple orifice of 1.2 cmdiameter.10. A 5-cm pipe carries water. It is planned to filter out plane waves traveling inthe water by using sections of pipes 10 cm in diameter to serve as expansionchambertype of filters.(a) Find the minimum length of the filter section that will most effectivelyfilter out a sound of 1 kHz.(b) If it is desired to have a filtering action lessen the level of intensity by35 dB, how many sections must be used? Disregard the effects of anyinteraction between the individual filter sections.11. Find the transmission coefficient of a low-pass filter that consists of a circularpipe of 3.0 cm radius mated with a circular expansion chamber that is 12.0 cmlong. Assume that wavelength λ is 30 cm (recall that k = 2π/λ).

9Sound-Measuring Instrumentation9.1 IntroductionAcoustic measurements constitute an essential step in order to establish the status ofan acoustic environment and to develop a systematic approach toward modifyingthe environment and to set up criteria for improvements. Other more recentlydeveloped methodologies of acoustical measurements entail the studies of materialproperties and medical diagnoses. A wide variety of instrumentation exists, rangingfrom a simple sound pressure level (SPL) meter to real-time spectral analyzersinterfaced with sophisticated computer systems. Instruments may be portable forfield use; and recorded field data can be later evaluated in more elaborate systems.New instruments are continually being developed, and with new advances in digitaltechnology arriving on the scene on a daily basis, it is not inconceivable thatmore versatile and “user-friendly” devices will become available at even lowerprices.This chapter deals principally with instruments intended for the audio rangeof frequencies. Much of the salient aspects of instrumentation in this chapter,such as the principal performance requirements and methodologies for evaluatingdata, also apply to the specialized underwater instruments and ultrasonic sensinginstruments which are described in Chapters 15 and 16, respectively.9.2 Principal Characteristics of Acoustical InstrumentsThe most important performance characteristics of acoustical instruments are thefrequency response, dynamic range, crest factor capability, and response time. Itis also desirable that a measuring device or system has a negligible (or at leastpredictable) effect or influence on the variable being measured.Frequency response refers to the range of frequencies that an instrument is capableof correctly measuring the relative amplitudes of the subject variable withinacceptable limits of accuracy. Measurement accuracy depends on the instrumentationtype, the quality of design and manufacture, and calibration. A typical limit forthe flatness of the response for microphones may be ±2 dB or better; in contrast,173

174 9. Sound-Measuring Instrumentationthe response curve of a high-quality loudspeaker may deviate ±5 dB over its ratedfrequency range.Dynamic range defines the range of signal amplitudes that an instrument iscapable of handling in the process of responding and measuring accurately. Asound level meter (SLM), for example, which can measure a minimum of 10 dBand a maximum of 150 dB, covers a dynamic range of 140 dB.Crest factor capability denotes an instrument’s capacity to measure and distinguishinstantaneous peaks. Crest factor itself is the ratio of the instantaneous peaksound pressure to the root-mean-square sound pressure.Response time refers to the rapidity with which a measuring instrument respondsto abrupt changes in signals. An oscilloscope display of a square-wave responsewill result in a trapezoidal display if the relayed signals came from a loudspeakerthat requires a longer reaction time to respond to a square-wave pulse.9.3 MicrophonesMicrophones serve as transducers by receiving and sensing pressure fluctuationsand converting them into electrical signals that are relayed to other electronic components.The quality of a microphone determines the accuracy of a measurementsystem. A top-caliber measuring (or a sound reproduction) system can be underminedby the use of a microphone that is of a lesser caliber. Four principal typesof microphones are used in measurement procedures, namely, dynamic, ceramic,electret, and condenser microphones.Dynamic microphones produce an electric signal through the motion of a coilconnected to a diaphragm in a magnetic field. They are in effect loudspeakersworking in reverse, accepting an acoustic signal, and converting it into an electricpulsation rather than the other way around. Because of their low impedance,they can be used in applications entailing the use of long cables connected toauxiliary instrumentation. But they cannot be used in the vicinity of devices thatemit magnetic fields (e.g., transformers and motors). Also they generally havelonger response times, more limited frequency response, but can be constructed towithstand rough handling and high humidity.Ceramic microphones consist of a piezoelectric (ceramic) element attached tothe rear of a diaphragm. Sound pressure causes the diaphragm to vibrate, exertinga varying force on the ceramic element. The piezoelectric crystal generates anelectric signal from the oscillating strains imparted by the diaphragm. These microphonesare rugged, relatively inexpensive and reliable, have high capacitanceand good dynamic range, and do not require a polarizing voltage that electret andcondenser microphones need. But the high-frequency response may be lacking,and the operating temperature range may be limited.The electret (or electret-condenser) microphone, illustrated in Figure 9.1, consistsof a self-polarized metal-coated plastic diaphragm. Sound pressure causes thediaphragm to move relative to a back plate, varying the capacitance, and producinga signal. While relatively impervious to high humidity, this type of microphone

9.3 Microphones 175Figure 9.1. A cut-away view of an electret microphone showing the principal components.A thin electret polymer foil is suspended over a perforated backplate.does not equal the frequency response of a condenser microphone and are notlikely to withstand temperature extremes very well. An alternate form of the electretmicrophone is created by depositing the electret film onto the stationary plateand using the thin metal foil as a moving diaphragm.Condenser microphones are capacitance-varying devices; each one consisting oftwo electrically charged plates with an air gap between them. The thin diaphragmserves as the plate that deflects under the influence of changes in sound pressure,causing the gap to vary. The resulting change in the capacitance is converted intoan electric signal. Figure 9.2 illustrates the principal components of a condensermicrophone. A capillary tube behind the plates provides the air bleed to provide airpressure equalization with the ambient. These devices have relatively low capacitance,require a polarizing voltage supplied by a preamplifier which also providesFigure 9.2. Principal elements of a condenser-microphone cartridge. (Courtesy of Brüel& Kjær.)

176 9. Sound-Measuring Instrumentationthe appropriate impedance for connecting the microphone to a measuring system.Because of their long-term stability, superb high-frequency response, insensitivityto vibration excitation, condenser microphones are generally preferred for precisionmeasurements. They function well in extreme temperature and pressureenvironments, but they are affected adversely by high humidity (which causeselectrical leakage) and their diaphragms are fragile.9.4 SensitivityIn addition to frequency response, sensitivity constitutes one of the principal characteristicsof a transducer. In general there is a trade-off between sensitivity andfrequency response. Small microphones tend to have lower sensitivity but operate atboth low and higher frequencies, whereas large microphones possess high sensitivitybut are useful mainly at lower frequencies. Microphone diameters are typically1, 2, and 4 cm ( 1 / 4 , 1 / 2 , and 1 in.) in diameter. The frequency response of a 4-cmcondenser microphone is virtually flat to 20 kHz, while that of a 1-cm microphoneis fairly flat to approximately 100 kHz. We define microphone sensitivity S byS =electrical outputmechanical inputThe microphone sensitivity level L s , also called simply “sensitivity,” is defined as( / ) 2Eout pL s = 10 logdBV/μbar = 20 log ( / )E out p dBV/μbar (9.1)E rewhereE out = output voltage into the instrumentE re = reference voltage (1 V for an incident sound pressure of 1.0 μbar)p = rms pressure on the microphoneOne microbar (μbar) is equal to 0.1 Pa. Equation (9.1) can be rearranged to obtainthe output voltage:E out = p 10 L s/20(9.2)We recall the definition of Equation (3.6) for sound pressure level (SPL), L p andrecast Equation (9.2) asE out = 0.0002 × ( 10 L p/20 )( 10 L s/20 ) = 0.0002 × 10 (L p+L s )/20 V (9.3)Example Problem 1A microphone has sensitivity rating of L s =−50 dB V/μbar. Find the outputvoltage for a sound pressure level of 85 dB.

SolutionEquation (9.3) is applied to yield9.5 Selection and Positioning of Microphones 177E out = 2 × 10 −4 × 10 (85−50)/20 V = 0.0112 V = 11.2 mVMicrophone sensitivities typically range between 0.5 μV/μbar (−125 dB V/μbar)and 3 mV/μbar (−50 dB V/μbar). A microphone used for general sound levelpressure measurements in the frequency range of 10 Hz–20 kHz might have asensitivity of 3 mV/μbar (−50 dB V/μbar). But if it were desired to measurein the frequency range extending beyond 100 kHz, a special microphone designedfor this purpose would have a sensitivity of only 0.5 μV/μbar (−125 dBV/μbar).9.5 Selection and Positioning of MicrophonesFigure 9.3 illustrates a flow chart for the selection of microphones and theirorientation during their use on the basis of the nature of the sound field. TwoFigure 9.3. Flow chart for the selection of microphones and their orientations. (Courtesyof Brüel & Kjær.)

178 9. Sound-Measuring Instrumentationstandards apply here, one by the American National Standards Institute (ANSI)which calls for microphones with random-incidence response and the other bythe International Electrotechnical Commission (IEC) which specifies free-fieldmicrophones.The free field occurs as a region that is not subjected to reflected waves, as is thecase in an open field or in an anechoic chamber. The presence of a microphone inthe sound field disturbs the field. A microphone designed to compensate for thisdisturbance is called a free-field microphone. In order to obtain maximum accuracyin measurements, the free-field microphone should be pointed toward the noisesource [0 ◦ incidence, as shown in Figure 9.4(a)]. Microphone sensitivities are alsostated in terms of mV/Pa.The disturbance of the sound field by the presence of the microphone dependson the sound frequency, the direction of propagation, and the size and shape of themicrophone. At higher frequencies, where the wavelength of the sound is smallcompared with the principal dimensions of the microphone, reflections from themicrophone cause the pressure acting on the microphone diaphragm to differ fromthe actual free-field sound pressure that is supposed to be measured. Becausea wavelength of 1 inch corresponds to 13,540 Hz, a 4-cm microphone will notprovide accurate free-field measurements of noise in the frequency range in theneighborhood of 13 kHz and above. Even at 6 kHz, the error for a 4-cm microphonecan exceed 2 dB.The converse of the free field is the diffuse field which occurs as the resultof multiple reflections. A random-incidence microphone is utilized in measuringsound in diffuse fields; it is omnidirectional in that it responds uniformly to soundarriving from all angles simultaneously [cf. Figure 9.4(d)].Pressure microphones are designed to yield a uniform frequency response tothe sound field including the disturbance produced by the microphone’s presence.This type of microphone may also be used in diffuse fields. Using a free-fieldmicrophone in a diffuse field will result in lessened accuracy unless special circuitryin the measurement system provides compensating corrections. As shown inFigure 9.4(b), when a random-incidence microphone is used to measure sound in afree field, the unit should be placed at an incidence angle of 70 ◦ −80 ◦ to the source.A pressure microphone should be positioned at an incidence angle of 90 ◦ (oftenreferred to as the grazing incidence) in a free field [Figure 9.4(c)]. Microphoneplacement becomes more critical as the increasing sound frequencies approach theupper limits of accuracy.For a windy environment, special precautions should be taken to obtain theproper data. The wind rushing past a microphone produces turbulence, generatingpressure fluctuations resulting in low-frequency noise. In the use of A-weightedmeasurements, no precautions are necessary for winds below 8 km/h (5 mph),because the A-weighting network attenuates greatly at low frequencies. But forC-scale or linear sound level measurements, a windscreen should be employed forany sort of wind. The device should also be used for wind speeds above 5 mph inA-weighted measurements. Windscreens are typically open-celled polyurethane

9.5 Selection and Positioning of Microphones 179Figure 9.4. Microphone orientations with respect to the sound source.

180 9. Sound-Measuring InstrumentationFigure 9.5. Sound pressure level corrections for conditions that deviate from the standardatmospheric pressure and temperature of 760 mmHg and 20 ◦ C.foam spheres that are placed over the microphones. However, these may not betoo effective if the wind speed exceeds 30 km/h (20 mph).Changes in atmospheric conditions, namely, temperature and pressure, maynecessitate corrections. Figure 9.5 illustrates a chart for determining correctionsdue to deviations from 1 standard atmosphere (762 mmHg or 30 in Hg) and 20 ◦ C,based on the following expression:(√ ( ) ) (√ ( ) ) F + 460 30 C + 273.3 760C T,P = 10 log= 10 log528 B293.3 B ′ (9.4)whereC T,P = correction factor to be added to the sound pressure level, dBF, C = temperature in ◦ For ◦ C, respectivelyB, B ′ = barometric in inches Hg or mmHg, respectivelyExample Problem 2For a sound pressure level, a reading of 92 dB is taken at the standard conditionsof 1 atmosphere and 68 ◦ F, predict the readings at 10 ◦ F and at 110 ◦ F for the sameatmospheric pressure.

9.6 Vector Sound Intensity Probes 181SolutionApplying Equation (9.4)C T,P = 10 logC T,P = 10 log(√10 + 460528(√110 + 460528( ) ) 30=−0.25 dB (for 10 ◦ F)30( ) ) 30=+0.17 dB (for 110 ◦ F)30At the lower temperature the reading would be (92 − 0.25) = 91.8 dB and at thehigher temperature, (92 + 0.17) = 92.2 dB. Since the deviations are within theinstrument error, the application of such corrections is not very meaningful.The presence of reflecting surfaces affect measurements with the use of microphones.For example, the presence of a person near the microphone will disturbthe sound field, and it would be advisable to place the microphone on a tripod andmonitor the instrumentation from a distance. Also when a long cable is used toconnect the microphone to other instruments, care must be taken to avoid noisegenerated by the motion of different segments of the cable with respect to eachother. The cable should be constrained from moving and isolated from vibrationto the greatest degree possible. The cable should also be well shielded from strayelectromagnetic fields; but it is preferable that the tendency to noise generation becut down by incorporating the preamplifier and the microphone into a single unit,thereby yielding a considerably greater signal-to-noise ratio (S/N ratio). As a rule,stronger signals are more impervious to outside influences than would be the caseif the signals were not preamplified.9.6 Vector Sound Intensity ProbesA sound intensity probe that is used to measure vector sound intensity is illustratedin Figure 9.6(a), and a block diagram in Figure 9.6(b) shows the components of thisvector sound-measuring system. The device contains two microphones mountedface to face. Other modes of mounting are possible, including side by side andback to back. The probe measures sound pressure levels at two different pointssimultaneously. With the microphone spacing constituting a given factor, the soundpressure gradient can be determined and the particle velocity in a given directioncan be calculated. The intensity vector component in that direction may then beestablished. The microphone spacing should be considerably smaller than thewavelength of the sound being measured in order to yield valid results from twopointmeasurements. For example, in order to sustain an accuracy of ±1 dB, theupper frequency limit for 1-cm microphones set apart 12 mm from each otheris approximately 5 kHz. If the microphones are spaced apart by only 6 mm, thecorresponding upper frequency limit is 10 kHz.

182 9. Sound-Measuring InstrumentationFigure 9.6. Sound intensity probe. (a) View of a sound intensity probe and its components.(b) Block diagram of the vector intensity measurement device. (Courtesy of Brüel &Kjær.)9.7 The Sound Level MeterThe sound level meter (SLM), a most valuable means of assaying noise environments,amplifies the signal from a sensing microphone and processes the informationfor visual display or information storage. It is generally portable and batteryoperated. The quality ratings of sound level meters are specified by the AmericanNational Standards Institute (ANSI) and the International Electrotechnical Commission(IEC) according to the precision of these instruments. Types 1, 2, and 3are, respectively, termed “precision,” “general purpose,” and “survey.” In addition,Type 0 has been specified by IEC for laboratory reference standard (InternationalElectrotechnical Commission, 1985). ANSI includes the suffix S in its standard todesignate special-purpose meters, e.g., meters equipped with only A-weighting.Measurement precision depends on a number of factors, including meter calibration,method of surveying, and frequency content of the noise being measured.

9.8 Proper Procedures for Using the Sound Level Meter 183Type 1 meters generally should measure with 1 dB accuracy and are employedto obtain accurate data for noise control purposes. The corresponding error forType 2 meters, which are used for quick surveys, may not exceed 2 dB.Figure 9.7(a) shows a variety of digital-readout SLMs and Figure 9.7(b) illustratesan SLM with a bar-graph spectrum display. Note that the contour of a metercase slopes away from the microphone in order to minimize reflections from itssurfaces. Both Types 1 and 2 generally incorporate A-, B-, and C- weighting networks.The A and C networks are far more commonly used than B-weighting,particularly if low-frequency acoustic energy is present. Also a “fast” or “slow”response setting is generally available. The former setting, corresponding to a timeconstant RC ≈ 0.1 s, responds more quickly to changes in noise levels, but thereadings become more difficult to ascertain with very rapid fluctuations. The slowsetting (RC ≈ 1) reduces the response speed, as attested by the slower movementof the needle in the older-type analog-readout SLMs, and a better grasp of typicalsound levels can be obtained for rapidly fluctuating sounds. More elaborate versionsof sound level meters include 1/3-octave and octave filters to enable soundpressure measurements in various frequency bands.In the block diagram of Figure 9.8, a typical SLM is shown to contain the followingcomponents: a 4-cm or 2-cm microphone feeding a preamplifier (whichfunctions as a cathode follower), which in turn, relays to the one of the weightingnetworks that is selected by a switch. The weighted (or unweighted, if none ofthe weighting curves has been selected) signal then becomes amplified and thenpasses through a root-mean-square amplifier, becomes converted to logarithmic(i.e., decibel) form, and fed to either a digital or analog readout device. Some SLMmodels contain output jacks so that AC and DC signals from the meter can serve asinputs to other instruments, such as Fast Fourier Transform (FFT) analyzers, printers,or graphic plotters. By taking advantage of the PCMCIA modular technology,a meter can be designed to provide a variety of other functions such measuringL eq , L n , reverberation times, and so on.9.8 Proper Procedures for Using the Sound Level MeterTo ensure that a sound level meter is in proper working order, an acoustic calibratorshould be employed just prior to beginning a series of measurements and aftercompleting the series. The calibrator is a single-tone, battery-driven device thatfits over the microphone and produces a precise reference sound pressure level forcalibrating the meter. The calibrator, illustrated in Figure 9.9, employs a zenerstabilizedoscillator to provide impetus to a piezoelectric driver element that causesa diaphragm to vibrate at 1 kHz ±1.5%. The diaphragm produces a sound pressurelevel of 1 Pa (corresponding to rms SPL of 94 ± 0.3 dB) in the coupler volume.The cavity at rear of the diaphragm behaves as a Helmholtz resonator with anatural frequency of 1000 Hz. Calibrators operate at 1 kHz, the internationalreference frequency for weighting networks. Therefore, no correction is requiredfor calibrating instruments for weighted and unweighted measurements.

(a)(b)Figure 9.7. (a) Sound level meters. Some models come equipped with built-in octaveor 1/3-octave filters for band measurements or with provisions for attaching separatefilter units to perform such measurements. (b) Close-up of a sound level meter capableof providing an octave-band spectrum. Its compactness is apparent in this photograph.(Courtesy of Brüel & Kjær.)

9.8 Proper Procedures for Using the Sound Level Meter 185Figure 9.8. Block diagram of the principal elements of a sound level meter.Periodically, every 6 months to a year depending on frequency and rigor ofusage, both the SLM and the calibrator should be rechecked with calibration instrumentswhich, in turn, have been calibrated periodically in a manner traceableto direct comparison with the standards set up by the U. S. National Institute ofScience and Technology (NIST) or the appropriate standards bureau in anothernation. Updated certifications of calibration are necessary for both SLM and itscalibrator as proof of the accuracy and reliability of the measurement devices.While executing measurements and entering data, one should also record sufficientinformation to identify all measuring instruments at a later date should theFigure 9.9. Cross section of an oscillator-driven sound level calibrator. This model producesan 1-kHz tone at a SPL of 94 dB. (Courtesy of Brüel & Kjær.)

186 9. Sound-Measuring Instrumentationneed arise to prove the accuracy of the instruments at the time of its questionedusage.One of the simplest case of noise measurements is that of steady noise, say, amotor running at steady speed in a factory. In this situation it would be necessaryto take only a few sound level readings in dB(A) near the worker’s ear and to checkthe measured values with established permissible levels. In the more common caseof fluctuating levels, moving noise sources or receivers, etc., it may be necessaryto take more readings over longer time intervals. Other types of meters or measurementprocedures may be more suitable for sustained noise exposures, such asthe use of the integrating sound level meter and the dosimeter which are describedin the following sections.9.9 The Integrating Sound Level MeterIn Chapter 3 the equivalent sound level was defined asL eq = 10 log 1 10 L/10 dtT 0or in terms of N sound level measurements taken during Nequal intervals:()1N∑L eq = 10 log 10 L 1/10N∫ Ti=1(9.5)Integrating sound level meters are based on the application of Equation (9.5) inmeasurements of fluctuating noise over a considerable interval. For example, ameter may be preset to measure in intervals of 1 s over a total time period of15 min, thus calling for 900 individual sets of measurements. A measurement over12 h may be programmed for 720 intervals of 1 min each. More elaborate metersdesigned for 24-h surveillances can compute the day–night equivalent sound levelin which a 10-dB penalty is automatically added to noise levels occurring betweenthe hours of 10 p.m. and 7 a.m. The integrating sound level meter is also capableof measuring the sound exposure level (SEL), which characterizes a single eventon the basis of both the pressure level and the duration. This parameter is definedby(∫ TSEL = 10 log0prms2pref2) (∫ T)dt = 10 log 10 L/10 dt0(9.6)where T is measured in seconds. If we consider a 2-s burst of sound at therms pressure of 1 Pa, use of Equation (9.6) will indicate an SEL of 97 dB.Comparison of Equations (9.5) and (9.6) leads to the relationship between SELand L eq :SEL = L eq + 10 log T (9.7)

Example Problem 39.10 Dosimeters 187What is the SEL for an equivalent sound level pressure of 98 dB(A) lasting 8 s?SolutionFrom Equation (9.7)SEL = 98 + 10 log 8 = 107 dB(A)Equation (9.6) provides the means of computing SEL when the integrating soundlevel meter does not offer an automatic SEL mode.9.10 DosimetersIt is not generally a practical matter to trail continuously an industrial workerperforming his or her duties in order to gauge the amount of noise exposure,particularly if that individual moves from place to place and is exposed to varyingdegrees of noise levels in the course of the day. A more convenient method ofmeasuring the total exposure is to have this person wear either a personal soundexposure meter or a noise dose meter (also called a noise dosemeter or noisedosimeter). The personal sound exposure meter, which is schematically describedin Figure 9.10, consists of a small microphone attached to a small extension cordso that it may be mounted close to the ear, a tiny amplifier that incorporates anA-weighting network, and a circuit that squares and integrates the electrical signalwith respect to time. The unit normally includes a sound exposure indicator whichcan come separate from the wearable unit. This device should be able to respondto a wide range of frequencies and sound levels without the presence of a manualcontrol. A latching overload indicator is also incorporated to provide a warningthat excessive sound pressure levels are occurring within the frequency range of theinstrument. The unit must be small, battery-powered, tamperproof, and constructedto withstand harsh environment.Figure 9.10. Schematic of a personal noise-exposure meter.

188 9. Sound-Measuring InstrumentationFigure 9.11. Schematic of a noise dosimeter.The noise dosimeter is a device designed to measure the percentage of the maximumdaily noise dose permitted by regulations. Its functional elements are givenin Figure 9.11. Both the noise dosimeter and the personal sound exposure metershare the same requirements for operation under severe operating conditions; andboth types of instruments include a microphone, an amplifier, A-weighting network,a squaring device, a time integrator, and an indicator. The personal soundexposure meters contain a latching overload but the noise dosimeter may includea latching upper-limit indicator. Additionally, the dosimeter must include exponentialtime-weighting usually incorporating the slow-response characteristic anda manufacturer-specified threshold sound level, 1 neither of which is found in thepersonal sound exposure meter.American National Standard ANSI S1.25 gives the specifications for a noisedosimeter. Slow-time weighted, A-weighted sound pressure is integrated with a5-dB exchange rate 2 in accordance with the regulations of U.S. OccupationalSafety and Health Administration (OSHA, 1983) and Mine Safety and Health1 Threshold sound level, stated in decibels, is the A-weighted sound level specified by the manufacturerof the noise dosimeter below which the instrument provides no significant indication. The thresholdsound level should be at least 5 dB less than the criterion sound level.2 The terminology exchange rate, expressed in decibels, refers to the change in decibels required todouble (or halve) the exposure time in order to maintain the same amount of noise exposure. Forexample, with the 5 dB exchange rate, exposure to 90 dB for 8hisequivalent to exposure to 95 dBfor4horto100dBfor2h.

9.11 Noise Measurement in Selected Frequency Bands, Band Pass Filters 189Administration (MSHA, 1982). The exchange rate may also be 3 or 4 dB dependingon the application. ANSI S1.25 also specifies limits for the effects of changes inair pressure, temperature, vibration, and magnetic field on the noise dosimeter.The British standard BS 6402 calls for a personal sound level meter to measuresound exposure, i.e., the timed-integrated A-weighted sound pressure level witha 3-dB exchange rate and without exponential time weighting (British StandardsInstitution, 1983). A latching overload indicator is also mandated by the Britishstandard to provide warning that the sound level received at the microphone hasexceeded the measuring range of the instrument (up to 132 dB peak).An IEC international standard for personal sound exposure meters defines awider operating range for a Type-2 integrating averaging SLM. Under this specification,measurements are made of the exposures produced by impulsive, fluctuating,and intermittent sounds over a range of A-weighted sound levels from 80 dBto 130 dB. Excessive input sound levels should trigger the mandatory overloadindicator.9.11 Noise Measurement in Selected Frequency Bands,Band Pass FiltersA valuable means of analyzing noise is the evaluation of sound in each frequencyband. If a machine emits a noise that indicates it is malfunctioning, the analysisof the sound output according to frequency can provide vital clues as to whichcomponent of the machine is defective. This situation calls for the use of aspectrum analyzer, which is a device that analyzes a noise signal in the frequencydomain by electronically separating the signal into various frequency bands. Thisseparation is executed through the use of a set of filters. A filter is a two-portelectrical network with a pair of terminals at each port. A filter can be constructedwith as few as two passive electrical components (e.g., a resistor and a capacitor),or it can be more complex involving a large number of passive components, or acombination of passive elements operating in conjunction with active components(e.g., op amps). Analog filters embody electronic circuitry tuned to pass certainfrequencies, whereas digital filters make use of active electronic elements.Figure 9.12 illustrates the effects of ideal and real filters. An ideal bandpassfilter is a circuit that transmits only that part of the input signal within its bandpass( f 1 ≤ f ≤ f 2 ) and completely attenuates all of the components at all frequenciesoutside of the bandpass ( f ≤ f 1 and f ≥ f 2 ). The ideal low-pass filter passes allsignals up to frequency f lp and rejects all signals having frequencies above f lp .The ideal high-pass filter passes all signals above frequency f hp and rejects all ofthe contents of the input signal below f hp . In the real world, filters alter the shapesof the input signals to some degree. The amplitude and phase characteristics of thefilter can be ascertained by computing the transfer function (or filter response),which is the ratio of the filter output to the filter input for all possible values offrequency. Both ideal and actual frequency responses are shown in Figure 9.12.The actual filters have characteristics of the form shown on the right-hand side

190 9. Sound-Measuring InstrumentationFigure 9.12. Real and ideal filters.of Figure 9.12. In the case of the low-pass filter of Figure 9.13(a), the amplitudetransfer function for the circuit is expressed as∣ |H(ω)| =E o ∣∣∣ 1∣ = √ (9.8)E i 1 + (ωRC)2where the subscripts i and o to voltage E denote the input and output values,respectively; R is the resistance in ohms and C the capacitance in farads. Equation(9.8) can be written to yield in decibels the characteristic of the form shown, plottedon the semi-logarithmic plot [Figure 9.13(b)] as follows:∣∣H ′ (ω) ∣ ∣ ∣ []∣∣∣ E o ∣∣∣ 1= 20 log = 20 log √ dBE i 1 + (ωRC)2=−20 log √ 1 + (2π fRC) 2 dB (9.9)

9.11 Noise Measurement in Selected Frequency Bands, Band Pass Filters 191Figure 9.13. (a) Circuit for a single, low-pass filter and (b) frequency response of thelow-pass filter.Equation (9.9) becomes∣ H ′ (ω) ∣ ∣ ≈−20 log 1 = 0 for (ωRC ) 2 ≪ 1that is, for relatively small frequency values the amplitude transfer function is 0 dB.In the case of large values of frequency, i.e., (ωRC) 2 ≫ 1, Equation (9.9) becomes∣∣H ′ (ω) ∣ ∣ ≈−20 log(2π fRC)The amplitude characteristic therefore exists as a slope of 20 dB per decade. Theintersection f 1 of the two asymptotes is the cutoff frequency given byf 1 = 12π RCAccording to Equation (9.9) the amplitude decreases by 1/ √ 2 at the cutoff frequency,or 3 dB. Because power is proportional to the square of the pressure amplitude,the cutoff frequency is also the half-power point. The amplitude responsesof complex filters can be obtained in a similar manner.Filters may be grouped in bands so that serial analysis is rendered possibleby switching manually or automatically from one band to the next. The octaveband analyzer constitutes an example of a commonly used serial analyzer. Thisdevice is so called because it is capable of resolving the noise-signal spectruminto frequency bands that are one octave in width. We have already seen in Chapter3 that the center frequency of an octave band is √ 2 times the lower cutofffrequency, and the upper cutoff frequency is twice the lower cutoff frequency. Theoctave bands in the audio range are designated by their center frequencies and are

192 9. Sound-Measuring InstrumentationFigure 9.14. Filter response typical of an octave-band filter set.standardized internationally at 31.5, 63, 125, 250, 500, 1000, 2000, 4000, 8000,16,000 Hz. Figure 9.14 illustrates the frequency response typical of an octave-bandfilter.In order to obtain more details of the noise spectrum, filters with bandwidthsnarrower than one octave must be used. Narrow band analyzers constitute excellenttools for diagnosing noises in industrial environments. Typical narrow-bandanalyzers measure in 1/3-octave and 1/12-octave bandwidths, with the formerbeing more commonly used. These analyzers resolve the noise spectrum into thirdoctaves and twelfth octaves. Denoting f 1 as the lower cutoff frequency, f 2 theupper cutoff frequency, and f 0 the center frequency, the relationship between theupper and lower cutoff frequencies for a 1/nth octave filter isf 2 = 2 1/n f 1 (9.10)The center frequency is the geometric mean of the product of the upper and lowercutoff frequencies:and the bandwidth bw is expressed as follows:f 0 = √ f 1 f 2 (9.11)bw = f 2 − f 1 = (2 12n − 2− 12n ) f0 (9.12)The smaller the bandwidth, the more detailed the analysis, and it obviouslyfollows that the equipment becomes more costly. The advantages of narrower-band

9.12 Real Time Analysis 193analyzers become more apparent in the detection of prominent components ofextremely narrow-band noise.9.12 Real Time AnalysisReal time analysis entails the evaluation of a signal over a specified numberof frequency bands simultaneously. This type of parallel process is schematicallyillustrated in Figure 9.15. Most real time analyzers rely on digital filteringof a sampled time series. A fluctuating noise signal is converted by a microphoneassembly into an electric signal that is entered simultaneously into aparallel set of filters, detectors, and storage components. The scanned output isdisplayed as a bar graph of sound pressure levels versus frequency on a cathoderay tube (CRT) monitor or another type of display such as liquid crystal display(LCD) or plasma screen. The display must be refreshed several times persecond as the level and frequency content of the input signal change. The scanninginterval may be adjusted to meet different conditions, becoming longer tocover the cycling of a specific machine operation, or shortened to evaluate impactsounds.A considerable variety of features are available on current real-time analyzers.Some features may include alphanumeric displays, choice of weighting and linear(unweighted) sound levels, linear and exponential averaging, time constants, spectrumstorage for recall and comparison with other data, and integrated or externalsoftware packages for further analysis of analyzer output. A number of real timeanalyzers operate on batteries, and models are available that provide a choice ofbattery or AC operation.Figure 9.15. Schematic of the processing of a signal through a parallel bank of filters ina real-time analyzer.

194 9. Sound-Measuring Instrumentation9.13 Fast Fourier Transform AnalysisThe Fast Fourier Transform (FFT) technique, rendered feasible by the advent ofmicrocomputers, employs both digital sampling and digitization. Instead relyingon bandpass filters to measure the analog amplitudes that formulate a signal’sspectrum, the FFT analyzer executes an efficient transformation of the signal fromthe time domain to the frequency domain. As it is capable of executing nearly anyanalysis function on the signal at high speed, the FFT technique is an extremelypowerful analytical method. The FFT analyzer captures a block of sampled dataof finite length (generally 1024 or 2048 samples) for a processing interval. Intransforming from the time domain to the frequency domain, the Fourier transformrelates a function of time g(t) to a function of frequency F(ω) in the followingmanner:F(ω) =∫ +∞−∞g(t)e −iωt dt (9.13)In measuring noise, a microphone assembly generates a voltage proportional tosound pressure. A time series is formed when the voltage is sampled at equal intervals,as shown in Figure 9.16. In order to transform this series into the frequencydomain, Equation (9.13) must be reformulated into the discrete Fourier transform(DFT), given byF(k) = 1 Nn=N−1 ∑n=0g(n)e −2πikn/N (9.14)The matrix format of Equation (9.14) (Randall, 1977) isF = 1 N[A] g (9.15)Figure 9.16. Sound pressure sampling at discrete intervals.

9.13 Fast Fourier Transform Analysis 195Figure 9.17. The effect of aliasing in generating a “false” wave.where F is a column array of N complex frequency components, [A] is a squarematrix of unit vectors, and g a column array of N samples in the time series.Direct evaluation of the discrete Fourier transform obviously requires a largeamount of number crunching, and the DFT methodology was rendered possibleonly in recent years by the development of efficient data processors. The FFTalgorithm, developed by Cooley and Tukey, originally for the implementationon mainframe computers but now utilized even in portable FFT analyzers andlaptops, cuts down on the number of calculations required to find a discrete Fouriertransform (Cooley and Tukey, 1965). The Cooley–Tukey algorithm rearranges the[A] matrix of Equation (9.15) by interchanging rows and factoring, with the resultthat reduces memory requirements and saves computing time. Computing timecan be cut even more so by tabulating sine and cosine values.In the FFT process, the signal is modified in three ways, giving rise to the threepotential traps of the FFT, these being the aliasing, leakage, and the so-calledpicket fence effect. Aliasing is the apparent measurement of a false or incorrectfrequency. Higher frequencies (after sampling) appear as lower ones, as in Figure9.17 in which a solid line represents a sound wave pressure with a period T . Thewave is sampled at intervals of T s corresponding to the sampling frequency off s = 1/T s . The frequency of the dotted line that results from the sampling rate canthen be incorrectly identified as the frequency of the solid line. This ambiguity canbe avoided by having a sampling rate at least twice the frequency of the highestfrequency present in the signal, i.e.,f s > 2 f maxThis minimum sampling frequency 2 × f max is called the Nyquist frequency. Theuse of very steep antialiazing filters (typically 120 dB/octave) prevents frequenciesthat cannot be adequately sampled from being analyzed and renders it possibleto utilize a major portion of the computed spectrum (e.g., 400 lines out of 512calculated). In most FFT units this is done automatically when the frequencyrange is selected (This step also sets the appropriate sampling frequency).

196 9. Sound-Measuring InstrumentationFigure 9.18. Illustration of the picket fence effect. If the frequency coincides with a line,it is indicated at its full level. Otherwise, it is represented at a lower level if the frequencyfalls between the lines.The effect of leakage, or window error, becomes apparent when the power ina single-frequency component appears to leak into other frequency bands. Twoprimary causes exist for window error: (1) the signal is not fully contained withinthe observation window or (2) it is not periodic within the window. The simplestcase is that of a monofrequency sinusoidal wave, which should yield only onefrequency component in the FFT analysis if there are an integral number of periodsof the sine wave in the finite record length. But a nonintegral number ofperiods will generally occur, and the cyclic repetition will yield a signal whosespectrum covers a range of frequencies. Leakage may be counteracted by forcingthe signal in the data window to correspond to an integral number of period ofall significant frequency components through a process called order or trackinganalysis, where the sampling rate is related directly to basic frequency of the noisegenerating process (such as the shaft speed of a machine) and in modal analysismeasurements where the analyzer cycle synchronizes with the periodic excitationsignals.The picket fence effect (Figure 9.18), which is not unique to FFT analysis, occursin any set of discrete fixed filters. The magnitude of the amplitude error occurringfrom this effect depends on the degree of overlap of adjacent filter characteristics,a consideration that influences the selection of a data window.9.14 Data Windows and Selection of Weighting FunctionsIf a steady pure-tone of an unknown frequency were to be analyzed, it wouldusually be sampled over a short-time interval that is termed window duration. Arectangular window will allow passage of a portion of the input signal without adjustment.That short segment obtained is presumably representative of the originalsignal, and this would also hold true if that segment embodies an integer numberof periods of the original signal. The effect of window duration arising from thescanning of (usually) noninteger number of periods is depicted in Figure 9.19.Such a signal would be analyzed as if the segment iterated itself as in the figure.

9.14 Data Windows and Selection of Weighting Functions 197Figure 9.19. Effect of window deviations in FFT scanning of noninteger number ofperiods.The discontinuities at the ends of the segments will engender a frequency spectrumdiffering from the true frequency content of the original signal. Aperiodic signalsmay also be misinterpreted on account of the finite length of sampling.The weighting or window function modifies the shape of the observation windowby tapering off both the leading and trailing edges of the data. Because theweighting function is used to obviate the effects of duration limiting, it increasesthe effective bandwidth. This increase is stated in terms of the noise bandwidthfactor (NBF) defined asEffective bandwidth with window functionNBF =(9.16)Effective bandwidth without window functionThe ideal value of NBF is unity. Another important consideration in the use ofweighting functions is the side-lobe ratio (SLR) or highest side-lobe level, expressedasSLR =Most sensitive out-of-band responsedB (9.17)Center of bandwidth responseA greater negative value of SLR (in decibels) is more desirable. But the selectionof the two parameters given by Equations (9.16) and (9.17) requires a compromise,i.e., a tradeoff between the steepness of the filter characteristic on one hand and theeffective bandwidth on the other. Table 9.1 lists the popular types of data windowsand their parametric values. The Hanning window (one period of a cosine-squaredfunction) constitutes a good choice for stationary signals, because that functionhas a zero value and slope at each end and thus renders a gradual transition overthe discontinuity of data. Figure 9.20 illustrates the rectangular, Hanning, andHamming weighting functions.

198 9. Sound-Measuring InstrumentationTable 9.1. Properties of Data Windows.Maximum Side lobe Noise bandwidth Maximumside lobe, falloff, (relative to line amplitudeWindow type dB dB/decade spacing) error, dBRectangular −13.4 −20 1.00 3.9Hanning −32 −60 1.50 1.4Hamming −43 −20 1.36 1.8Kaiser-Bessel −69 −20 1.80 1.0Truncated Gaussian −69 −20 1.90 0.9Flattop −93 0 3.70 0.19.15 ResolutionThe smallest increment in a parameter that can be displayed by a measurementsystem is the resolution for that system. In FTT analysis, the resolution is expressedbyβ = f s NN= 2 f RN(9.18)Figure 9.20. Typical weighting functions for FFT analysis, with corresponding noise bandfactor (NBF) and side lobe ratio (SLR).

9.16 Measurement Error 199whereβ = the resolution which is the frequency increment between lines in aspectrum, Hzf s = sampling frequency, the reciprocal of sampling period T sN = number of samples in the original time seriesf R = frequency range, normally from 0 to the Nyquist frequencySome versions of FFT analyzers incorporate a “zoom” capability which displaysselected portions of a spectrum with finer reduction. Normally, in order to increasethe resolution (which means that β must decrease) the number of samplesmust be larger. This is not always feasible since either the processing time willincrease or the block size is limited by machine memory capacity. The zoomcapability overcomes this limitation by permitting the spectrum analyzer to concentrateits entire resolution, whether it be 200 or 800 or 1000 lines, on a smallfrequency interval selected by the user. While arbitrarily fine resolution is achievable,a compromise must be effected between resolution and the required samplingtime, as attested by Equation (9.18). Note that according to this equation, if thefrequency resolution for a signal is 10 Hz, the sampling time is 0.1 s, but if thefrequency resolution is changed to 0.1 Hz, the corresponding sampling time will be10 s.A number of FFT analyzers incorporate large digital processing capacitiesthat can present measurement results in the format of three-dimensional plots(or “waterfall” or “cascade” plots) with the vertical coordinate representing theamplitudes of the spectra as functions of the other two coordinates, one representingthe frequency f and the other time t. The scan presented on a displaycan show a “running” plot that moves in the direction of increasing t. This typeof display is most useful in observing the behavioral characteristics of transientsounds.9.16 Measurement ErrorNoise is usually random in nature, i.e., its sound pressure level cannot be predictedfor any instant. But statistical means can be used to describe random noise. If thenoise is relatively constant in level and frequency content then it may be deemeda stationary random process, one in which statistical parameters are invariantwith respect to time. A machine operating in a constant cyclic manner may emitdifferent levels of sound, with corresponding changes in frequency content, foreach successive instant of the cycle, but a measurement interval over a group ofcycles will yield a consistent spectral distribution over time. Analysis data basedon a very short interval that is less than the length of a single cycle is certain toyield misleading results.Let us consider noise with an idealized Gaussian or normal probability distribution.The standard deviation ε, which is the uncertainty in the rms signal dividedby the long-term average rms signal, relates to the ideal filter bandwidth bw and

200 9. Sound-Measuring Instrumentationthe averaging time T , according to1ε =2 √ bw × TThe measurement uncertainty is twice that given by Equation (9.19).(9.19)Example Problem 4A normal distributed noise is to be analyzed in 20 Hz bands, with the measurementuncertainty not to exceed 4%. Find the necessary averaging time T .SolutionUsing Equation (9.19) modified to give the measurement uncertainty, we obtainT =9.17 Sound Power1bw × ε = 12 20 × 0.04 = 31.3s2Sound power denotes the rate per unit time at which sound energy is radiated. Thisrate is expressed in watts. The sound power L w level defined by( ) WL w = 10 logW 0is given in decibels. W is the power of the sound energy source in watts and W 0 =1 pW is the standard reference power in watts.The principal advantage of using sound power level rather than the sound pressurelevel given by Equation (3.22) to describe noise output of stationary equipmentis that the sound power output radiated by a piece of equipment is independent ofits environment. The sound energy output of the machine will not change if thatunit is moved from place to place, provided it operates in the same manner. Soundpower is primarily used to describe stationary equipment, but it is not generallyused to rate mobile equipment since the operational situations may be too highlyvariable, as is the case with construction equipment.Sound power may be measured directly using spund intensity instrumentationor indirectly, either determined from the rms sound pressures at a number ofmicrophone locations spatially averaged over an appropriate surface enclosingthe source in a free field over a reflecting plane (as exemplified by the use ofa semianechoic chamber) or in a totally free field (full anechoic chamber), oraveraged over the volume of a reverberation chamber in which the measurementsare conducted.Some sources are omnidirectional, i.e., they radiate sound uniformly in all directions.Most sources are highly directional, radiating more sound energy in somedirections than in others. Hence, the directivity,ordirectional characteristic, constitutesan important descriptor of a sound source. In a free field or anechoic

9.18 Measurement of Sound in a Free Field over a Reflecting Plane 201chamber, the directivity is readily apparent, owing to the absence of reflections.But in a highly reflective environment, such as that of an echo chamber, the multiplereflections that occur render the directivity less important and the sound fieldbecomes more uniform.9.18 Measurement of Sound in a Free Field over aReflecting Plane (Semianechoic Chamber)We can summarize the measurement procedure in a free field over a reflectingplane as follows:1. The source is surrounded with an imaginary surface of area S, either a hemisphereof radius r or a rectangular parallelepiped.2. The area of the hypothetical surface is calculated. S = 2πr 2 in the case ofa hemispherical surface, and S = ab + 2(ac + bc) in the case of the parallelepipedhaving length a, width b, and height c.3. The sound pressure is measured at designated points of the imaginary surface.4. The average sound pressure level ¯L p is computed from the measured results ofthe previous step. This is found from[¯L p = 10 log1NN∑10 L i /10i=1](9.20)where N is the total number of measurements and L i denotes the measuredvalue of the SPL at the designated point i.5. The sound power level is then calculated from the following:L w = ¯L p + 10 log(S/S 0 ) (9.21)where S 0 is the reference area of 1 m 2 .The above procedure applies only if the source is not too large, i.e., the radiusr of the hypothetical hemisphere should be at least 1 m and at least twice thelargest dimension of the source (or the perpendicular distance between the sourceinside the imaginary parallelepiped and a measurement surface is 1 m), and thebackground noise level is more than 6 dB below that of the source. The rectangularparallelepiped setup is preferred for large rectangular sources.Figure 9.21 shows the designated points on the hemispherical surface wherethe microphones are located. The corresponding points for the rectangular parallelepipedare given in Figure 9.22. These designated points are associated withequal areas on the surface of the hemisphere or the rectangular parallelepiped. TheSPL is usually measured at the designated points with A-weighting or in octaveand partial octave bands, with the meter set in the slow-response mode.The applicable international standards for acceptable sound power measurementtechniques under semianechoic conditions are given in ISO 3744 and ISO 3745.Adjustments in the values of the measured SPL should be made for the presenceof background noise. Equations (9.20) and (9.21) are used to convert the measurementsinto the desired values of averaged SPL, ¯L p , and sound power level L w .

202 9. Sound-Measuring InstrumentationFigure 9.21. Locations of the microphones on the surface of an imaginary hemispheresurrounding a source whose sound power level is to be measured according to ISO 3744.The directivity index DI of the source may be computed from measurements ina semianechoic chamber from the following equation:DI = L pi − ¯L p + 3 dB (9.22)where L pi is the sound pressure level measured at point i, located on the measuringsurface and defining the direction along which the DI is desired at a distance fromthe source.9.19 Measurement of Sound Power Level in a Free Field(Full Anechoic Chambers)The procedure for measuring sound power in a free field is basically the sameas that for a free field with a reflecting surface (semianechoic condition), withsome modifications. In this case, the source is centered in a hypothetical sphereof radius rand surface area S = 4πr 2 . The sound pressure levels are measured atspecific points on the spherical surfaces; these points are stipulated by ISO 3745and shown in Figure 9.23, and defined in terms of Cartesian coordinates in Table9.2. Equations (9.20) and (9.21) also apply to yield the average sound pressurelevel ¯L p and the sound powerL w . The surface area ratio S/S 0 in Equation (9.21)now refers to two spheres, the hypothetical one (of radius r) used for placementof measuring sensors and the other a reference sphere with an area of unity, with

9.20 Sound Power Measurement in a Diffuse Field 203Figure 9.22. Locations of the microphones on the surface of an imaginary parallelepipedsurrounding a source whose sound power level is to be measured according to ISO 3744.the result that this ratio is equal to simply 4πr 2 . Equation (9.21) becomes simplyL w = ¯L p + 20 log r + 10.99 (9.23)where all dimensions are stated in SI units. The rectangular measurement surfaceis not generally used for measurements in a free field. As specified by ISO 3745,the radius of the test sphere should be at least twice as large as the major sourcedimension, but never less than 1 m. A large source will necessitate a very largeanechoic chamber for measurement purposes.9.20 Sound Power Measurement in a Diffuse Field(Reverberation Chamber)Reverberation chambers are rooms with extremely reflective walls, ceilings, andfloors. If the surfaces are made nonparallel to each other, standing waves canbe avoided. When a steady noise source is operating, the sound field is diffusedeverywhere in the room except in the immediate vicinity of the source, as the

204 9. Sound-Measuring InstrumentationFigure 9.23. The locations of microphones on an imaginary spherical surface surroundingthe source in a free field, according to ISO 3755.

9.20 Sound Power Measurement in a Diffuse Field 205Table 9.2. Microphones Array Positions a in a Free Field According toISO 3745.No. x/r y/r z/r1 −0.99 0 0.152 0.50 −0.86 0.153 0.50 0.86 0.154 −0.45 0.77 0.455 −0.45 −0.77 0.456 0.89 0 0.457 0.33 0.57 0.758 −0.66 0 0.759 0.33 −0.57 0.7510 0 0 1.011 0.99 0 −0.1512 −0.50 0.86 −0.1513 −0.50 −0.86 −0.1514 0.45 −0.77 −0.4515 0.45 0.77 −0.4516 −0.89 0 −0.4517 −0.33 −0.57 −0.7518 0.66 0 −0.7519 −0.33 0.57 −0.7520 0 0 −1.0a This table lists the Cartesian coordinates (x, y, z) with the origin located at the centerof the source. The vertical axis is perpendicular to the horizontal plane z = 0.result of sound waves reflecting back and forth between room surfaces until theydie out. A steady noise source generates sound and builds up the sound pressurelevel from ambient level in the room, until an equilibrium sound pressure levelis reached. This occurs when the new acoustic energy emanating from the sourceoffsets the dissipation of reflected sound energy in the slightly elastic reflectingsurfaces (however hard they may be) and in the slightly viscous air of the chamberand through energy leakage from the room. The applicable international standardsfor measuring the sound power levels in reverberation chambers are ISO 3741and ISO 3742, the former giving details for broadband noise and the latter fordiscrete-frequency and narrow-band sources. The relationship between the soundpower of a source and the average reverberant sound level can be determined andallows calculation of the sound power.The procedure for measuring the sound power level in an echoic chamber maybe summarized as follows:1. The reverberation time T 60 is measured by using a standard technique prescribedby ISO 354. 33 Reverberation time T 60 for an enclosure is the time interval required for a sound pressure level todrop 60 dB after the sound source has been stopped. More detail treatment of T 60 is rendered inChapter 11.

206 9. Sound-Measuring Instrumentation2. The room volume V and the total surface area A of the test chamber are calculatedfrom its internal dimensions. The barometric pressure B is measuredbut this constitutes only a small influence on the sound power level of thesource.3. The average sound pressure level ¯L p in the room is obtained by sweeping amicrophone at steady speed over a path at least 3minlength while its outputsignal is measured and averaged on a root-mean-square-pressure basis. Thesweep time should be at least 10 s for the 200 Hz band and higher and 30 s forlower frequencies. Measurements in the near field (close to the source) and verynear the chamber surfaces should be avoided. The microphone should not belocated closer to a surface than one half the wavelength of the lowest pertinentfrequency. Alternatively, the average value may be obtained by averaging theoutput of an array of three fixed microphones spaced a distance of λ/2 apart(wavelength λ corresponds to the lowest frequency of interest). ¯L p constitutesthe average band pressure level corrected for the background noise (the soundlevel that exists in the measurement chamber when the source is not operating).4. The contribution of each frequency band to the sound power level L w of thesource, is calculated by using the following equation:L w = ¯L p − 10 log(−10 log(T601sB1000 mbar) ( V+ log1m 3 )+ log(1 + Sλ )8V)− 14 dB (9.24)where wavelength λ corresponds to the center frequency of the frequency band ofinterest. The A-weighted sound power level L w A may be computed from octavebandor one-third-octave band levels, according to ISO 3741, Annex C.9.21 Substitution (or Comparison) Method for MeasuringSound Power LevelA simpler method than the direct method described above can be used to determinethe sound power level L w of an unknown source by comparing two measurements,without the necessity for knowing the reverberation time of the test chamber. Infact, this method does not even require the use of a special laboratory chamber,and it can be applied in situations where it would be impracticable to move alarge piece of machinery to a laboratory. Commercially available reference sourcesprovide known values of sound power level L wr for each octave band and one-thirdoctave band. Reference sound sources are classified into three types: aerodynamicsources, electrodynamics sources, and mechanical sources. Aerodynamic sources,the most prevalent type, consist of a specially designed fan or blower wheel drivenby a motor. Technical requirements are listed in international standard ISO 6926.2.The comparison method, defined by ISO 3741, is as follows:

9.23 The Addition Method for Measuring Sound Power Level 2071. The procedures for measuring the average of the sound pressure levels ¯L p areconducted in the test room in the same manner described above for the directmethod. ¯L p is measured for the unknown source, and then the unknown soundsource is turned off.2. The reference source is turned on, and ¯L pr is measured for the reference source.The reference sound sources should be mounted on the floor at least 1.5 m awayfrom another sound reflecting surface, such as a wall or the unknown sourcebeing evaluated.3. The sound power level for the source undergoing measurement can be simplycomputed fromL w = ¯L p + (L wr − ¯L pr ) dB (9.25)9.22 Alternation Method for Measuring Sound Power LevelA reference source with adjustable sound power—for example, a wideband sourcethat generates pink noise (i.e., the same level in each band) or an octave bandfiltered noise—is used in the alternation method for measuring sound power. Areadout meter indicates the reference source power output. The procedure is asfollows:1. The noise source being tested is operated in a diffuse field. The spatial averagesound power level ¯L p is measured in each octave band.2. The reference source replaces the noise source. The reference source is adjusteduntil it produces the same sound level as the tested source did in the first octaveband. The sound power level L w indicated on the reference source meter isrecorded. This procedure is repeated for each of the other octave bands.3. The sound power levels noted in the reference source meter are the sound powerlevels for the unknown source for each octave band. The total sound power canbe computed from¯L w = 10 logN∑i=110 ¯L wi /10where ¯L wi is the spatial average sound power level for the ith octave.(9.26)9.23 The Addition Method for Measuring SoundPower LevelThe addition method is useful for situations where the machine under test cannotbe conveniently shut down, for example, a power station generator. As with thealternation method, an adjustable reference source is used. The procedure is asfollows:

208 9. Sound-Measuring Instrumentation1. The spatial average sound power level ¯L p generated by the subject machine ismeasured in each octave band.2. The reference source is placed near the machine and both are operated simultaneously.The reference source is adjusted until it produces an additional 3 dBin the sound pressure level in each octave band than when the machine wasoperating by itself. The reference source is then producing a sound power levelin that band equal to that generated by the machine under test. The sound powerlevel indicated on the meter of the reference power source is recorded. Theprocedure is repeated for each of all the other octave bands.3. The total sound power level can now be calculated from the use of Equation(9.26).9.24 Data Acquisition SystemsFor a number of years, magnetic recorders provide a permanent record of noisedata taken in situ for subsequent analyses on instruments located elsewhere and forarchival purposes. Currently, digital recorders are being used for acoustical dataacquisition. Very few, if any, analog recorders (either AM or FM) are currentlysold. Computer technology has advanced to the stage where measured data andits analytical results can be acquired and digitally stored in computer memoryor on recordable CD-ROMs, DVD-R, and removable cartridges or disks. A CD-ROM can hold approximately 700 MB of data, but the DVD (which can be usedto contain digital data and not just video programs) holds nearly ten times asmuch. Newer storage disks are soaring past the gigabyte range to encroach inthe terabyte territory. We are witnessing the rapid demise of the magnetic taperecorder in favor of computerized acquisition devices and digital storage units. Therecent introduction of flash disks provides the potential for even greater portabilityof data. Transient sounds, analyzed on a cascade-type FFT analyzer, are moreconveniently archived in a nonvolatile memory medium such as a random-accessremovable cartridge than they would be on a magnetic tape (which obviously doesnot provide random access), for the purpose of later retrieval.9.25 Integration of Measurement Functions in ComputersAdvances in computer technology and software development tools make it possibleto integrate measurement functions into a personal desk computer or a laptopequipped with the appropriate acquisition printed circuit boards, high quality soundcards, and the applicable sensors. A specially equipped personal computer can executeDSP-based (diagnostic signal processing) signal generation, filtering, andspectrum analysis. Hence, a single computer can replace a whole rack of dedicatedanalog units linked together by BNC cables. In testing the performance ofa loudspeaker through traditional analog means, a sine wave generator provides asignal to the loudspeaker and a calibrated microphone picks up the loudspeaker’s

References 209acoustical output and feeds it to a spectrum analyzer, which then relays the datato a display or a recorder. The dedicated functions of the traditional hardware—signal generation, filtering, analysis, and data handling—can now be performedby software in such a manner that the personal computer essentially constitutesthe test platform. The user interface can even be made to emulate familiar analoginstrumentation controls using standardized Windows TM controls.Measurement procedures with a PC require a high-quality D/A (digital to analog)converter for transforming the digital representation of a sine sweep created by theprogram into analog signal, and an equally good A/D converter for transformingthe measured analog signals back into the digital domain for analytical purposes.Most sound cards provide at least 16-bit resolution, and some sound cards can evenmeasure frequency response from DC to 20 kHz with an accuracy of ±0.25 dBand distortion as low as 0.003%.The SoundCheck TM PC-based electroacoustical measurement system, essentiallya software package, requires only a computer, sound card, amplifier, microphone,and microphone power supply. The SoundWare TM software featuresa family of “virtual instruments” that perform the functions of a signal generator,voltmeter, oscilloscope, spectrum analyzer, and real-time analyzer. The PC isrendered capable of measuring frequency, time, phase response, total harmonicdistortion (THD), impedance, as well as performing other electrical tests. Sucha system can be applied to evaluating loudspeakers, microphones, telephones,hearing aids, headsets, and other communication devices. Programming of testsequences allows the PC to be used not only for research and development work,but also in high-speed production testing and inspection procedures.ReferencesAmerican National Standard Institute. 1966. ANSI Standard Specification for Octave, Halfoctaveand Third-Octave Band Filter Sets, S1.11-1966.American National Standard Institute. 1978. ANSI Standard Specification for PersonalNoise Dosimeters, ANSI S1.25-1978.British Standards Institution. 1983. Personal Sound Exposure Meters, BS6402.Cooley, J. W. and Tukey, J. W. 1965. An algorithm for the machine calculation of complexFourier series. Mathematics of Computation 19:297–301.Crocker, Malcolm J. (ed.). 1997. Encyclopedia of Acoustics, Vol. 4. New York: John Wiley& Sons: Part XVII, pp. 1837–1879, and Part XVIII: 1933–1944.Fraden, Jacob. 1993. AIP Handbook of Modern Sensors: Physics, Design and Applications.New York: American Institute of Physics, Chapters 3, 4, 9, and 11. (While thisuseful handbook covers devices for sensing temperature, light, humidity, and chemicalprocesses, the chapters cited here refer to topics applicable to acoustic measurements.)Harris, Cyril M. (ed.). 1991. Handbook of Acoustical Measurements and Noise Control,3rd ed., New York: McGraw-Hill, Chapters 5, 8, 9, 11–15.Hixson, Elmer L. and Busch-Vishniac, Ilene. 1997. In: Crocker, Malcolm J., ed. Transducerprinciples. Encyclopedia of Acoustics, Vol. 4. New York: John Wiley & Sons, Part VVIII,Chapter 59.

210 9. Sound-Measuring InstrumentationInternational Electrotechnical Commission. 1985. IEC 804: Integrating-Averaging SoundLevel Meters. CH-1211 Geneva 20, Switzerland: International Electrotechnical Commission.International Organization for Standardization. 1981. ISO 3744: Engineering Methodsfor Free-Field Conditions over a Reflecting Plane, CH-1211 Geneva 20, Switzerland:International Organization for Standardization.International Organization for Standardization. 1977. ISO 3745: Precision Methods forAnechoic and Semi-Anechoic Rooms. CH-1211 Geneva 20, Switzerland: InternationalOrganization for Standardization.International Organization for Standardization. 1975. ISO 3741: Precision Methods forBroad-Band Sources in Reverberation Rooms. CH-1211 Geneva 20, Switzerland: InternationalOrganization for Standardization.International Organization for Standardization. 1975. ISO 3742: Precision Methods forDiscrete-Frequency and Narrow-Band Sources in Reverberation Rooms. CH-1211Geneva 20, Switzerland: International Organization for Standardization.International Organization for Standardization. 1985. ISO 354: Measurement of SoundAbsorption in a Reverberation Room. CH-1211 Geneva 20, Switzerland: InternationalOrganization for Standardization.International Organization for Standardization. 1990. ISO 6926.2: Requirements on thePerformance and Calibration of Reference Sound Sources. CH-1211 Geneva 20,Switzerland: International Organization for Standardization.International Organization for Standardization. 1979. ISO 3746: Survey Method. CH-1211Geneva 20, Switzerland: International Organization for Standardization.Randall, Robert B. 1977. Application ofB&KEquipment to Frequency Analysis, 2nd ed.Nærum, Denmark: Brüel and Kjær.Sessler, G. M. and West, J. E. 1973. Journal of the Acoustical Society of America pp. 1589–1600.Temme, S. F. 1998. Virtual Instruments for Audio Testing Presented at the 105thConvention of the Audio Engineering Society. Audio Engineering Society Preprint 4894(J-8).U.S. Department of Labor, Occupational Safety and Health Administration (OSHA). 1983.Occupational Noise Health Standard. Code of Federal Regulations, title 29, part 1910,sec. 1910.95 (29 CFR 1910.95), Federal Register 48:29687–29698.U.S. Mine Safety and Health Administration (MSHA). 1982. Code of Federal Regulations.title 30, part 70, subpart F, and part 71, subpart I, Federal Register 47:28095–28098.Wilson, Charles E.. 1989. Noise Control. New York: Harper & Row, Chapter 3 (probablythe most outstanding chapter in this text).Wong, George S. K. and Embleton, Tony F. W. 1995. AIP Handbook of Condenser Microphones,Theory, Calibration, and Measurements, New York: American Institute ofPhysics. (A great reference for those interested in the history of condenser microphonesand in learning further details on their uses. Chapter 2 describes the classic WesternElectric 640AA, the de facto standard for all microphones.)Problems for Chapter 91. If a microphone has a sensitivity of −60 dB V/μbar and an output voltage of21.6 mV is measured, what is the sound pressure level (SPL) responsible forthat output voltage?

Problems for Chapter 9 2112. Determine the value of the output voltage of a microphone with a sensitivitylevel of −100 dB V/μbar if it is exposed to a sound pressure level of85 dB?3. A sound pressure level reading of 95 dB is taken at the standard conditions of1 atmosphere and 68 ◦ F. What will be the readings at 20 ◦ F and at 95 ◦ F.4. Some laboratory notes indicated that a noise level reading resulted in L eq =95 dB(A) and an SEL of 107 dB(A). How long did this noise level readinglast?5. Assuming a sixth octave analyzer could be made available, determine the firstthree center frequencies after 16 Hz. Also determine the bandwidth.6. Calculate the lower and upper cutoff frequencies and the bandwidth for theoctave band where the center frequencies are, respectively, f 0 = 63 Hz andf 0 = 500 Hz.7. If it is desired to analyze noise in 10 Hz bandwidths with an uncertainty of3%, what must be the necessary averaging time?8. If we are sampling through the FFT process, a signal that ranges from 15 Hzto 30 Hz, what must be the minimum sampling frequency?9. Why is a spectrum analyzer necessary and possibly more useful in dealingwith noise machinery?10. In an FFT analysis, it is desired to have a resolution of 1 Hz. If there are 1000samples in the time series, what must be the minimum frequency samplingrate?

10Physiology of Hearingand Psychoacoustics10.1 Human HearingThe mechanism of the human ear has been a source of much wonder for physiologists,who are progressing well beyond the fragmentary knowledge of the pastby continually uncovering layers of new marvels of how the human ear reallyfunctions. Our hearing mechanism is a complex system that consists of many subsystems.The mechanism of the brain’s processing of auditory stimuli is probablybeginning to be understood, with the relatively recent discovery that hearing ismetabolic in nature. Much of the pioneering work was performed by Georg vonBékésy (1899–1972) who received the Nobel Prize in Medicine or Physiologyfor his investigations, particularly those entailing the frequency selectivity of theinner ear. His work indicated that the frequency selectivity ranked far poorer thanthe ear actually exhibits, but William Rhode found much greater selectivity in hiswork with live animals (von Békésy worked with dead animals, which most likelyaccounted for the difference). It is now realized that frequency selectivity of theinner ear fades within minutes after the metabolism ceases.A young, healthy human is capable of hearing sounds over the frequency rangeof 20 Hz–20 kHz, with a frequency resolution as small as 0.2%, Thus, we candiscern the difference between a tone of 1000 Hz and one of 1002 Hz. With normalhearing, a sound at 1 kHz that displaces the eardrum less than 1 Å (angstrom) canbe detected, in fact, less than the diameter of a hydrogen atom! The intensity rangeof the ear spans extremes from threshold at which softest sounds can be detected tothe roar of a fighter jet taking off, thus covering an intensity range of approximately100,000,000–1. The ear acts as a microphone in the process of collecting acousticsignals and relaying them through the nervous system into the brain. The ear(cf. Figure 10.1) subdivides into three principal areas: the outer, middle, and innerear.The outer ear consists of a pinna that serves as a sound-collecting horn and theauditory canal that leads to the inner ear. The collected sound enters the ear throughthe opening (the meatus) into the auditory canal that forms a tube approximately0.75 cm in diameter and 2.5 cm length. The canal terminates at the tympanicmembrane (eardrum). Under the impetus of the sound the eardrum vibrates, causing213

214 10. Physiology of Hearing and PsychoacousticsBrain8552 3Cochlea5External auditory canal1476Outer EarMiddle Ear9Inner Ear1. Eardrum2. Malleus3. Incus4. Stapes5. Semicircular canals6. Auditory nerve7. Vestibular nerve8. Endolymphatic sac9. Eustachian tubeFigure 10.1. Coronal section of the right ear. (From Internet site of the Center forSensory and Communication Disorders, Northwestern University, funding by U.S.National Institute of Health.)three bones linked in an ossicular chain—namely, the malleus (or hammer), theincus (anvil), and the stapes (stirrup)—to oscillate sympathetically. At the lowestresonance (3 kHz) of the auditory canal, the sound pressure level at the eardrum isabout 10 dB greater than it is at the entry into the canal. Because a resonance curvetends to be broad, human hearing tends to be more sensitive to sound in the rangeof approximately 2–6 kHz, as the consequence of the resonance being centered at3 kHz. The diffraction of sound waves inside the head has the effect of causing thesound pressure level at the eardrum to exceed the free-field sound pressure levelby as much as 20 dB for some specific frequencies.The eardrum itself is a thin, semitransparent diaphragm that completely sealsoff the canal, marking the inner boundary of the outer ear and the outer boundaryof the middle ear. This membrane is quite flexible at its center and is attached at itsperimeter at the terminus of the auditory canal, thus demarcating the entrance tothe middle ear. The middle ear, lined with a mucous membrane, constitutes an airfilledcavity of about 2 cm 3 in volume, which contains the three ossicles (bones),namely, the malleus, incus, and the stapes forming a bony bridge from the externalear to the middle ear. These bones are supported by muscles and ligaments. Themalleus is attached to the eardrum; the incus connects the malleus and the stapes.The last bone in the chain, the stapes, covers the oval window. The Eustachiantube, which is normally closed, opens in the process of swallowing or yawning toequalize the air pressure on each side of the eardrum; this is a tube approximately

10.1 Human Hearing 215Figure 10.2. The membranous semicircular canals showing the cristae within theampullae. (From “The Internal Ear,” What’s New, 1957, Abbot Laboratories. Reproducedwith permission of the publisher.)37 mm in length that connects the middle-ear cavity with the pharynx at the rearof the nasal cavity.Just below the oval window lies another connection between the middle andinner ears, the membrane-covered round window. Between the oval and roundwindows is a rounded osseous projection, formed by the basal turn of the cochlea,called the promontory. A canal encasing the facial nerve is situated just above theoval window.The structures to the right of the oval and round windows shown in Figure 10.1are collectively called the inner ear (also called labyrinth), which comprisesa number of canals hollowed out of the petrous portion of the temporal bone.These interconnecting canals contain fluids, membranes, sensory cells, and nerveelements. Three principal parts exist in the inner ear: the vestibule (an entrancechamber), the semicircular canals, and the cochlea. The vestibule connects withthe middle ear through the oval and the round windows. Both of these windows areeffectively sealed, by the action of the stapes and its support on the oval windowand the presence of a thin membrane in the round window, thus preventing theloss of the liquid filling the inner ear. The semicircular canals play no role in theprocess of hearing but they do provide us with a sense of balance. The cochlea,

216 10. Physiology of Hearing and PsychoacousticsFigure 10.3. Cross section of the organ of Corti in the cochlear canal. (From H. Gulick.1971. Hearing. Physiology and Psychophysics. New York: Oxford University Press.Reproduced with permission of the publisher.)shown in enlarged detail in Figure 10.2, is the sensory system that converts thevibratory energy of sound into electrical signals to the brain for the detection andinterpretation of that sound. The cochlea can be described as a 3.5-cm long tubeof roughly circular cross section, wound about 2 1 / 2 times in a snail-like coil. Thistube’s cross-sectional area decreases in a somewhat uneven manner from its baseto its apex. Its total volume is about 5 (10) –2 cm 3 .The coils of the cochlea surrounds an area called the modiolus; and the membranouslabyrinth of the cochlear sector of the inner ear divides into three ducts orgalleries (scalae). The cochlear duct (ductus cochlearis) runs the length of the spiralingcochlea, and because it occupies the central portion of the cochlea’s interior,it has been termed the scala medi (i.e., the middle gallery), whose walls effectivelypartition the cochlea into two longitudinal channels, the scala vestibuli (or uppergallery) and the scala tympani (lower gallery). The only communication betweenthe two galleries occurs through the helicotrema, a small opening at the apex ofthe cochlea. The other ends of the upper and lower galleries terminate in the ovaland round windows, respectively.Figure 10.3 shows an enlarged view of the cochlear duct. This duct is bounded byReissner’s membrane, the basilar membrane, and the stria vascularis. The basilarmembrane extends from the bony spiral lamina, a ledge extending from the centralcore of the cochlea, to the spiral ligament. The length of the basilar membraneis about 32 mm long, from the base to the apex of the cochlea; the width varies

10.2 The Mechanism of Hearing 217from about 0.05 mm at the base to about 0.5 mm at the apex; and the membranegradually becomes thinner as it nears the apex. Positioned on the basilar membraneis the organ of Corti. This organ, shown in detail in Figure 10.4, consists of somestructural cells (e.g., Dieter’s cells and Hensen’s cells), the rods and tunnels ofCorti, and two types of hair cells on top of which lies the tectorial membrane. Thetunnel of Corti, isolated from the endolymph, contains the fluid cortilymph.The hair cells constitute the sensory cells for hearing. The inner hair cells arearranged in a single row on the modiolar side of the tunnel of Corti, and the outerhair cells exist in three parallel rows on the strial side of the tunnel of Corti. Theinner hair cells are round and squat; their upper surfaces contain about 50–70 hairscalled stereocilia. The outer hair cells, which look more reedlike than do the innerhairs, contain about 40–150 stereocilia arranged in a W-shaped pattern. There are3000–3500 ciliated cells in the single row of inner hair cells and a total of 9000–12,000 ciliated cells in the three rows of the outer hair cells. The hair cells areconnected to some 24,000 transverse nerve fibers in a complex network leadinginto the central core of the cochlea. The nuclei of these nerve fibers form the spiralganglion, which unite to form the cochlear branch of the VIIIth nerve.The cochlear branch joins with the vestibular branch to form the VIIIth cranialnerve, also called the auditory or vestibulocochlear nerve. The VIIIth cranial nervealong with the VIIth (facial) cranial nerve proceeds in a helical fashion throughthe internal auditory meatus to nuclei in the brain stem. From the brain stemthe auditory pathway extends through various nuclei to the cerebral cortex in thetemporal lobes of the brain.The VIIIth cranial nerve is primarily as sensory nerve, i.e., it conveys sensoryinformation from the cochlea and the vestibular system to the brain.10.2 The Mechanism of HearingSound waves are directed by the pinna into the auditory canal. The longitudinalchanges in air pressure of the sound wave propagate to the eardrum, causing itto vibrate. Because the handle of the malleus is imbedded in the eardrum, theossicular chain is set into vibration. These tiny bones vibrate as a unit, elevatingthe energy from the eardrum to the oval window by a factor of 1.31 to 1. Soundenergy is further enhanced by the difference in area between the eardrum and thestapes footplate by a factor of approximately 14. Multiplying this effective arealdifference of 14 by the lever action of the ossicular chain (1.31) yields an energyincrease of 18.3 to 1, which translates into an amplification factor of 25.25 dBon the sound pressure level (SPL) scale. The middle ear acts as a transformer, bychanging the energy collected by the eardrum into greater force and less excursion,thereby matching the impedance of the air to the impedance of the inner ear’s fluid.Because the fluid of the inner ear is virtually incompressible, provision for reliefof the pressure produced by the movement of the footplate of the tapes is providedby the interaction of oval window and the round window, with the fluid motionfrom the oval to the round window being transmitted through the cochlear duct.

218 10. Physiology of Hearing and PsychoacousticsFigure 10.4. A cross section of the organ of Corti: (A) low magnification; (B) highermagnification. (From A. T. Rasmussen, 1947. Outlines of Neuro-Anatomy, 3rd edition.Dubuque, IA: Wm. C. Brown Company.)

10.2 The Mechanism of Hearing 219When the footplate of the stapes pushes into the perilymph of the scala vestibuli,the vestibular membrane (or membrane of Reissner) bulges into the cochlear duct,causing movement of the endolymph within the cochlear duct and displacementof the basilar membrane. Von Békésy’s experiments with cochlear models led himto formulate a theory that the displacement of the basilar membrane is in the formof a traveling wave that proceeds from the base to the apex of the cochlea. Themaximum amplitude of the wave occurs at a point along the basilar membranecorresponding to the frequency of the stimulus, i.e., each point of the basilarmembrane corresponds to a specific value of the simulating frequency.The cilia of the hair cells are embedded in the gelatinous tectorial membrane,so that when the basilar membrane is displaced, it generates a “shearing” forceon the cilia. The sidewise motion of the cilia creates an alternating electricalcurrent, also referred to as the cochlear microphonic (CM), cochlear potential,orthe Wever-Bray effect 1 . The deflection of the hair cells triggers responses in theneurons connected to the hair cells. Impulses are borne along the nerve fibers tothe main trunk of the cochlear portion of the VIIIth nerve and onward to the brain.This is how the cerebral cortex eventually “hears” the vibrations that strike theeardrum.Research over the past decade indicates that the cochlea does not act passively.Active processes occur that indicate that energy is being added to the cochleathrough mechanisms that are not yet fully understood. More energy is contained inthe cochlea than that from the sound going into it. One phenomenon that has beenidentified is that of otoacoustic emission, which is a sound in the external ear canalbelieved to have originated from vibrations within the cochlea and propagatedback through the middle ear. Otoacoustic emissions can be measured by placinga miniature microphone in the ear canal. A spontaneous otoacoustic emissionis identified as a constant low-level sound that occurs spontaneously in half ofnormal ears. When a high-level click is introduced to the ear, an evoked otoacousticemission occurs some 5 msec later as a low-level sound. Also, when two differenttones are presented at high levels to a normal ear, the otoacoustic emission occursin the form of new tones generated at frequencies other than the two originalfrequencies. These new tones are termed distortion products.When the sound striking the eardrum is sufficiently loud, the middle ear musclescontract reflexively. This acoustic reflex occurs as a contraction in the stapediusmuscle, which results in a pull against the ossicular chain and a reduction in theenergy transmitted through the oval window into the perilymph in the vestibule.The largest amount of reduction in sound due to acoustic reflex—ranging approximatelyfrom 20 to 30 dB—occurs for low frequencies. Above 2 kHz this acousticreflex is fairly negligible.As we mentioned previously, a line of hair cells in the organ of Corti sensessense the movements of basilar membrane. Each hair cell holds fine rods of protein,i.e., the stereocilia, hinged at their respective ends. When the basilar membrane1 So named after the two investigators who discovered in 1930 that the speech delivered to a cat’s earcould be understood when the CM signal was picked up from the cochlear nerve and amplified.

220 10. Physiology of Hearing and Psychoacousticsmoves, these stereocilia sway under the impact. The deflections initiate a chain ofelectrochemical events that generate electrical spikes or action potentials in thecells of the spiral ganglion. These signals are transmitted to a relay station calledthe cochlear nucleus in the brain stem. The information flows through other nucleialong the path to the auditory cortex, the portion of the brain that processes auditoryinformation.Within this cellular circuitry, the frequency of a sound is encoded by two mechanisms.The first mechanism is a place code that indicates the location along thetapered basilar membrane that moves the greatest distance. Stereocilia on the haircells respond to this displacement and produce action potentials among the nearestspiral-ganglion neurons. The other mechanism is a temporal code that is createdwhen neurons become synchronized (or phase-locked) into the period of an acousticwave. In normal hearing, neural responses can readily match frequencies upto about 1 kHz, but this phase-locking capability declines progressively at higherfrequencies. It is conceivable that perception of frequency is based on some combinationof phase and temporal orders, with the temporal code being effective onlyat lower frequencies.10.3 Hearing LossNearly a quarter of the population between the 15 and 75 years of age suffer hearingimpairment. Impaired hearing, which is often caused by infectious diseases oroverexposure to loud noise or simply the process of aging, is common enough tobe on a par with the onset of poor vision. When hearing loss occurs in early childhood,its consequences become more obvious than when it occurs in adulthood. Achild’s progress in learning and developing social relationships may be hinderedand the child may even be deemed “not too bright” if professional help and guidanceare not forthcoming. The primary problem of hearing loss, regardless of theage of the affected individual, is a diminution of a person’s ability to understandspeech.Even milder forms of hearing loss early in life can generate great difficulty,particularly for children who developed within normal limits but are not doingwell in school, due to their being inattentive. Such moderate hearing losses arenot uncommon and may even be on the increase due to heightened exposure to“rock” music. When a mild hearing loss is corrected, the child often becomes“like a different person.” Fortunately, many of the hearing impaired can be helpedthrough the use of hearing aids.The gradual waning of hearing loss affects adults in a more underhand manner.Most people with age-induced or noise-induced hearing impairment first losehearing acuity at high frequencies, making it difficult for them to distinguish consonants,especially s versus f , and t versus z. Such persons must strain harder tounderstand conversations. Going to the movies, listening to lectures, conversingwith friends and other pleasures become stressful chores. This can result in anindividual’s becoming withdrawn from his friends and relatives. Some of these

10.3 Hearing Loss 221patients can be helped through counseling and rehabilitation, but no cure exists formost cases of sensorineural deafness.Hearing loss falls into two principal categories: conductive and sensorineural.Conductive hearing loss occurs from any condition that impedes the transmissionof sound through the external or the middle ear. Sound waves are not transmittedeffectively to the inner ear because of some blockage in the auditory canal,interference in the eardrum, the ossicular chain, the middle ear cavity, the ovalwindow, the round window, or the Eustachian tube. For example, damage to themiddle ear, which carries the task of transmitting sound energy effectively, or to theEustachian tube, which sustains equal air pressure between the middle ear cavityand the external canal, may result in mechanical deficiency of sound transmission.In pure conductive hearing loss, no damage exists in the inner ear or the neuralsystem. Conductive hearing losses are generally treatable.Sensorineural deafness, which is a far more accurate term than the ambiguousterms “nerve deafness” and “perceptive deafness,” describes the effect of damagethat lies medial to the stapedial footplate—in the inner ear, the auditory nerve,or both. In the majority of cases sensorineural deafness is not curable. The term“sensory hearing loss” is applied when the damage is localized in the inner ear.Applicable synonyms are “cochlear” or “inner-ear” hearing loss. “Neural” hearingloss is the proper terminology to describe the result of damage in the auditory nerveproper, anywhere between its fibers at the base of the hair cells and the auditorynuclei. This category also encompasses the bipolar ganglion of the eighth cranialnerve.Mixed hearing loss results from conductive hearing loss accompanied by asensory or a neural (or a sensorineural) in the same ear. Otologic surgery mayhelp in cases of mixed hearing loss in which the loss is primarily conductiveaccompanied by some sensorineural damage of a lesser degree.Functional hearing loss, which occurs far less frequently than the hearing losstypes described above and presents a greater diagnostic challenge to clinics, denotesthe condition in which the patient does not seem to hear or to respond, yetthe handicap cannot be attributable to any organic pathology in the peripheral orthe central auditory pathways. The basis for this type of hearing difficulty may becaused by entirely emotional or psychological etiology. Psychiatric or psychologicaltherapy may be called for, rather than otological treatment.Central hearing loss, orcentral dysacusis, remains mystifying to otologists,although information about this type of hearing defect is accumulating. Patientssuffering this type of condition cannot interpret or understand what is being said,and the cause of the difficulty does not lie in the peripheral mechanism but somewherein the central nervous system. In central hearing loss, the problem is nota lowered pure-tone threshold but the patient’s ability to interpret what he or shehears. It is obviously a more complex task to interpret speech than to respond topure-tone signals; consequently, the tests necessary to diagnose central-hearingimpairment must stress measuring the patient’s ability to process complex auditoryinformation. It requires an extremely skilled, intuitive otologist to make anaccurate diagnosis.

222 10. Physiology of Hearing and Psychoacoustics10.4 Characteristics of HearingIf the sound is audible, the amplitude of the sound is said to be above threshold; andif the sound is inaudible, the amplitude is considered to be below threshold. Theamplitude of the sound at the transition point between audibility and inaudibility isdefined as the threshold of hearing. When sound amplitude exceeds threshold, thesound is processed and perceived as having certain qualities including loudness,pitch, and a variety of other perceptive traits such as information. The study ofauditory perception in relation to the physical characteristics of sound defines thefield of psychoacoustics.SensitivityThe ear is not equally sensitive to all frequencies. The absolute sensitivity of theear, defined by its threshold, depends on a variety of factors, the most important ofwhich is the sound pressure level and the frequency of the sound. The resonanceof the ear canal, the level effect of the ossicles, and the difference between thesurface area of the eardrum and that of the stapes footplate all affect the intensityof the sound that actually penetrates the cochlea.An audiometer which generates signals of varying frequency and intensity isused to measure an individual’s hearing sensitivity. The signals produced by theaudiometer can be directed either to earphones or to a loudspeaker in an anechoicchamber. As it is far more difficult to ascertain the intensity of the sound at thelevel of the cochlea, and such a determination would not accurately representhow well an individual hears under normal circumstances, we generally specifyhearing sensitivity in terms of thresholds for sounds of various frequencies ofwhich sound pressure levels were determined in a sound field without the listenerpresent. Figure 10.5 maps the hearing sensitivity of the normal young humanear over a range of frequencies. The solid curve, referred to as the minimumaudible field, or MAF, describes the minimum intensities that can be detectedwhen the listener is positioned before a loudspeaker at a prescribed distance. Bothof the listener’s ears are stimulated simultaneously by the sound source (i.e., theloudspeaker).However, most clinical work in audiology entails measurements in reference toa single ear rather than to both ears. This is usually performed by directing the testsignals to the appropriate earphone of a headset rather than to a loudspeaker. Theuse of a headset as opposed to exposure to a loudspeaker considerably modifies thelistening situation. For example, the resonant frequency of the ear canal is shiftedbecause both ends of the canal are sealed in contrast to the situation when the canalis open to the sound field. Moreover, the placement of the earphones may give rise tounwanted physiological noise that can interfere with the detection of low-frequencysounds. Also, the method of calibrating sound from a loudspeaker differs from thatfor calibrating sound from an earphone. Because of these differentiating and otherfactors, the measurement of thresholds through the use of headphones is calledminimum audible pressure, or MAP. The MAP measurements are contrasted with

10.4 Characteristics of Hearing 223Figure 10.5. The area of human audibility. The two lower curves represent the lowest(best) thresholds of hearing of young adults. The solid curve is the minimum audible field(MAF) and the dashed curve is minimum audible pressure (MAP). (From American NationalStandard Specification for Audiometers, ANSI S3.6–1969, revised 1989, AmericanNational Standards Institute. New York: Acoustical Society of America.) The dotted curverepresents the current standard for the audiometric zero. The upper three lines representaverages for sensations of discomfort, tickle, and pain. The ordinates define intensity interms of pressure in dynes/cm 2 , sound pressure level in dB, and power flow in W/cm 2 .MAF measurements in Figure 10.5. The threshold curve for the MAP conditionappears to be several dB higher (i.e., showing lower sensitivity) than the MAFcurve, a situation referred to as the “missing 6 dB.” This can be attributed to thefact that using both ears in a sound field enhances sensitivity, in contrast to listeningwith only one ear under an earphone, and other factors occur such as the diffractionof sound around the head in a sound field, the different resonances of the externalear canal, and so on.The two curves of Figure 10.5 represent thresholds that are two standard deviationsbelow the mean, i.e., the curves represent the thresholds of approximately2.5% of young adults (16–25 years of age) determined by examination to be otologicallynormal. These curves are based on data given in two separate studiesperformed 4 years apart conducted by the National Physical Laboratory in GreatBritain. Tables 10.1 and 10.2 list the mean and standard deviations reported inthese two studies and the data points (two standard deviations below the means)on which the MAP and MAF curves are based.The intensities defining an audiometric zero at each of the standard frequencieson a pure-tone audiometer are represented by the dotted curve of Figure 10.5. Theseintensity values were established by international agreement among scientists as

224 10. Physiology of Hearing and PsychoacousticsTable 10.1. The Means and Standard Deviations and the Data Points (TwoStandard Deviations Below the Means) on Which MAP Curves of Figure 10.5Were Based.Frequency,SPL at 2σHz or kHz Mean SPL, dB σ, dB below mean ANSI, 196980 Hz 61.0 8.0 45.0125 45.5 6.8 31.9 45.5250 28.0 7.3 13.4 24.5500 12.5 6.5 −0.5 11.01 kHz 5.5 5.7 −5.9 6.51.5 8.5 6.1 −3.7 6.52 10.5 6.1 −1.7 8.53 7.0 5.9 −4.8 7.54 9.5 6.9 −4.3 9.06 10.5 9.1 −7.7 8.08 9.0 8.7 −8.4 9.510 17.0 9.0 −1.012 20.5 9.6 1.315 39.0 10.7 17.618 74.0 21.9 a 30.2a Calculated from reported standard of error of the mean. Data on mean sound-pressurelevels and standard deviations (σ ) for 80 Hz through 15 kHz taken from Dudson and King(1952). The data for 18 kHz were taken from Harris and Myers (1971).Table 10.2. The Means and Standard Deviations and the DataPoints (Two Standard Deviations Below the Means) on Which theMAF Curves of Figure 10.5 Were Based. aFrequency,SPL at 2σHz or kHz Mean SPL, dB σ, in dB below mean25 Hz 63.5 8.0 47.550 43.0 6.5 30.0100 25.0 5.0 15.0200 15.0 4.5 6.0500 5.5 4.5 −3.51 kHz 4.5 4.5 −4.52 0.5 5.0 −9.53 −1.5 6.0 −13.54 −5.0 8.0 −21.06 4.5 8.5 −12.58 13.5 8.5 −3.510 16.5 11.5 −6.512 13.0 11.5 −10.015 24.5 17.0 −9.5a Data on mean sound-pressure levels and standard deviations (σ ) taken fromRobinson and Dudson (1956).

10.4 Characteristics of Hearing 225being representative of the average minimum audible sound pressure levels foryoung adult ears and have been incorporated in the standards for audiometriccalibration throughout most of the world. This standard as the result of beingadopted in 1969 by the American National Standards Institute, is referred to asANSI-1969 standard and it is listed with the MAP values in Table 10.1 (AmericanNational Standards Institute, 1969).LoudnessWhile the sensation of loudness correlates to the amplitude of the sound abovethreshold, loudness is not perceived by the human ear in equal measures as theamplitude increased over different frequencies above the threshold. Individualjudgment constitutes the deciding factor in ascertaining the degree of loudness.This had led to the development of equal loudness contours which are curvesconnecting SPL points of equal loudness for a number of frequencies, as judgedby tested listeners. These curves, also called phon 2 curves, are constructed byasking subjects to judge when tones of various frequencies are considered equal inloudness to a 1-kHz tone at a given SPL. The official definition (ANSI, 1973) of thephon specifies binaural (two-ear) listening to the stimuli in a sound field (AmericanNational Standards Institute, 1986). Equal-loudness contour curves are given insteps of 10 phons in Figure 10.6, with the dashed MAF curve from Table 10.2included in the plot as a threshold reference.As an example of how humans perceive sound, consider a 30-Hz tone at 95 dBSPL. It would be judged by a typical listener as being as equally loud as a 1000-Hztone at 70 dB SPL or a 5000-Hz tone at 65 dB SPL. As sound is steadily increasedin intensity above the threshold, it will eventually cause the listener to experiencephysiological discomfort. A further increase in the intensity produces a ticklingsensation in the ear, and an additional increase in intensity causes the listener toexperience pain. These three levels constitute, respectively, the thresholds of discomfort,tickle, and pain, which are represented by the upper three lines of Figure10.5. While these threshold values of 120, 130, and 140 dB SPL represent statisticalaverages for young adult ears, different individuals have different tolerancethresholds, but these values do not differ markedly from the statistical averages.Thus, in Figure 10.5, the region between the discomfort and the audiometric zeroconstitutes the usable dynamic range of hearing for humans.PitchThe sensation of pitch is obviously related to the frequency of the tone. The actualpitch of a sound is affected by other factors, including the sound pressure level andthe presence of component frequencies. Pitch perception is a complex process, onethat is not yet fully understood. Pitch elicited by some sounds may evoke the same2 A phon is a unit of loudness that, at the reference frequency of 1 kHz, is equated to the decibel scale.

226 10. Physiology of Hearing and PsychoacousticsFigure 10.6. Equal loudness contours. (From Peterson, A. P. G. and E. E. Gross. Handbookof Noise Measurement. 1980. Concord, MA: General Radio.)aural response whether or not the fundamental frequency is present. The averageadult male voice carries a fundamental between 120 and 150 Hz and that of atypical adult female lies between 210 and 240 Hz. Yet we generally find it easyto distinguish between male and female voices even though the telephones doesnot transmit frequencies much lower than 300 Hz. Somehow we are able to atoneaurally for the fundamental frequency missing from the signal passing through thetelephone receiver.The spectrum of a sound generates a psychological sensation of quality. Thispermits us to distinguish the difference, say, between a trumpet and an Englishhorn playing the same note. This is because of the differences in their respectivesound spectra (i.e., the frequency content or the presence of overtones), which, inturn, are functions of the complex vibrations and the resonance modes inherentin their respective structures. We are also able to discern different speech soundsbecause of the differences in the sound spectra. Even over the telephone, individualvoices are recognizable because of the differences in their sound spectra.MaskingMasking is said to have occurred when the audibility of a sound is interfered withby the presence of noise or other background sound. The “cocktail party” effect,

10.5 Prediction of Speech Intelligibility: The Articulation Index 227which makes it difficult to carry on a private conversation against a backdrop ofother people’s chatter, is a familiar example of masking. Speech becomes unintelligibleby the presence of excessive background noise. Although masking is almostalways undesirable, broadband noise may be purposely introduced into an officeenvironment to make conversation unintelligible to potential eavesdroppers in anadjacent office. Because most of the intelligence in speech is generally containedin the frequency range between 200 Hz and 6 kHz, noise in that frequency rangeis most objectionable in terms of speech masking. But excessively loud noise inany frequency band can adversely affect speech intelligibility by causing such anoverload of the auditory system that one cannot effectively discriminate speechfrom the prevailing total signal. Consonants essential to conveying verbal informationtend to be pronounced softly, so they become readily indiscernible in thepresence of noise.A person speaking normally produces an unweighted sound level of 55–70 dBat 1 m. It is more taxing for that person to speak more loudly for a sustained time.A typical maximum voice effort, in the form of a shout, produces about 90 dB at1 m. Speech intelligibility generally improves when the speaker and the listenerare near each other and if the speaker increases the signal-to-noise (S/N) ratio bytalking louder. Maximum intelligibility usually can be obtained if the unweightedlevel of the speech is between 50 and 75 dB at 1 m from the speaker. Speaking moreloudly does not always guarantee greater intelligibility, even though the S/N ratio(defined as the intensity of the signal divided by the intensity of the noise) maybe increased, because the formation of speech sound above 75 dB may degradesufficiently that there is little or no improvement in intelligibility. If a listener isfamiliar with the words and the dialect used, intelligibility will be greater. It is forthis reason that critical communications, particularly those of air controllers, arebased on a limited vocabulary. In ordinary face-to-face conversation, the listenerhas the additional luxury of making out the context of the words by observing thespeaker’s facial expressions and gestures.10.5 Prediction of Speech Intelligibility:The Articulation IndexIn order to assess the effect of noise on speech communication, it is necessaryto conduct speech-intelligibility tests with actual speakers and listeners in thepresence of interfering noise. The test materials may be sentences, digits, disyllabicwords, monosyllabic words, or nonsense syllables. The listeners are scoredaccording to the percentage of the speech materials heard correctly. The backgroundinterfering noise is generally recorded and played back in the testinglaboratory.From such experiments came the realization that speech intelligibility is a functionof the intensity and the frequency characteristics of the interfering noise.Regarding the S/N ratio, Licklider and Miller stated that the S/N should exceed

228 10. Physiology of Hearing and PsychoacousticsFigure 10.7. The effect of white noise on the thresholds of detectability and intelligibilityof running speech (Hawkins and Stevens, 1950).6 dB for satisfactory communication, although the presence of speech may bedetected for S/N as low as −18 dB. If the intensity of the signal (speech) exceedsthe noise, the sign of the S/N value is plus; conversely, a negative value of S/Nindicates that the noise is more intense than the signal.Figure 10.7 maps the effect of white noise on the thresholds of detection andintelligibility of running speech. According to this figure, which was developedby Hawkins and Stevens based on extensive tests making use of running speechand white noise (Hawkins and Stevens, 1950), the threshold of intelligibility occurswhen the level of the speech exceeds the noise level by about 6 dB (S/Nof 6 dB). As the sound pressure level of the noise is increased above this value,the threshold of intelligibility is proportionally increased so that the S/N value of−6 dB remains fairly constant over a wide range of intensities. For other kindsof speech materials and different masking noises, the relationship between thethreshold of intelligibility and the level of the interfering noise may not necessarilyremain the same. At an S/N of −18 dB, running speech can be detected but notunderstandable.Other but simpler methods have been developed for measuring the effect ofinterfering noise on the intelligibility of speech. A principal method of predictingspeech intelligibility is the articulation index, or AI, which is a value that rangesfrom 0.0 to 1.0 and represents the proportion of the speech spectrum that occursabove the noise. French and Steinberg of the Bell Laboratories developed theconcept of articulation index on the basis of the assumption that most of the intelligencein speech is contained in the frequency bands between 200 and 6100 Hz.The articulation index can be calculated from the levels of the masking signal andthe speech level in the frequency bands. The contribution of each frequency bandto speech intelligibility is defined as 12 dB plus the sound level of the speech

10.5 Prediction of Speech Intelligibility: The Articulation Index 229Table 10.3. Weighting Factors as a Function of Center Frequencyfor One-Third Octave Band-Based Calculation of ArticulationIndexes. Speech Levels Given in the Second Column are Those for aTypical Male Voice at 1 m.Center frequency (Hz) Speech level (+12 dB) Weighting factor200 67 4250 68 10315 69 10400 70 14500 68 14630 66 20800 65 201000 64 241250 62 301600 60 372000 59 372500 57 343150 55 344000 53 245000 51 20less the masking level. Each frequency band’s contribution is limited to the rangebetween 0 and 30 dB. The sound level of the speech signal is based on a long-termenergy average in each frequency band, and each frequency band contribution ismultiplied by a weighting factor. The sum of the weighted contributions dividedby 10,000 yields the AI.Table 10.3, based on the division of the speech spectrum into one-third octaves,provides the data necessary to calculate AI. The first column lists the center frequencyof the one-third octave, the second column gives the typical male voicelong-term average speech level plus 12 dB at 1 m distance. The weighting factorfor each one-third octave band is listed in the third column. If the masking noisehas been measured only in full octave bands, Table 10.4 may be used instead ofTable 10.3.Table 10.4. Weighting Factors for One Octave Band-BasedCalculation of Articulation Indexes. This is for the Typical MaleVoice Level at 1 m.Center frequency (Hz) Speech level (+12 dB) Weighting factor250 72 18500 73 501000 78 752000 63 1074000 58 83

230 10. Physiology of Hearing and PsychoacousticsTable 10.5. Calculation for the Articulation Index in the Sample Problem.Center Speech Weighting Noise SL − N = WF ×frequency (Hz) level (SL) (+12 dB) factor (WF) (N) DIFF DIFF250 72 18 47 25 450500 73 50 47 26 13001000 78 75 47 30 3 22502000 63 107 47 16 17124000 58 83 47 11 913Example Problem 1: Calculation of AILet us compute the articulation index of a male voice speaking at a normal level1 m from the listener in the presence of pink noise that contributes 47 dB in eachoctave band.SolutionTable 10.4 is used for this calculation. The 47-dB octave-band noise level is subtractedfrom the values in the second column; and the difference (up to a maximumof 30 dB) is multiplied by the weighting factors of the third column. The resultingweighted contributions are added and divided by 10,000, yielding the articulationindex. Table 10.5 below gives details of the calculations, with the values given inthe fifth column produced by subtracting 47 dB from the speech level of the secondcolumn, and each of the values in the fifth column is multiplied by the weightingfactor of the third column to yield the figures listed in the sixth column.Articulation index (AI) = (450 + 1300 + 2250 + 1712 + 913)/10000 = 0.6625The articulation index is 0.6625, or 66.25%.10.6 Speech-Interference Level (SIL)Measurements to obtain data for articulation indexes require special laboratoryequipment for determination of S/N in a number of frequency bands. A simplerprocedure for estimating the effect of noise on verbal communication makes useof octave-band levels as measured in a typical noise survey. The parameter thatis called the speech-interference level, abbreviated SIL, can be obtained by computingthe arithmetic average of octave-band levels in the three octave bands of600–1200, 1200–2400, and 2400–4800 Hz. However, the current practice usesthe arithmetic level in the “preferred” octave bands with center frequencies at 500,1000, and 2000 Hz. Speech-interference level defined thusly is referred to as PSIL.3 The value of the following expression (Speech Level + 12 dB—Noise Level) must fall between0 and 30.

10.7 Prosthetics for Hearing 231Table 10.6. PSIL (in dB) at Which Effective Speech Communication isBarely Possible.Normal Raised Very Loud ShoutingDistance (m) M F M F M F M F0.3 68 63 74 69 80 75 86 811 58 53 64 59 70 65 76 712 52 47 58 53 64 59 70 653 48 43 54 49 60 55 66 614 46 41 52 47 58 53 64 59The speech-interference level (PSIL = 68 dB) has been identified as the levelat which reliable speech communication is barely possible in a normal male voiceat a distance of 0.3 m (or 1 ft) outdoors. If a male speaker talks in a raised voice, avery loud voice, or in a shout, the speech interference levels have been identified,respectively, as PSIL = 74, 80, and 86 dB. A female speaker, on the average, hasPSIL levels 5 dB less than the corresponding values for a male. Table 10.6 liststhe PSIL (in dB) at which effective speech communication is barely possible. Thetable is based on minimally reliable communication, at which about 60% of thecommunication of uttered numbers and words out of context can be discerned. Inorder to roughly approximate PSIL in terms of dBA, 7 dB can be added to thevalues of PSIL.Example Problem 2: SILBackground noise levels for an industrial plant were measured to be 62, 65, and74 dB, respectively, in the 500-, 1000-, and 2000-Hz center-frequency bands. Whatare the implications for speech interference at a distance between a speaker and alistener standing 1 m apart?SolutionTo solve this problem, the arithmetic average of the noise level in three bandsare first determined. This will be (62 + 65 + 74)/3 = 67 dB. From Table 10.6 weestablish that reliable speech is barely possible for a male, speaking in a raisedvoice or a female speaking in a very loud voice.10.7 Prosthetics for HearingHearing AidsA conventional hearing aid works by amplifying sound and delivering that amplifiedsound to the eardrum. Sound is received by a miniature microphone thatconverts acoustic signals into electrical signals that is amplified and then relayed to

232 10. Physiology of Hearing and Psychoacousticsa receiver (i.e., a mini-loudspeaker) that converts the signals into amplified sound.Current hearing aids evolved from a long history of continuously improving technology,from the bulky electrical hearing aids of the 1930s (which were poweredby rather large multiple batteries) to the smallest models inserted into the ear canal.The electronic part of a hearing aid can be based on different technologies: analog,programmable, and digital. Each of these technologies has advantages anddisadvantages, but the choice rests on the optimum benefit to the user.In the analog models, variable resistors perform the trimming adjustments, andany changes of resistor value will change the properties of the circuit accordingly,thus providing a change in the frequency response, usually prescribed by a qualifiedaudiologist, to meet the needs of the user. In principle, the same applies toa programmable hearing aid. In this case the total variable resistor componentconsists of a ladder of resistors, each attached to a digital switching element. Settingthe individual switches either “on” or “off,” thereby bypassing some of theresistors in the ladder changes the total resistor value. These switches or gates arethe fundamental building blocks of all digital circuits. CMOS are used to producegates that function as long as there is electrical power. But in order to avoid losingthe switch settings (and hence the adjustment), a common solution is to constructa secondary set of switches, using a technology that enables the switch settingsto be retained even when there is no battery attached. This technology known asE2PROM (programmable read only memory) provides the nonvolatility necessaryto retain the settings.By their very nature digital adjustments are discreet, i.e., the resistor can onlytake on a finite number of values. Although there is theoretically no limit onhow fine the resolution can be, the penalty is a larger circuit using more power.The resolution is therefore set to be sufficiently adequate for the parameter to beadjusted, typically in the region of 0.5–5 dB. Programmable hearing aids mayalso use digital switching to select functions (ON/OFF, M/T). The M/T functionrepresents a choice between the microphone (M) and telephone (T) mode. In thetelephone mode, induction signals are received from the telephone receiver ratherthan the acoustic signal, so as to provide better coupling and eliminate feedback.A digital volume control may be used to work in steps, typically 1 dB.In digital hearing aids, an analog-to-digital converter reads the analog voltagesignal from the microphone and converts it to a digital signal. The digital signalrepresents the original sound as a series of number, which then can be manipulatedmathematically by the digital signal processor (DSP). Once digitized, the signal ismore robust and is no longer subject to electronic noise and distortion. To obtainthe highest possible signal quality the input signal is sampled at a very high rate(500 kHz–1 M Hz) before the digitization. The sampling rate at the input affects thesignal quality, so as the sampling rate increases, so does the quality of the signal.Another important parameter for sound quality is the resolution, which determinesthe initial precision of the signal and the precision that later can be achieved duringthe mathematical manipulations that constitute signal processing. After the sampling,the converted signal is then ready for manipulation by a specially designeddigital signal processor (DSP). The DSP is programmed by an audiologist on a

10.7 Prosthetics for Hearing 233computer to provide the frequency equalization curve on the basis of the resultsfrom testing the patient’s hearing. Algorithms in the DSP can continuously dividesounds into frequency channels and emphasize the higher frequencies containingvital consonant sounds in speech over the distracting rumble of low-frequencynoise. Algorithms also manage noise: owing to the fact that speech sound intensitiescan change radically in a millisecond whereas noise is more acousticallystable over a comparatively longer time. On a time basis, DSP can reduce the levelsof continuous sounds such as traffic noise and household appliances. And it simultaneouslyreadjusts when changes occur, restoring amplification when shorterduration sounds are detected. In relatively quiet surroundings, the digital algorithmcan detect the consistency of softer environmental sounds from ventilationsystems and appliances, and it also automatically reduces amplification in theappropriate frequency range, immediately restoring proper levels when the soundpattern changes. Digital hearing aids carry the disadvantage of considerably highercosts and may not provide enough amplification as the other types of hearing aids.Implantable hearing aids, referred to by the acronym, IHA, can be used bypeople having sensorineural hearing loss but a healthy middle ear. The IHA, whichis still in the stages of final development, converts sound to vibrations inside themiddle ear. This gives the IHA several advantages of traditional hearing aids. Ifthe bones of the middle ear can be directly used, sound quality may be improvedalong with much less feedback. Also, there are no external components to the IHA,which is, after all, completely implantable.Three versions of IHA are being tested in human trials. Either the receiver,which collects the sound energy, or the entire hearing device is surgically insertedinto the middle ear. The surgical procedure is as follows: skin and tissues aroundthe ear are laid back, and a magnet (which may no may not include electronics) isnestled into the bone behind the ear. A component is then attached to one of themiddle ear bones. One prototype is called the Envoy Totally Implantable HearingRestoration System. This device utilizes the eardrum as the microphone that sendssound energy to the piezoelectric crystal in the middle ear. Two other prototypesuse electromagnetic technology. The Vibrant Soundbridge (made by SymphonixDevices, Inc.) has already received an Investigational Device Exemption (IDE)from the Federal Drug Administration (FDA). IDE represents the preliminary stagein receiving FDA approval. Another system, Otologies, is also in the IDE stage ofthe FDA process. Both of these systems are partially implantable; only the receiver,consisting of the magnet and transducer coil, is implanted. The microphone andamplifier are worn externally. These devices may be more suitable for a widerrange of hearing losses than the one using piezoelectric technology, which can beused by those having not more than a moderate (50 dB) hearing loss.Cochlear ImplantsIn Section 10.2 we have examined how the cochlea produces signals to the cochleanucleus. When air cells become so damaged that they cannot be stimulated cellsof the spiral ganglion, hearing is lost. Without regular activity, the portion of

234 10. Physiology of Hearing and Psychoacousticsthe ganglion that receives signals, the dendrite, may atrophy and the cells maydie. However, and fortunately, even in the case of complete hearing loss, someganglion cells survive and remain connected to the appropriate frequency receptorsin the cochlear nucleus. If the electrical action from implant electrodes can causeaction potentials among the remaining cells, then hearing can be restored. Also,if multiple groups of neurons can be compelled to respond to low-, middle-, andhigh-frequency components of the cochlea, then perception of speech will berestored.The next question is: how many channels of portions of the frequency spectrumare necessary to encode speech? Dorman, Louizon, and Rainey determined throughthe use of bandpass filters that as few as four channels of simplified audio signalsto normal-hearing listeners were required to achieve a 90% comprehension rateof words spoken in simple sentences (Dorman Wilson, 2004). Eight channelsallowed these listeners to identify 90% comprehension of spoken isolated words.With background noise, more channels were needed to retain this performance,and the more channels used, the better the comprehension.It is obvious that hair-cell failures engender a roadblock between the peripheraland central auditory systems, resulting in deafness. Cochlear implants restores thelink, though bypassing the hair cells to stimulate the direction of the cell bodies inthe spiral ganglion. A cochlear implant consists of five components, only two ofwhich are inside the body. In Figure 10.8, an external microphone picks up soundsand directs them to a sound processor that is enclosed in a case behind the ear. Theprocessed signals are sent on to a high-bandwidth RF transmitter, which then relaysthe information through a few millimeters of skin to a receiver/stimulator that hasbeen surgically implanted in the temporal bone above the ear. The signals pass onto an array of electrodes inside the cochlea. Target cells on the spiral ganglion aresegregated from the electrodes by a bony partition.Continuous interleaved sampling, or CIS, is a strategy used to covert signalsinto a code for stimulating the auditory nerve. It begins by filtering a signal intofrequency bands (16 bands or more). For each band, the CIS algorithm converts theslow changes of the sound envelope into amplitude-modulated groups of biphasic(i.e., having both positive and negative values) pulses at the electrodes. The processorsenses the information from low-frequency channels to the electrodes in theapex and information for high-frequency channels to electrodes in the base of thecochlea. Thus, this setup sustains the logic of the frequency mapping in a normalcochlea.The efficacy of the cochlea, however, depends on a number of factors, amongthem are the number and location of the surviving cells in the ganglion, the spatialpattern of current flow from the electrodes, and the manner in which the neurons inthe brainstem and cortex can encode frequency. If the period of deafness is a longone and only a few cells survive in the spiral ganglion, the electrode stimulation isless likely to convey frequency-specific information to the cochlear nucleon andcortex. Then there is a possibility of surviving cells clustered at one location in theganglion at the cortex, which results in the lack of range of frequencies necessaryfor speech understanding.

10.7 Prosthetics for Hearing 235external transmitterimplantedreceiver/stimulatormicrophone, battery packand speech processorFigure 10.8. The components of a cochlear implant. There are five main components,only two of which are inside the body. A microphone above the ear receives sound wavesthat are directed to a tiny computer behind the ear. The computer transforms the inputinto specifications for stimuli to be conveyed to the implanted electrodes. The disk-shapetransmitter uses high-bandwidth radio waves to send these instructions to a receiver justunder the skin. The receiver converts the instructions into electrical stimuli and relays themto the appropriate electrode in the cochlea, which, in turn, excites neurons in the auditorynerve. (Courtesy of Michael F. Dorman and American Scientist.)Adults who lost their hearing and later receive a cochlear implant can associatethe new stimulation patterns with their recall of what speech should sound like.Children born deaf do not have this advantage, but it has been found that congenitallydeaf children who received cochlear implants during their first or second yearcan learn spoken language as well as, or almost as well as, children with normalhearing. Children receiving implants later in life have greater difficulty in copingwith signals from an implant.Ongoing cochlear implant research is now focused on the combination of electricand acoustic stimulation (EAS). A number of hearing-impaired people have someability to hear low frequencies but little or no sensitivity to higher frequencies.When an electrode array is inserted about two thirds of the way into the cochlea,hearing at 1 kHz and above may possibly be restored by electrical stimulation.

236 10. Physiology of Hearing and PsychoacousticsGene Therapy: Formation of New Cochlear Hair CellsIn 1998 the geneticist Huda Y. Zoghbi of the Baylor College of Medicine discoveredthe key to generation of new hair cells in a gene called Atohl, first discovered infruit flies. Variations of this gene have since been discovered in almost all speciesof animals. During fetal development, the gene converts some cells in the ear intohair cells. In other ear cells, called supporting cells, its activity is suppressed.Researchers have demonstrated that working in laboratory dishes, the gene couldconvert supporting cells into hair cells.In 2003, Yehoash Raphael and Kobei Kawamoto at the University of MichiganMedical School reported that inserting the gene into live guinea pigs producedthousands of new hair cells (Kawamoto et al., 2003; University of Michigan HealthSystem, 2003). But in those experiments, the researchers did not deafen the animalsfirst. Later they did, using toxic chemicals to kill the hair cells in both ears of tenguinea pigs. Microscopic images taken 3 days later confirmed that all the hair cellswere destroyed.On the fourth day, they used gene therapy with a viral vector to insert the Atohlgene into the guinea pigs’ left ears. Within 2 months, new hair cells appearedin treated ears, but not in the untreated right ears. In order to determine whetherthe new hair cells were functional, the team applied tests of auditory brainstemresponse to measure the guinea pigs’ ability to hear sounds. In effect, they observedincreases in brain activity when they exposed the animals to noises, which indicatesthat their ability to hear was at least partially restored.Raphael and his colleagues are presently trying to determine how good therestored hearing is. To indicate whether the guinea pigs can hear and how well,they are working with a psychologist who is an expert at training animals todisplay various behaviors. They are trying to determine, for example, whetherthe animals can differentiate between loud and soft sounds and between differentfrequencies. They are also studying animals that have been deafened by othermeans, older animals and animals that have been deaf for longer periods of timebefore treatment begins. If these experiments turn out to be successful, the studiesnecessary to ensure safety and efficacy must be conducted before the techniquecan be tried on humans, and this phase may take the better part of a decade.Direct Drive Hearing System (DDHS)A conventional hearing aid operates by amplifying sound and delivering the soundto the eardrum. From there, the amplified sound passes through the three hearingbones on its way to the cochlea. The direct drive hearing system (DDHS) undergoingtesting at the Department of Otolaryngology and Virginia Merrill BioedelHearing Research Center at the University of Washington replaces this acousticpath with an electromagnetic one (Von Ilberg et al., 1999). Instead of producinghigh-volume sound, this hearing aid is attached to an electromagnetic coil that fitsinside the earmold. The coil is used to drive a magnet that is attached to the thirdhearing bone. This system effectively bypasses the ear canal, the eardrum, and

10.8 Hearing in Animals 237the hearing bones. It is claimed that the fidelity at the third ear bone is improvedthrough the use of this system. Outpatient surgery is required to attach the magnetto the hearing bone. The surgery is performed under local anesthetic and requiresabout an hour.One of the advantages of the DDHS is that it eliminates the feedback problemthat often occurs with a loose earmold of a conventional hearing aid. There is nofeedback because the DDHS does not produce amplified sound. It is also claimedthat sound quality is improved by the electromagnetic system and that earmoldsdo not have to be fitted so tightly. The original study with five volunteers wassuccessful, and FDA authorized a larger study entailing 100 patients. The DDHSis initially targeted at patients with a moderate hearing loss.Drug TherapyDrug delivery systems are being integrated into newer designs of cochlear implants.The purpose of these drugs is to arrest the shriveling or demise of remaining haircells and neural structures in the cochlea and to promote the growth of neuraltentacles (called neurites) from spiral ganglion cells toward the electrodes. Theneurites help each electrode to function as an independent channel of stimulation.Recent experiment with deafened guinea pigs demonstrated that injecting brainderivedneurotrophic factor and ciliary neurotropic factor could increase the survivaland, more importantly, the sensitivity of spiral ganglion cells.A second approach is to block apoptosis, the normal process of cell death followinginjury. Self-destruct messages can be triggered by a number of events, forexample, acoustic trauma or ototoxic drugs that work through mitogen-activatedprotein kinase (MAPK) signaling pathway. The pathway can be blocked at variouspoints, thanks to a protein called c-Jun N-terminal kinase (JNK). A peptideinhibitor, developed by a multinational team at the University of Miami, targetsthis enzyme. By blocking JNK, this team headed by Jing Wang and Thomas VanDe Water, prevented hair-cell death and hearing loss following acoustic trauma oradministration to the ototoxic antibiotic neomycin (Wang et al., 2003).10.8 Hearing in AnimalsBecause their lives depend on the acuity of their hearing, many animals hear amuch wider range of frequencies than humans do (cf. Figure 10.9). The averagehearing range for humans is about 20 Hz–17 kHz, but killer whales have respondedto tones with the frequency range of approximately 0.5–125 kHz with a peak sensitivityat about 20 kHz. Odontocetes (toothed whales) can produce sounds for twooverlapping functions: communicating and navigating. Higher frequency clicksprobably function primarily in echolocation. Most sound reception, or hearing,seems to take place through the lower jaw. A killer whale may also receive soundthrough soft tissue and bone surrounding the ear.

238 10. Physiology of Hearing and Psychoacoustics100,000FREQUENCY RANGE, HERTZ10,0001,000100100BAT CAT DOG DOLPHIN GRASSHOPPER HUMANTYPE OF ANIMALFigure 10.9. Range of hearing in animals. Many animals hear a much wider range offrequencies than the human ear can sense.The term echolocation refers to an ability that odontocetes (and a few otheranimals such as bats) possess that enables them to locate and discriminate objectsby projecting high-frequency sound waves and listening for the echoes. The hearingof bats spans a 10 Hz–100 kHz range. Big brown bats (Eptesicus fuscus) producethe frequency range of 10–100 kHz for sonar or for acoustic social communicationand they also hear those ultrasonic frequencies. They have a lower frequency regionof auditory sensitivity from 10 Hz to 5 kHz and may use these lower frequenciesto detect insect prey by passive hearing of the insect’s own sounds. The hearingis tuned to 0.7–1.3 kHz indicating that some specialization of the auditory systemmay underlie the capacity to hear these lower frequencies.An Indian elephant is sensitive to low-frequency tones and could hear as lowas 16 Hz at 65 dB. However, the elephant is insensitive to high-frequency tonesand it generally could not hear above 12 kHz. The high-frequency hearing abilityis the poorest of any mammal yet tested and the failure of the elephant to hearmuch above 10 kHz demonstrates that the inverse correlation between the headsize (i.e., the interaural distance) and high-frequency hearing acuity is valid evenfor the largest of terrestrial mammals.Psychophysical investigations in a number of avian (bird) species over the pastthree decades have added significantly to the knowledge of hearing capabilitiesof this vertebrate group. Behavioral measurements of absolute auditory sensitivityin a wide variety of birds show a region of maximum sensitivity between 1 and5 kHz, with a rapid decrease in sensitivity at higher frequencies. Data accumulatedto date suggest that, in the region of 1–5 kHz, birds show a level of hearing

References 239sensitivity similar in most respects to that found for the most sensitive membersof the mammal class, with avian performance clearly inferior above and belowthis range of frequencies. Possible exceptions to this general picture include theecholocating oilbird (Steatornis caripensis) and growing evidence that pigeons(Columba livia) are sensitive to infrasound at moderate intensity levels.ReferencesAmerican National Standards Institute. 1969. ANSI 1969: Methods for Calculation of theArticulation Index (ANSI S3.5-1969, revised 1989). New York: Acoustical Society ofAmerica.American National Standards Institute. 1989. ANSI 1969: American National StandardSpecification for Audiometers (ANSI S3.6-1969, revised 1989). New York: AcousticalSociety of America.American National Standards Institute. 1986. ANSI 1973: American National PsychoacousticTerminology (ANSI S3.20-1973, revised 1986). New York: Acoustical Societyof America.Crocker, Malcolm J. (ed.). 1997. Encyclopedia of Acoustics, Vol. 2. Parts XI and XII.Dorman, Michael F. and Wilson, Blake S. 2004. The design and function of cochlearimplants. American Scientist 92:436–445.Dudson, R. S. and King, I. H. 1952. A determination of the normal threshold of hearing andits relation to the standardization of audiometers. Journal of Laryngology and Otology66:366–378. [Reproduced in “Forty Germinal Papers in Human Hearing,” Harris, J.Donald (ed.). Groton, C. T. Journal of Auditory Research, 1969; pp. 600–601.]French, N. R. and Steinberg, J. C. 1947. Factors governing the intelligibility of speechsounds. Journal of the Acoustical Society of America 19:90–119.Harris, J. Donald and Myers, C. K. 1971. Tentative audiometric hearing threshold levelstandards from 8 through 19 kilohertz. Journal of the Acoustical Society of America49:600–601.Hawkins, J.E., Jr. and Stevens, S. S. 1950. The masking of pure tones and speech by whitenoise. Journal of the Acoustical Society of America 22:12.Kanzaki, S., Beyer, L. A., Canlon, B., Meixner, W. M., and Raphael, Y. The cytocaud: Ahair cell pathology in the waltzing guinea pig. Audiology & Neural Otology.Kawamoto, Kohei, Brough, Douglas E., et al. 2003. Journal of Neuroscience.Kinsler, Lawrence E., Frey, Austin R., Coppens, Alan B., and Sanders, James V. 1982.Fundamentals of Acoustics, 3rd ed. New York: John Wiley & Sons, Chapter 11.Kleine, Ulrich and Mauthe, Manfred. 1994. Designing circuits for hearing instruments.Siemens Review R& D Special.Kryter, Karl D. 1970. The Effects of Noise on Man. New York: Academic Press.Lipscomb, David M. 1974. Noise: The Unwanted Sound. Chicago, IL: Nelson-Hall.Manci, Karen M., Gladwin, Douglas N., Villella, Rita, and Cavendish, Mary G. 1988.Effects of aircraft noise and sonic booms on domestic animals and wildlife: a literaturesynthesis. Report NERC 88/29 AFE TR 88.14. Washington DC: U.S. Department of theInterior.Miyamoto, R. T., Houston, D. M., Kirk, K. I., Perdew, A. E., and Svirsky, M. A. 2003.Language development in deaf infants following cochlear implantation. Acta Otolaryngologica123:241–244.

240 10. Physiology of Hearing and PsychoacousticsNewby, Hayes A. and Popelka, Gerald R. 1992. Audiology, 6th ed. New York: PrenticeHall.Robinson, D. W. and Dudson, R. S. 1956. A re-determination of equal loudness relationsfor pure tones. British Journal of Applied Physics 7:166–181. [Reproduced by: FortyGerminal Papers in Human Hearing.” Harris, J. Donald (ed.). Groton, CT: The Journalof Auditoroy Research, 1969:600–601.]Sandlin, R. E. 1988. Handbook of Hearing Aid Amplification. Vol. I: Theoretical andTechnical Considerations. Boston: College Hill Press.Sandlin R. E. 1990. Handbook of Hearing Aid Amplification. Vol. II: Clinical Considerationsand Fitting Practices. Boston: College Hill Press.Sataloff, Robert T. and Sataloff, Joseph, 1993. Hearing Loss, 3rd ed. New York: MarcelDekker, Inc.Schow, Ronald L. and Nerbonne, Michael A. 1989. Introduction to Aural Rehabilitation,2nd ed. Boston: Allyn and Bacon.Silman, Shlomo and Silverman, Carol A. 1991. Auditory Diagnosis—Principles and Applications.New York: Academic Press.University of Michigan Health System. 2005. June 2, 2003. Gene therapy triggergrowth of new auditory hair cells in mammals. [On line] http://www.med.umich.edu/opm/newspage/2003/haircells.htmVon Békésy, Georg. 1978. Experiments in Hearing. Huntington, NY: Kreiger.Von Ilberg, C. J., Kiefer, J., Tillein, T., et al. 1999. Electro-acoustic stimulation of theauditory system: New technology for severe hearing loss. ORL: Journal for Oto-Rhino-Laryngology and Its Related Specialties 61:334–340.Wang, J. T. R., Van De Water, C. Bonny, et al. 2003. A peptide inhibitor of c-Jun N-terminalkinase protects against both aminoglycoside and acoustic trauma-induced auditory haircell death, and hearing loss. Journal of Neuroscience 23:8596–8607.Problems for Chapter 101. Find the articulation index for a male speaker at a normal level 1 m from alistener if we have the following background noise spectrum:Center Frequency, HzNoise Level, dB200 42250 39315 44400 46500 48630 38800 301000 261250 201600 232000 182500 153150 144000 105000 10

2. Find the PSIL from the following octave-band noise spectrum.Problems for Chapter 10 241Center FrequencyBackground Noise Level, dB500 551000 632000 723. Find the PSIL from the following octave-band noise spectrum:Center frequency, Hz 63 125 250 500 1k 2k 4k 8kBand-pressure level, dB 52 56 57 62 55 51 50 454. Determine the voice level necessary to effectively communicate of a distanceof 4 ft with a background of(a) 63 dB(b) 72 dB(c) 85 dB5. Why does the U.S. Occupational Safety and Health Administration prohibitimpact noises of 130 dB or more even if the overall sound pressure level is lessthan 90 dB during the course of an 8-h day?6. Express 35 dB I terms of pressure in pascals and in terms of intensity (W/cm 2 ).At 62.5 Hz, is this considered audible for normal hearing? If so, how muchabove the MAP is the sound pressure level?7. How loudly must one speak, in terms of decibels, with a white noise backgroundof 50 dB in order to be understood? How loud must the speech be in order tobe detected, if not necessarily understood?8. At 500 Hz, a 40-dB tone sounds as equally loud as a 5-kHz tone. What is thedB level of that 5 kHz tone?

11Acoustics of Enclosed Spaces:Architectural Acoustics11.1 IntroductionAlthough people have gathered in large auditoriums and places of worship sincethe advent of civilization, architectural acoustics did not exist on a scientific basisuntil a young professor of physics at Harvard University accepted an assignmentfrom Harvard’s Board of Overseers in 1895 to correct the abominable acousticsof the newly constructed Fogg Lecture Hall. Through careful (but by present-daystandards, rather crude) measurements with the use of a Gemshorn organ pipeof 512 Hz, a stopwatch, and the aid of a few able-bodied assistants who luggedabsorbent materials in and out of the lecture hall, Wallace Sabine establishedthat the reverberation characteristics of a room determined the acoustical natureof that room and that a relationship exists between quality of the acoustics, thesize of the chamber and the amount of absorption surfaces present. He defined areverberation time T as the number of seconds required for the intensity of thesound to drop from a level of audibility 60 dB above the threshold of hearing to thethreshold of inaudibility. To this day reverberation time still constitutes the mostimportant parameter for gauging the acoustical quality of a room. The originalSabine equationT = ∑0.049ViSi α iis deceptively simple, as effects such as interference or diffraction and behaviorof sound waves as affected by the shape of the room, presence of standing waves,normal modes of vibration, are not embodied in that equation. Here V is the roomvolume in cubic feet, S i the component surface area and α i the correspondingabsorption coefficient. On the basis of his measurements Sabine was able to cutdown the reverberation time of the lecture hall from 5.6 s through the strategicdeployment of absorbing materials throughout the room. This accomplishmentfirmly established Sabine’s reputation, and he became the acoustical consultantfor Boston Symphony Hall, the first auditorium to be designed on the basis ofquantitative acoustics.243

244 11. Acoustics of Enclosed Spaces: Architectural AcousticsIn this chapter we shall examine the behavior of sound in enclosed spaces, anddevelop the fundamental equations that are used in optimizing the acoustics ofauditoriums, music halls, and lecture rooms. We shall also study the means of improvingroom acoustics through installation of appropriate materials. This chapterconcludes with descriptions of a number of outstanding acoustical facilities.11.2 Sound FieldsThe distribution of acoustic energy, whether originating from a single or multiplesound sources in an enclosure, depends on the room size and geometry and on thecombined effects of reflection, diffraction, and absorption. With the appreciablediffusion of sound waves due to all of these effects it is no longer germane toconsider individual wave fronts, but to refer to a sound field, which is simply theregion surrounding the source. A free field is a region surrounding the source,where the sound pattern emulates that of an open space. From a point source thesound waves will be spherical, and the intensity will approximate the inverse squarelaw. Neither reflection nor diffraction occurs to interfere with the waves emanatingfrom the source. Because of the interaction of sound with the room boundaries andwith objects within the room, the free field will be of very limited extent.If one is close to a sound source in a large room having considerably absorbentsurfaces, the sound energy will be detected predominantly from the sound sourceand not from the multiple reflections from surroundings. A free field can be simulatedthroughout an entire enclosure if all of the surrounding surfaces are linedwith almost totally absorbent materials. An example of such an effort to simulatea free field is the extremely large anechoic (echoless) chamber at Lucent TechnologiesBell Laboratories in Murray Hill, New Jersey, shown in the photographof Figure 11.1. Such a chamber is typically lined with long wedges of absorbentfoam or fiberglass and the “floor” consists of either wire mesh or grating suspendedover wedges installed over (and covering entirely) the “real” floor underneath. Preciselycontrolled experiments on sound sources and directivity patterns of soundpropagation are rendered possible in this sort of chamber.A diffuse field is said to occur when a large number of reflected or diffractedwaves combine to render the sound energy uniform throughout the region underconsideration. Figure 11.2 illustrates how diffusion results from multiple reflections.The degree of diffusivity will be increased if the room surfaces are notparallel so there is no preferred direction for sound propagation. Concave surfaceswith radii of curvature comparable to sound wavelengths tend to cause focusing,but convex surfaces will promote diffusion. Multiple speakers in amplifying systemsauditoriums are used to achieved better diffusion, and special baffles may behung from ceilings to deflect sound in the appropriate directions.Sound reflected from walls generates a reverberant field that is time dependent.When the source suddenly ceases, a sound field persists for a finite interval asthe result of multiple reflections and the low velocity of sound propagation. Thisresidual acoustic energy constitutes the reverberant field. The sound that reaches a

11.2 Sound Fields 245Figure 11.1. Photograph of the large anechoic chamber at the Lucent Bell Laboratoryin Murray Hill, NJ. Dr. James E. West, a former president of the Acoustical Society ofAmerica, is shown setting up test equipment. (Courtesy of Lucent Technologies.)listener in a fairly typical auditorium can be classified into two broad categories: thedirect (free field) sound and indirect (reverberant) sound. As shown in Figure 11.3,the listener receives the primary or direct sound waves and indirect or reverberantsound. The amount of acoustic energy reaching the listener’s ear by any singlereflected path will be less than that of the direct sound because the reflected path islonger than the direct source–listener distance, which results in greater divergence;and all reflected sound undergo an energy decrease due to the absorption of eventhe most ideal reflectors. But indirect sound that a listener hears comes from a greatnumber of reflection paths. Consequently, the contribution of reflected sound tothe total intensity at the listener’s ear can exceed the contribution of direct soundparticularly if the room surfaces are highly reflective.Figure 11.2. Sound diffusion resulting from multiple reflections.

246 11. Acoustics of Enclosed Spaces: Architectural AcousticsFigure 11.3. Reception of direct and indirect sound.The phases and the amplitudes of the reflected waves are randomly distributedto the degree that cancellation from destructive interference is fairly negligible. If asound source is operated continuously the acoustic intensity builds up in time until amaximum is reached. If the room is totally absorbent so that there are no reflections,the room operates as an anechoic chamber, which simulates a free field condition.With partial reflection, however, the source continues to add acoustic energy to theroom, that is partially absorbed by the enclosing surfaces (i.e., the walls, ceiling,floor and furnishings) and deflected back into the room. For a source operating ina reverberation chamber the gain in intensity can be considerable—as much as tentimes the initial level. The gain in intensity is approximately proportional to thereverberation time; thus it can be desirable to have a long reverberation time torender a weak sound more audible.11.3 Reverberation EffectsConsider a sound source that operates continuously until the maximum acousticintensity in the enclosed space is reached. The source suddenly shuts off. Thereception of sound from the direct ray path ceases after a time interval r/c, wherer represents the distance between the source and the reception point and c the soundpropagation velocity. But owing to the longer distances traveled, reflected wavescontinue to be heard as a reverberation which exists as a succession of randomlyscattered waves of gradually decreasing intensity.The presence of reverberation tends to mask the immediate perception of newlyarrived direct sound unless the reverberation drops 5–10 dB below its initial levelin a sufficiently short time. Reverberation time T , the time in seconds required forintensity to drop 60 dB, offers a direct measure of the persistence of the reverberation.A short reverberation time is obviously necessary to minimize the maskingeffects of echoes so that speech can be readily understood. However, an extremelyshort reverberation time tends to make music sound harsher—or less “musical”—while excessive values of reverberation time T can blur the distinction between

11.4 Sound Intensity Growth in a Live Room 247Figure 11.4. Typical reverberation times for various auditoriums and functions.individual notes. The choice of T, which also depends on the room volume, thereforerepresents an optimization between two extremes.Figure 11.4 represents the accumulation of optimal reverberation time data asfunctions of intended use and enclosure volume. Lower values of T occur fromincreased absorption of sound in the surfaces of the enclosures. Hard surfacessuch as ceramic tile floors and mirrors tend to lengthen the reverberation time. Inaddition to reverberation time, the ability of a chamber or enclosure to screen outexternal sound minimizes annoyance or masking effects. The acoustic transmissionof walls, treated in Chapter 12, constitutes a major factor in enclosure design. Ashort reverberation time with its attendant high absorption tends to lessen theambient noise level generated by external sounds that penetrate the walls of theenclosure.11.4 Sound Intensity Growth in a Live RoomWe now apply the classic ray theory to deal with a sound source operating continuouslyin an enclosure, which will yield results in fairly good agreement with

248 11. Acoustics of Enclosed Spaces: Architectural AcousticsFigure 11.5. Geometric configuration for setting up the relationship between energydensity and intensity of sound.experimental measurements. The process of absorption in the medium or the enclosingsurfaces prevents the intensity from becoming infinitely large. Absorptionin the medium is fairly negligible in medium- and small-sized enclosures, so theultimate intensity depends upon the absorption power of the boundary surfaces. Ifthe enclosure’s boundary surfaces have high absorption the intensity will quicklyachieve the maximum which exceeds only slightly the intensity of the direct ray.If the enclosure has highly reflective surfaces, i.e., low absorption, a “live” roomensues; the growth of the intensity will be slow and appreciable time will haveelapsed for the intensity to reach its maximum.After a sound source is started in a live room, reflections from the wall becomemore uniform in time as the sound intensity increases. With the exception of closeproximity to the source, the energy distribution can be considered uniform andrandom in direction. In reality a signal source having a single frequency will resultin standing-wave patterns, with resultant large fluctuations from point to point inthe room. But if the sound consists of a uniform band of frequencies or a pure tonewarbling over at least a half octave, the interference effects of standing waves areobliterated.Referring to Figure 11.5, we establish the relationship between intensity (whichrepresents the energy flow) and energy density of randomly distributed acousticenergy. In the figure dS represents an element of the wall surface and dV the volumeelement in the medium at a distance r from dS. The distance r makes an angle θwith the normal NN ′ to dS. Let the average acoustic energy density E (in W/m 3 )beassumed uniform throughout the region under consideration. The acoustic energy

11.5 Sound Absorption Coefficients 249in incremental volume dV is EdV. The surface area of the sphere of radius rencompassing dV is 4πr 2 . The projected area of dS on the sphere is cos θ dS. Theportion of the total energy contained in dV is given by the ratio dS cos θ/4πr 2 . Theenergy from dV that strikes dS directly becomesdE = EdVdScos θ . (11.1)4π r 2Now consider the volume element dV as being part of a hemisphere shall ofradius rand thickness dr. The acoustic energy rendered to S by the complete shellis found by assuming a circular zone of radius r sin θ (with θ treated as a constant)in Figure 11.5 and integrating over the entire surface of the shell. The volume ofthe resultant element is 2πr sin θ rdrdθ. From θ = 0toθ = π/2, and Equation(11.1) yieldsE = EdSdV2∫ π/20sin θ cos θ dθ = EdSdr4This energy arrives during time interval t = dr/c. Hence, the rate of acoustic energyimpinging dS from all directions isEt= EcdS4or Ec/4 per unit area, which is therefore the intensity I of the diffused sound atthe walls. This is also equal to one fourth of a plane wave of energy intensity Iincident at a normal angle onto a plane. The intensity I of the diffuse sound at thewall becomesI = Ec(11.2)411.5 Sound Absorption CoefficientsAll materials constituting the boundaries of an enclosure will absorb and reflectsound. A fraction α of the incident energy is absorbed and the balance (1 – α) isreflected. Reflection is indicated by the reflection coefficient r defined asamplitude of reflected waver =amplitude of incident waveBecause the energy in a sound wave is proportional to the square of the amplitude,the sound absorption coefficient α and the reflection coefficient are related byα = 1 − r 2The value of the sound absorption coefficient α will vary with the frequency of theincident ray and the angle of incidence. Materials comprising room surfaces aresubject to sound waves that impinge upon them from many different angles as aresultof multiple reflections. Hence, published data for absorption coefficients

250 11. Acoustics of Enclosed Spaces: Architectural AcousticsTable 11.1. Absorption Coefficients.Octave-Band Center Frequency (Hz)125 250 500 1000 2000 4000Brick, unglazed 0.03 0.03 0.03 0.04 0.05 0.07Brick, unglazed, painted 0.01 0.01 0.02 0.02 0.02 0.03Carpet on foam rubber 0.08 0.24 0.57 0.69 0.71 0.73Carpet on concrete 0.02 0.06 0.14 0.37 0.60 0.65Concrete block, coarse 0.36 0.44 0.31 0.29 0.39 0.25Concrete block, painted 0.10 0.05 0.06 0.07 0.09 0.08Floors, concrete or terrazzo 0.01 0.01 0.015 0.02 0.02 0.02Floors, resilient flooring 0.02 0.03 0.03 0.03 0.03 0.02on concreteFloors, hardwood 0.15 0.11 0.10 0.07 0.06 0.07Glass, heavy plate 0.18 0.06 0.04 0.03 0.02 0.02Glass, standard window 0.35 0.25 0.18 0.12 0.07 0.04Gypsum, board 0.5 in. 0.29 0.10 0.05 0.04 0.07 0.09Panels, fiberglass, 1.5 in. thick 0.86 0.91 0.80 0.89 0.62 0.47Panels, perforated metal, 4 in. thick 0.70 0.99 0.99 0.99 0.94 0.83Panels, perforated metal with 0.21 0.87 1.52 1.37 1.34 1.22fiberglass insulation, 2 in. thickPanels, perforated metal with 0.89 1.20 1.16 1.09 1.01 1.03mineral fiber insulation, 4 in. thickPanels, plywood, 3/8 in. 0.28 0.22 0.17 0.09 0.10 0.11Plaster, gypsum or lime, rough 0.02 0.03 0.04 0.05 0.04 0.03finish on lathPlaster, gypsum or lime, smooth 0.02 0.02 0.03 0.04 0.04 0.03finish on lathPolyurethane foam, 1 in. thick 0.16 0.25 0.45 0.84 0.97 0.87Tile, ceiling, mineral fiber 0.18 0.45 0.81 0.97 0.93 0.82Tile, marble or glazed 0.01 0.01 0.01 0.01 0.02 0.02Wood, solid, 2 in. thick 0.01 0.05 0.05 0.04 0.04 0.04Water surface nil nil nil 0.003 0.007 0.02One person 0.18 0.4 0.46 0.46 0.51 0.46Air nil nil nil 0.003 0.007 0.03Note: The coefficient of absorption for one person is that for a seated person (m 2 basis). Air absorptionis on a per cubic meter basis.are for “random” incidence as distinguished from “normal” or “perpendicular”incidence.The angle–absorption correlation appears to be of somewhat erratic nature,but at high frequencies the absorption coefficients in some materials is roughlyconstant at all angles. At low frequencies the random-incidence absorption tendsto be greater than for normal incidence. However, as Table 11.1 shows, α variesconsiderably with frequency for many materials, and the absorption coefficientsare generally measured at six standard frequencies: 125, 250, 500, 1000, 2000,and 4000 Hz. Absorption occurs as the result of incident sound penetrating andbecoming entrapped in the absorbing material, thereby losing its vibrational energy

11.6 Growth of Sound with Absorbent Effects 251that converts into heat through friction. Ordinarily the values of α should fallbetween zero for a perfect reflector and unity for a perfect absorber. Measurementsof α>1.0 have been reported, owing possibly to diffraction at low frequenciesand other testing condition irregularities.Let α 1 , α 2 , α 3 , ...α i denote the absorption coefficient of different materials ofcorresponding areas S 1 , S 2 , S 3 ,....S i forming the interior boundary planes (viz.the walls, ceiling and floor) of the room as well as any other absorbing surfaces(e.g. furniture, draperies, people, etc.). The average absorption coefficient α for anenclosure is defined byα = α 1 S1 + α 2 S2 + α 3 S3 +···+α i Si= A (11.3)S1 + S2 + S3 +···+Si Swhere A represents the total absorptive area ∑ α i S i , and S the total spatial area.11.6 Growth of Sound with Absorbent EffectsThe rate W of sound energy being produced equals the rate of sound energy absorptionat the boundary surfaces of the room plus the rate at which the energyincreases in the medium throughout the room. This may be expressed as a differentialequation governing the growth of acoustic energy in a live room:V dEdt + AcE ≡ W (11.4)4The solution for E in Equation (11.4) isE = 4W Ac(1 − e−(Ac/4V )t ) (11.5)with the initial condition that the sound source begins operating at t = 0. From therelationship of Equation (11.2) the intensity becomesI = W A(1 − e−(Ac/4V )t ) (11.6)and from Equation (3.58) the energy density isE =(11.7)2 ρ 0 c 2The mean square acoustic pressure becomesp 2 = 4W ρ 0 c (1 − e−(Ac/4V )t ) (11.8)AEquation (11.8) is analogous to the one describing the growth of direct current inan electric circuit containing an inductance and a resistance. The time constant ofthe acoustic process is 4V/Ac. If the total absorption is small and the time constantis large, a longer time will be necessary for the intensity to approach its ultimatep2

252 11. Acoustics of Enclosed Spaces: Architectural Acousticsvalue of I ∞ = W/A. The ultimate values of the energy density and mean squareacoustic pressure are given byE ∞ = 4W Ac , p2 ∞ = 4W ρ 0cAA number of caveats pertain to the use of Equation (11.8). In order that the assumptionof an even distribution of acoustic energy be cogent, a sufficient time t musthave elapsed for the initial rays to undergo several reflections at the boundaries.This means approximately 1/20 of a second should have elapsed in a small chamber;and the time must approach nearly a full second for a large auditorium. Thefinal energy density, being independent of the size and shape of the room, shouldbe the same at all points of the room and dependent only upon the total absorptionA. But Equation (11.6) does not hold for spherical or curved rooms which canfocus sounds; neither is Equation (11.8) applicable to rooms having deep recessesnor to oddly shaped rooms or rooms coupled together by an opening, and nor torooms with some surfaces of extraordinarily high absorption coefficients α (thesecause localized lesser values of energy densities).11.7 Decay of SoundWe can now develop the differential equation describing the decay of uniformlydiffuse sound in a live room. The sound source is shut off at time t = 0, meaningW = 0 at that instant. E 0 denotes the uniformly distributed energy density at thatinstant. From Equation (11.4)AcEdt = dE (11.9)4Vand the solution to Equation (11.9) becomesE = E0 e −(Ac/4V )t (11.10)The intensity I at any time t after the cessation of the sound source is related tothe initial intensity I 0 byI= e −(Ac/4V ) (11.11)I 0Applying the operator 10 log to both sides of Equation (11.11) results inIL = 10 log e −(Ac/4V )t = 102.3 ln e−(Ac/4V )t =− 1.087Act (11.12)Vwhere IL denotes the intensity level change in decibels. The intensity level in alive room decreases with elapsed time at a constant decay rate D (in dB/s),D = 1.087Ac .V

11.7 Decay of Sound 253Following Sabine’s definition, we define the reverberation time T as the timerequired for the sound level in the room to decay by 60 dB:T = 60 D = 55.2V(11.13)AcExpressing volume V in m 3 and area S used to compute A in m 2 , and setting soundpropagation speed c = 343 m/s, Equation (11–13) becomesT = 0.161V(11.14)AEquation (11.14) becomes for English unitsT = 0.049V(11.15)Awhere volume V is rendered in ft 3 and A in ft 2 (or sabins, with 1 sabin equal to1ft 2 of absorption area αS). (One metric sabin is equal to 1 m 2 of absorption area.)It becomes apparent here that the reverberation time for a room can be controlledby selecting materials with the appropriate acoustic absorption coefficients. Theabsorption coefficient of a material can be measured by the introduction of a definitearea of the absorbent material in a specially constructed live room or reverberation(or echo) chamber. A photograph of such a chamber is given in Figure 11.6.Figure 11.6. Photograph of a reverberation chamber. (Courtesy of Eckel Industries, Inc.)

254 11. Acoustics of Enclosed Spaces: Architectural AcousticsExample Problem 1: Reverberation PredictionA room 8 m long, 4 m wide, and 2.8 m high contains four walls faced with gypsumboards. The only exceptions to the wall area are a glass window1mby0.5mandaplywood-paneled door 2.2 m by 0.6 m. In addition the door has a gap underneath,1.5 cm high. In order to estimate the reverberation time of the room at 500 Hz wemake use of the data in Table 11.1. Predict the reverberation time T .SolutionThe absorption area (in m 2 ) is found as follows:A = Si α i = [2(8 × 2.8) + 2(4 × 2.8) − 2.2(0.6) − (0.015)(0.6) − 1.0(0.5)]× 0.05 + (2.2)(0.6)(0.17) + (0.015)(0.6)(1) + 1.0(0.5)(0.18)+ (4)((8)(0.81) + (4)(8)(0.81) = 32.70 m 2 .Applying Equation (11.14):0.161 × 8 × 4 × 2.8T = = 0.82 s.32.7The gap at the bottom of the door is treated as a complete sound absorber with acoefficient of unity. From the above estimated value of the reverberation time of0.44 s and a chamber volume of 89.6 m 3 , the room may be suitable for use as aclassroom according to Figure 11.4.11.8 Decay of Sound in Dead RoomsThe derivation of Equation (11.14) was based on the assumption that a sufficientnumber of reflections occur during the growth or decay of sound and also thatthe energy of the direct sound and the energy of the fractional amount of soundreflected were both sufficient to ensure a uniform energy distribution. In the caseof anechoic chambers, where the absorption coefficient of the materials constitutingthe boundaries is very close to unity, it is apparent that the derivations of thepreceding equations for growth and decay of sound are not applicable. The onlyenergy present is the direct wave emanating from the sound source. The reverberationtime must be zero, whereas application of Equation (11.14) would yielda finite reverberation time of 0.161V/S, where S is simply the total area of theinterior surfaces of the chamber. Thus, it is apparent that Equation (11.14) wouldbe increasingly in error as the average sound absorption coefficient increases. Ifthe average value of the absorption coefficient exceeds 0.2, Equation (11.14) willbe in error by approximately 10%.A different approach to ascertaining the decay of sound in a dead room, whichwas developed by Eyring (1930), is to consider the multiplicity of reflections as aset of image sources, all of which are considered to exist as soon as the real sourcebegins. Let ᾱ, found from the relationship ᾱ = ( ∑ α i S i )/ ∑ S i denote the average

11.8 Decay of Sound in Dead Rooms 255sound absorption coefficient of the room’s boundary materials. The growth ofacoustic energy at any point in the room results from the accumulation of successiveincrements from the sound source, from the first-order (single reflection) imageswith strengths W (1 – ᾱ), from the second-order (secondary reflection) imageswith strengths W (1 – ᾱ) 2 , and so on, until all the image sources of appreciablestrengths have rendered their contributions. When the true sound source is stopped,the decay of the sound occurs with all the image sources stopped simultaneouslyalong with the source. The energy decay in the room occurs from successive lossesof acoustic radiation from the source, then from the first-order images, the secondorderimages, and so on.Eyring derived the following equation for the growth in acoustic energy density:4W [E =−1 − ecS ln (1−α)t/4V ] (11.16)cS ln (1 − α)The above equation is very similar to Equation (11.5) excepting that the total roomabsorption is given byα =−S ln(1 − α) (11.17)Here S is the total area of the boundary surfaces of the room. In a like fashion theanalogy to Equation (11.6) for the decay of sound energy is given bycS ln(1−α)t/4VE = E0 eand the decay rate in dB/s is expressed as1.08cS ln(1 − α)D =−Vwith the reverberation time expressed byT = 0.161V−S ln(1 − α)For small values of absorption (α ≪ 1) the term ln(1 – α) may be replaced by α,the first term in an infinite series. This results in recovering the Sabine formulafor live rooms. It should also be noted that the coefficient 0.161 for the Sabineand the Eyring formulas, which is based on the speed of sound at 24 ◦ C, will varyaccording to air temperature. The coefficient becomes somewhat higher at lowerair temperatures and vice versa.Another formula for determining the reverberation time of a room lined withmaterials of widely ranging absorption coefficients was developed by Millingtonand Sette (Millington, 1932; Sette, 1993). The Millington–Sette theory indicatesthat the total room absorption is given byA = ∑ −S i ln(1 − ᾱ i )which yields the reverberation timeT =0.161V∑ − Si ln(1 − ᾱ i )

256 11. Acoustics of Enclosed Spaces: Architectural Acoustics11.9 Reverberation as Affected by Sound Absorptionand Humidity in AirWe have previously not considered the effect of absorption of sound and humidity inair on reverberation times. The volume of air contained in very large auditoriumsor a place of worship can absorb an amount of acoustic energy that cannot beneglected as in the case with smaller rooms. If a room is small, the number ofreflections from the boundaries is large and the amount of time the sound wavespends in the room is correspondingly small. In this situation acoustic energyabsorption in the air is generally not important. In very large room volumes thetime a wave spends in the air between reflections becomes greater to the extentthat absorption of energy in air no longer becomes negligible. The reverberationequations must now include the effect of air absorption, particularly at higherfrequencies (>1 kHz).Sound waves lose some energy through viscous effects during the course oftheir propagation through a fluid medium. The intensity of a plane wave lessenswith distance according to the equationI = I 0 e −2βx = I 0 e −mx .Here m = 2β represents the attenuation coefficient of the medium. Some texts useα rather than β to denote the attenuation constant of the medium; we eschew itsuse in order to avoid confusion with α used in this chapter to denote the absorptioncoefficient of a surface. During time interval t, a sound wave travels a distancex = ct, and the preceding equation may be revised to read(β/4V +m)ctI = I0 eThe expression for the reverberation time becomesT = 0.161V(11.18)A + 4mVwhere the constant mis expressed in units of m –1 . The total surface absorption Ais given either by Equation (11.3) or (11.17) depending whether that room fits intothe category of being an acoustically live or dead chamber. As the room volume Vbecomes larger, the second term in the denominator of Equation (11.18) increasesin magnitude, as air absorption becomes more significant, due to increasing pathlengths between the walls. Since m also increases with frequency, air absorptionalso becomes more manifest at higher frequencies (above 1 kHz) than at lowerfrequencies. The values of m are given in Figure 11.7 1 as a function of humidityfor various frequencies at a normal room temperature of 20 ◦ C. More details, alsogiven in tabular form, for a range of air temperatures and humidities are givenin the NASA report (1967), prepared by Cyril M. Harris, listed at the end of thischapter. It is seen from Figure 11.7 that the effect of humidity reaches a maximum1 The plot of Figure 11.7 applies to indoor sound propagation, not to outdoor propagation that includesmeteorological effects not present indoors.

11.9 Reverberation as Affected by Sound Absorption and Humidity in Air 257Figure 11.7. Values of the total attenuation coefficient m versus percent relative humidityfor air at 20 ◦ C and normal atmospheric pressure for frequencies between 2 kHz and12.5 kHz. Values are rendered in both SI units and U.S. Customary System units. (AfterHarris, 1966, 1967.)in each of the given frequencies in the 5–25% relative humidity range and thentrails off at higher humidities.Example Problem 2Find the air absorption at a frequency of 6300 Hz and 25% relative humidity for aroom volume of 20,000 m 3 .Solution and Brief DiscussionFrom Figure 11.7 the value of m is equal to 0.026 m –1 . The air absorption A air =4mV in Equation (11.18) is equal to 4 × (20,000 m 3 ) × 0.026 m –1 = 2080 m 2 .Ifwe consider absorption at 500 Hz the effect of air absorption would be negligiblein comparison.

258 11. Acoustics of Enclosed Spaces: Architectural Acoustics11.10 Early Decay time (EDT10)A modification of the reverberation time T 60 is the early decay time, or EDT10,which represents the time interval required for the first 10 dB of decay to occur,multiplied by 6 to produce an extrapolation to 60 dB decay Originally proposed byJordan (1974), EDT10 is based on early psychoacoustical research, and accordingto Cremer and Muller (1982), “the latter part of a reverberant decay excited bya specific impulse in running speech or music is already masked by subsequentsignals once it has dropped by about 60 dB.”11.11 Acoustic Energy Density and DirectivityIn order to account for uneven distribution of sound in some sources, we expressthe sound intensity I (W/m 2 ) due to a point source of power W (W) in the directfield (i.e., reflections are not considered) asI = WQ(θ,φ)(11.19)4π r 2where r is the distance (m) from the source and Q(θ,φ)isthedirectivity factor. Thedirectivity factor Q(θ,φ) equals unity for an ideal point source that emits soundevenly in full space. For an ideal point source above an acoustically reflectivesurface, in an otherwise free half-space, Q(θ,φ) equals 2. The sound or acousticenergy density is the sound energy contained per m 3 at any instant. In the directfield in full space, the direct sound energy density D D (W s/m 3 )isgivenbyDD = I c = WQ(θ,φ)(11.20)4π r 2 cwhere c is the speed of sound in m/s.11.12 Sound Absorption in Reverberant Field:The Room ConstantThe product IS gives the rate of acoustic energy striking a surface area S; and IScos θ gives that rate for the incidence angle θ. In an ideal reverberant field, withequal probability for all angles of incidence, the average rate of acoustic energystriking one side of the surface is given by IS/4. The power absorbed by the surfacehaving an absorption coefficient α isPower absorbed = αIS4 = αcD R S4where D R denotes the reverberant sound field density. In a fairly steady statecondition the power absorbed is balanced by the power supplied by the source tothe reverberant field. This is the portion of the input power W that remains after

11.13 Sound Levels due to Direct and Reverberant Fields 259one reflection:Power supplied = W (1 − α)The steady-state condition results inc DR Sα= W (1 − α)4which we rearrange to obtain the energy density in the reverberant field,DR = 4W 1 − ααcSwhere the room constant R is by definition= 4WcR(11.21)R =αS1 − α .In most cases, the boundaries of the actual enclosure and other objects inside theenclosure are constructed of different materials with differing absorption coefficients.The room constant R of the enclosure is then described in terms of meanproperties byR = ST α1 − αwhereR = the room constant (m 2 )S T = total surface area of the room (m 2 )ᾱ = mean sound absorption coefficient = ∑ α i S i /S T .11.13 Sound Levels due to Direct and Reverberant FieldsNear a point of nondirectional sound source, the sound intensity is greater thanfrom afar. If the source is sufficiently small and the room not too reverberant,the acoustic field very near the source is independent of the properties of theroom. In other words, if a listener’s ear is only a few centimeters away from aspeaker’s mouth, the room surrounding the two persons has negligible effect onwhat the listener hears directly from the speaker’s mouth. At greater distancesfrom the source, however, the direct sound decreases in intensity, and, eventuallythe reverberant sound predominates.If we are more than one-third wavelength from the center of a point source, theenergy density of a point r is given by Equation (11.20) for the direct sound field.Combining the Equations for the direct and the reverberant sound intensities, i.e.,Equations (11.20) and (11.21), we get the total sound intensity I given by[ Q(θ,φ)I = W + 4 ](11.22)4π r 2 R

260 11. Acoustics of Enclosed Spaces: Architectural AcousticsIt is assumed that reverberant sound comes from nearly all directions in a fairly evendistribution. The modes generated by standing waves must be rather insignificant;otherwise the assumption of uncorrelated sound is not valid and Equation (11.22)will not truly constitute the proper model for the actual sound field.The sound pressure level within the room can now be found from( ) [ ( I1L p = 10 log = 10 log WI ref 4πr + 4 2 R( 1= L W + 10 log4πr + 4 )2 R)]+ 120(11.23)for an ideal point source emanating equally in all directions. The reference soundintensity I ref is equal to 10 –12 W/m 2 , and the sound power level L W of the sourceis defined as( ) WL W = 10 log10 −12which is given in dB re 1 pW. For an ideal point source over an acousticallyreflective surface[ ( 1L P = 10 log W2πr + 4 )]( 1+ 120 = L 2 W + 10 logR2πr + 4 )(11.24)2 REquations (11.23) and (11.24) are based on the fact that the absorption coefficientsdo not vary radically from point to point in the room and the source is not closeto reflective surfaces. If the sound power and room absorption characteristics areassigned for each frequency band, the sound pressure level L P can be determinedfor each frequency band in dB/octave, dB/one-third octave, and so on. If the soundpower level is A-weighted, and if the room constant is based on frequencies in thesame range as the frequency content of the source, the sound power level will beexpressed in dB(A).Example Problem 3Predict the reading of a sound pressure level meter 12 m from a source havinga sound power level of 108 dB(A) re 1 pW in a room with a room constantR = 725 m 2 (at the source frequencies). The source is mounted directly on anacoustically hard floor.SolutionWe apply Equation (11.24) as follows:( 1L p = L W + 10 log2πr + 4 )2 R= 108 + 10 log[(2π × 12 2 m 2 ) −1 + 4/725 m 2 ]= 86.2 dB(A)

11.15 Concert Halls and Opera Houses 261If the meter would be placed at R = 3 m from the source, the SPL meter readingfor L p will increase to 91.7dB(A).11.14 Design of Concert Halls and AuditoriumsIdeally, the main objective of auditorium design is to get as many members of theaudience as close as possible to the source of the sound, because sound levels decreasewith increasing distances from the sound source. A good visual line of sightusually results in good acoustics, so stepped seating becomes desirable for largerrooms seating more than 100 people. Reverberation should be controlled in orderto provide optimum reinforcement and equalization of sound. For speech the roomdesign should provide more in the way of direct sound augmented by reflections,while the clarity of articulation of successive syllables must be sustained. Roomsfor music typically have longer reverberation times because the requirements forarticulation are not as stringent, and more enhancement of the sound is desirable.The aim of the design of a listening type of facility is to avoid the followingacoustic defects (Siebein, 1994). Echoes, particularly those from the rear walls of the facility. Echoes can belessened or eliminated by placing absorbent panels or materials on the reflectingwalls or introducing surface irregularities to promote diffusion of the sound. Excessive loudness can occur from prolonged reverberation. Again, the properdeployment of absorbent materials should alleviate this problem. Flutter echo results from the continued reflection of sound waves between twoopposite parallel surfaces. This effect can be especially pronounced in smallrooms; and this can be contravened by splaying the walls slightly (so as to avoidparallel surfaces) or using absorbent material on one wall. Creep is the travel of sound around the perimeter of domes and other curvedsurfaces. This phenomenon is also responsible for whispering gallery effects inolder structures with large domed roofs. Sound focusing arises when reflections from concave surfaces tend to concentratethe sound energy at a focal point. Excessive or selective absorption occurs when a material that has a narrow rangeof acoustical absorption is used in the facility. The frequency that is absorbed islost, resulting in an appreciable change in the quality of the sound. Dead spots occur because of sound focusing or poorly chosen reflecting panels.Inadequate sound levels in specific areas of the listening facility can result.11.15 Concert Halls and Opera HousesThree basic shapes exist in the design of large music auditoriums, namely (1)rectangular, (2) fan-shaped, and (3) horseshoe, all of which are illustrated in floorplans of Figure 11.8. A fourth category is the “modified arena”, nearly elliptical inshape. The Royal Albert Hall (constructed in 1871) in London, the Concertgebouw

262 11. Acoustics of Enclosed Spaces: Architectural AcousticsFigure 11.8. Three basic hall configurations: rectangular, fan shaped, and horseshoeshaped.(1887) in Amsterdam, the Sidney Opera House (opened in 1973), and the ColoradoSymphony’s Boettcher Concert Hall in Denver (opened in 1978 and acousticallyremodeled in 1993) are examples of this type of facility.The rectangular hall is quite traditional, and it has been built to accommodateboth small and large audiences. But these halls will always generate cross reflections(flutter echoes) between parallel walls. Sound can also be reflected fromthe rear walls back to the stage, depending on balcony layout and the degree ofsound absorption. These reflections can help in the buildup of sound and providesa reasonable degree of diffusion in halls of modest interior dimensions. A considerablylarger hall can result in standing wave resonances and excessive flutterechoes.It is interesting to note that the first music hall to be designed from a scientificviewpoint, by none other than Wallace C. Sabine, is the Boston Symphony Hall(1900), views of which are given in Figures 11.9 and 11.10. The structure containsa high, textured ceiling and two balconies extending along three walls. Volume is602,000 ft 3 ; seating capacity 2631; and the reverberation time in the 500–1000 Hz

Figure 11.9. A view of the Boston Symphony Hall from the stage. Wallace ClementSabine was the principal acoustical consultant for this facility. Reprinted with permissionfrom Leo Beranek, Concert Halls and Opera Houses: Music, Acoustics, and Architecture,2nd ed. (New York, NY: Springer, 2004), 48.Figure 11.10. A stage area of the Boston Symphony Hall. Reprinted with permissionfrom Leo Beranek, Concert Halls and Opera Houses: Music, Acoustics, and Architecture,2nd ed. (New York, NY: Springer, 2004), 48.

264 11. Acoustics of Enclosed Spaces: Architectural Acousticsrange is 1.8 s (occupied). Another example of a great rectangular hall is the venerableGrösser Musikvereinssaal (1870) in Vienna which has a reverberation time of2.05 (occupied) in a volume of 530,000 ft 3 . Its superior acoustics can be attributedto its relatively small size, high ceiling, irregular interior surfaces and the plasterinterior (Beranek, 2004).A fan-shaped hall accommodates, through its spread, a larger audience withincloser range from the sound source (stage). It features nonparallel walls thateliminate flutter echoes and standing waves; and most audience members canobtain a pleasing balance between direct and reflected sounds. A disadvantagein terms of early time delay gap is the distance from the side walls. Often it isnecessary to add a series of inner reflectors or canopies hanging from ceilings overthe proscenium area to maintain articulation and other acoustical characteristics.Many architects in the United States have resorted to the fan-shaped hall design inorder to accommodate larger audiences while retaining an appreciable degree ofboth visual and aural coupling to the stage area. Relatively modern examples ofthis design are the Dorothy Chandler Pavilion (1964) in Los Angeles; the OrchestraHall (completed in 1904 and most recently renovated in 1997, Chicago; the EastmanTheater (opened in 1922 as a movie theater and converted into a concert hallin 1930) in Rochester, New York; and the Kleinhaus Music Hall (1940, designedby Eliel and Eero Saarinen) in Buffalo, New York.Over a number of centuries horseshoe-shaped structures have been used as thepreferred design for opera houses and concert halls of modest seating capacity. Thisdesign provides for a greater sense of intimacy, and the textures of convex surfacespromote adequate diffusion of sound. The multiple balconies allow for excellentline of sight and short paths for direct sound. The La Scala Opera House (Figures11.11a, b) in Milan is probably the most notable example of the horseshoe design. Itwas opened in 1778. This edifice, formally known as Teatro alla Scala, was closedin 2001 for 3 years to undergo a badly needed renovation. A tubular structure anda 17-story fly tower designed by Mario Botta were added to provide stagecraftstorage, dressing rooms, and rehearsal rooms. In addition to repairing the ravagesof time on the structure, modern stage machinery and new wiring were installed,as well as a new heating, ventilation and air conditioning system. The acousticswere improved by the prominent acoustician, Higini Arau from Barcelona. Othercelebrated examples of the horseshoe design are the Carnegie Hall (completed1897, renovated 1983–1995) in New York City and the Academy of Music (thefirst opera house in the United States, opened in 1857) in Philadelphia.Nearly all concert halls have balconies, which were designed to accommodateadditional seating capacity within a smaller auditorium volume, so that listenerscan sustain an intimate relationship with the stage. The depths of the balconies generallydo not exceed more than twice their vertical “window” (opening) to the stage.In fact a smaller ratio is desirable to minimize undue sound attenuation at the rearwall. A rule of the thumb in contemporary acoustical design: the depth of the balconyshould not exceed 1.4 times its outlook to the stage at the front of the balcony.In all types of auditorium design, ceilings constitute design opportunities fortransporting sound energy from the stage to distant listeners. In Figure 11.12, it is

11.15 Concert Halls and Opera Houses 265(a)(b)Figure 11.11. (a) View looking toward the stage of the La Scala Opera House. The theaterwas closed down for three years for seriously needed renovations; (b) a classic exampleof the horseshoe configuration. (Courtesy of Dr. Antonio Acerba Cantier Escala)

266 11. Acoustics of Enclosed Spaces: Architectural AcousticsFigure 11.12. Transmission of sound to all areas of an auditorium through the ceiling andfloor profiling.shown how a ceiling can convey sound to the listeners without imposing a greattime difference between direct and ceiling-reflected sound. Floor profile is alsoimportant in establishing the proper ratio of direct to indirect sound. Splays on theside walls have proven effective in promoting diffusion and uniformity of loudness.Rear walls generally should be absorbent to minimize echoes being sent back tothe stage.Many concert halls have been built throughout the world, some with outstandingacoustics and others resulting in dismal sound. Among the features common to allof the aurally superior halls are a limited audience capacity (generally 2800 seatsor less), extreme clarity of sound so that the audience can clearly distinguished theindividual instruments of the orchestra without loss of fullness or blending of tonesassociated with reverberation. A good hall also allows the orchestra to hear itself.Table 11.2 contains a summary of the characteristics of a number of prominentmusical facilities all over the world. Figure 11.13 shows the stage view ofthe Orchestra Hall in Minneapolis (1974), which is patterned after the classicalrectangular design. The hall contains a slanted floor with 1590 seats and an additional983 seats are located in the three-stepped balcony tiers, making for a totalaudience capacity of 2573. A random pattern of plaster cubes covers the ceiling,providing effective diffusion of sound throughout the hall. This overlay of cubesalso continues down the back wall, behind the stage, as shown in Figure 11.13.Wood paneling partially covers the walls, and both the flooring of the stage andaudience areas are wood. This concert hall is notable for its clarity, dynamic range,and balance.Another acoustic success among the contemporary musical facilities is theKennedy Center for the Performing Arts which opened in Washington, D.C. in1971. The Center consists of a single structure that contains a 2759-seat concerthall, a 2319-seat opera house (Figure 11.14), and a 1142-seat theater. The locationwas environmentally challenging, for the Center is situated on a site near thePotomac River, in close proximity to the Washington Ronald Reagan NationalAirport on the other side of the river. Both commercial and private aircraft fly as

11.15 Concert Halls and Opera Houses 267Table 11.2. Reverberation Times of Leading Concert Halls and Auditoriums.Reverberation Time aVolume (ft 3 ) Seating Capacity Occupied UnoccupiedUnited StatesBaltimore, lyric Theatre 744,000 2616 1.47 2.02Boston Symphony Hall 662,000 2631 1.8 2.77Buffalo, Kleinhans Music Hall 644,000 2839 1.32 1.65Cambridge, Kresge Auditorium 354,000 1238 1.47 1.7Chicago, Aric Crown Theatre 1,291,000 5081 1.7 2.45Cleveland, Severance Hall 554,000 1890 1.7 1.9Detroit, Ford Auditorium 676,000 2926 1.55 1.95New York, Carnegie Hall 857,000 2760 1.7 2.15Philadelphia Academy of Music 555,000 2984 1.4 1.55Purdue University Hall of Music 1,320,000 6107 1.45 1.6Rochester, New York, Eastman Theatre 900,000 3347 1.65 1.82AustriaVienna, Grosser Musikvereinssaal 530,000 1680 2.05 3.6BelgiumBrussels, Palais des Beaux-Arts 442,000 2150 1.42 1.95CanadaEdmonton and Calgary, Alberta Jubilee Halls 759,000 2731 1.42 1.8Vancouver, Queen Elizabeth Theatre 592,000 2800 1.5 1.9DenmarkTivoli Koncertsal 450,000 1789 1.3 2.25FinlandTurku, Konserttisali 340,000 1002 1.6 1.95GermanyBerlin, Musikhochschule Konzertsaal 340,000 1340 1.65 1.95Bonn, Beethovenhalle 555,340 1407 1.7 1.95Great BritainEdinburgh, Usher Hall 565,000 2760 1.65 2.52Liverpool Philharmonic Hall 479,000 1955 1.5 1.65London, Royal Albert Hall 3,060,000 5080 2.5 3.7London, Royal Festival Hall 755,000 3000 1.47 1.77IsraelTel Aviv, Frederic R. Mann Auditorium 750,000 2715 1.55 1.97ItalyMilan, Teatro Alla Scala 397,300 2289 1.2 1.35NetherlandsAmsterdam, Concertgebouw 663,000 2206 2.0 2.4SwedenGothenburg, Konserthus 420,000 1371 1.7 2.0SwitzerlandZurich, Grosser Tonhallesaal 402,500 1546 1.6 3.85VenezuelaCaracas, Aula Magna 880,000 2660 1.35 1.8a At 500–1000 Hz.

268 11. Acoustics of Enclosed Spaces: Architectural AcousticsFigure 11.13. The stage area of the Minnesota Orchestra Hall in Minneapolis. (Courtesyof the Minnesota Orchestral Association.)low as a few hundred feet directly over the roof, and occasionally helicopters passby along the Potomac River at rooftop levels. In addition, vehicular traffic runsacross the river and directly beneath the plaza of the Center.To deal with these external noise sources, the Center was constructed as abox-within-a-box, so that each of the auditoriums is totally enclosed within anouter shell. The columns within each auditorium are constructed to isolate interiorFigure 11.14. The interior of the John F. Kennedy Opera House in Washington, DC.(Courtesy of the John F. Kennedy Center. Photograph by Scott Suchman.)

11.15 Concert Halls and Opera Houses 269ceilings, walls, and floors from both airborne and mechanical vibrations. Thedouble-wall construction generally consists of 6-in. solid high-density blocks separatedby a 2-in. air gap. The huge windows in the Grand Foyer facing the riverconsist of 1.27-cm (1/2-in.) and 0.64-cm (1/4-in.) thick glass sheets separated bya 10-cm (4-in.) air gap. Resilient mounts are used to isolate interior noise sources(e.g. transformers, air-conditioning units, etc.) The ductworks are acousticallylined, flexible connectors are used, special doors are installed at all auditoriumentrances together with “sound locks” between foyer and the auditoriums.The rectangular concert hall encompasses a volume of 682,000 ft 3 and accommodatesan audience of 2759. Large contoured wall surfaces and a coffered ceilingabet the diffusion of sound at low and high frequencies. The 11 massive crystalchandeliers, each weighing 1.3 metric ton, donated by the Norwegian government,also contribute to the diffusion. The balconies are purposely shallow to preventreduction of sound below the balcony overhang. Unoccupied, the concert hall hasa reverberation time of 2.2 s at 500 Hz and 2.0 s at 1 kHz; the corresponding valuesare 2.0 s and 1.8 s for the fully occupied hall.Located in downtown Seattle, Washington, Benaroya Hall opened in September1998, contains two spaces for musical performances: a 2500-seat main auditorium(Figure 11.15) and a more intimate 540-seat recital hall. The main auditorium, theS. Mark Taper Foundation Auditorium, is a classic rectangular configuration withthe stage enclosed in a permanent acoustic shell. LMN (Loschky, Marquartdt andFigure 11.15. Seattle Symphony performing in Benaroya Hall in Seattle, WA under thedirection of Music Director Gerard Schwarz. (Courtesy of the Seattle Symphony, photographby Craig Raymond.)

270 11. Acoustics of Enclosed Spaces: Architectural AcousticsNesholm) Architects and the acoustical consultant, Cyril M. Harris, combinedthe shoebox design with state-of-the-art materials to achieve maximum warmthand balance.The location of Benaroya Hall in a busy sector of Seattle posed special challengesto the designers. They had to contend with a railroad tunnel running diagonallybeneath the auditorium and a nearby underground bus tunnel. A slab of concretemore than 2m (6 ft) thick, 24 m (80 ft) wide, and 131 m (430 ft) long was pouredunder the hall to swallow the sound from the tunnels. In order to combat otherexterior noises, the designers essentially encased a building within a building. Theauditorium, weighing 12 million kilograms, rests on 310 rubber pads, which absorbvibration from the tunnels. The pads are 38 cm 2 and are composed of four layersof natural rubber sandwiched with 0.32-cm (1/8-in.) steel plates.All electrical, plumbing, and other noise-generating equipment are located outsidethe auditorium box. Any penetration of the box is made with flexible connections.Water is known to transmit sound very well, so the fire sprinkler system isleft dry and it will flood with water only when a fire is detected. The ventilationsystem is connected to the outside by a sound trap, which channels air throughnarrow openings between perforated aluminum boxes of sound insulation. Verylarge vents collect air below the floor and move it slowly behind the auditoriumto another sound trap and from thereon to fans. The basic idea of the ventilationsystem is to move a high volume (2400 m 3 /min) of air at low speed, eliminatingnoise created by fast-moving air in conventional systems.Instead of frame construction, the walls are built of precast concrete panels. Theheavy mass helps to cut down building vibration and provides a stiff, hard surface toreflect concert sound. Side walls, back walls, and ceiling are covered with panelingshaped like truncated pyramids to reflect sound at various angles to aid diffusion.Randomly spaced wood blocking behind the angled paneling creates framed soundboxes that reflect both high and low frequency sounds so that no tone is eliminatedfrom the music. Side walls are covered with particle boards veneered with a dense,fine-grained hardwood from a single makore tree. The ceiling is suspended fromthe roof by hundreds of metal strips. The ceiling is coated with 3.8 cm (1.5 in.)of plaster in irregularly shaped panels to diffuse sound. The plaster is sufficientlydense to prevent the ceiling from vibrating. House lighting is imbedded in theceiling to minimize sound leakage. Access to the light bulbs is achieved above theceiling through heavy, removable plaster caps.11.16 Band Shells and Outdoor AuditoriumsOver the past several decades there has been an increasing trend toward outdoorconcerts, either at band shells or in semi-open structures. These types of structuresare more economical to construct than full-fledged indoor concert halls, andthey also meet the criteria of providing an informal setting for audiences seekingentertainment in a usually rural environment, away from the metropolitanareas.

11.16 Band Shells and Outdoor Auditoriums 271Figure 11.16. The Hollywood Bowl in Los Angeles, California, after its reconstructionin 2004. (Courtesy of the Los Angles Philharmonic, photograph by Mathew Imaging.)It is generally not possible for a large orchestra to play effectively in open air.The use of a band shell becomes necessary, as this permits the members of amusical group to hear each other and directs the music toward the audience area.The band shell site should also be carefully selected. Ideally, the region shouldbe isolated from the noise of passing traffic and overhead aircraft. The topologyof the land also ranks important in providing the proper acoustics. If the land canbe contoured properly, there can be appreciably less attenuation of the sound thanwould be the case if the band shell were located on flat ground.The Hollywood Bowl (Figure 11.16) is an example of an orchestra shell strategicallylocated in a natural hollow. The only reflected sound is that reflected from theshell, but the stage distances are sufficiently short so that the sound is heard withoutany discernable time delay gap. However, shells can never equal the dynamic rangeof sound power and sonority that are achieved in an enclosed reverberant concerthall. The use of high quality amplification systems is therefore often necessary atmany outdoor concerts.The Bowl, the summer home of the Los Angeles Philharmonic, has undergonea number of changes throughout its years of operation. The previous shelldesign—the fourth since 1922—has been subjected to a great deal of criticism fromperformers and audiences alike. In 2004, the shell and stage area was reconstructedand made 30% larger to accommodate a full orchestra. The new shell is providedwith a set of adjustable reflectors above the orchestra, mounted on a 27.5 m ×18.2 m (90 ft × 60 ft) elliptical structure that also supports an improved lighting

272 11. Acoustics of Enclosed Spaces: Architectural AcousticsFigure 11.17. A view looking toward the interior of the Tanglewood Music Shed in Lenox,MA. The ceiling canopy reflects sound, adding dynamic range and brilliance. (Photographby Kim Knox Beckhius.)system. Advantage was also taken of the opportunity to install a completely newsound system.Another type of “outdoor” structure is the “music shed,” a semi-enclosed structurespecifically designed for musical performances. The Tanglewood Music Shedin the Berkshires region of Massachusetts (Figure 11.17) is the summer home ofthe Boston Symphony Orchestra, and the quality of its acoustics exceeds that ofany band shell for an audience of 6000 persons. An additional 6000 people onthe lawn adjacent to the shed can listen to the music through the open segmentsof the pavilion. The canopy of the interior projects and diffuses sound througha volume of 42,500 m 3 (1,500,000 ft 3 ). The ceiling is constructed of 5-cm thickwooden planks; the side and rear walls are of 2 cm fiberboard; and the floor issimply packed earth. When occupied, the shed embodies a reverberation time of2 s in the frequency range of 500–1000 Hz, which is quite excellent considering therustic nature of its construction. The Tanglewood shed is the precursor to similarstructures at Wolf Trap in Vienna, Virginia, the Performing Arts Center at SaratogaSpring, New York (the summer home of the Philadelphia Symphony Orchestra),and the Blossom Music Center near Cleveland, Ohio.On July 16, 2004, the Jay Pritzker Pavilion (Figure 11.18), a radical outdoorconcert facility located at Chicago’s Millennium Park near the banks of LakeMichigan, presented its first musical program. Working in concert with the eminentarchitect Frank Gehry, the consulting firm TALASKE of Oak, Park, Illinoisdealt with the acoustic challenge of outdoor orchestral performances. TALASKE

11.16 Band Shells and Outdoor Auditoriums 273Figure 11.18. The Jay Pritzker Pavilion in Chicago. The radical design also features atrellis system from which loudspeakers are suspended. (Courtesy of TALASKE, graphicsby DOXA.)established the overall shape and arrangement of the stage enclosure and, in thecourse of design, refined the shapes and finishes of the interior surfaces. The stageof the pavilion (Figure 11.19), the permanent home of the Grant Park Music FestivalOrchestra and Chorus, can accommodate a 100 plus member orchestra andFigure 11.19. The stage of the Pritzker Pavilion can accommodate an orchestra of morethan 100 members plus a 150-member chorus. (Photograph by Richard H. Talaske.)

274 11. Acoustics of Enclosed Spaces: Architectural Acousticsalso a 150-member chorus. The walls and the ceiling of the stage incorporatecontours and angles precisely planned to enhance cross-room reflections so thatmusicians can hear each other from one side of the stage to the other and fromfront to back. When an orchestra performs at high volume it is often difficult formusicians playing in the high register to hear instruments in the lower registers. Tocircumvent this problem, the orchestra risers feature a platform system, researchedand designed by the acousticians and musicians together, that allows the musiciansto feel the vibration created by the cellos and bass instruments. The riser system,essentially a floating floor with rigid interconnections and resilient materials justbelow the floor surface, acts to maximize cross stage vibrations. This enables amore precise ensemble and tonal balance.To compensate for the lack of reflected sound in an outdoor setting and the lackof envelopment, modern technology was recruited to improve the sound environmentby blending architecturally created and electronically processed sound. TheJay Pritzker Pavilion, which replaces the Petrillo Music Shell constructed in 1931and still in use on a more limited basis, is the first orchestral facility that distributessound by using an overhead steel trellis structure from which loudspeakers are suspended(Figure 12.20). The trellis area that incorporates the network of speakersis 100 m (325 ft) wide by 180 m (600 ft) long. These loudspeakers are strategicallysuspended at predetermined heights and orientations, in concentric circlesoutward from the stage. A distributed sound reinforcement system provides director “frontal” sound to the audience. A separate acoustic enhancement system,Figure 11.20. View from the stage of the Pritzker Pavilion. The trellis overhead of theaudience contains a distributed loudspeaker system.

11.17 Subjective Preferences in Sound Fields of Listening Spaces 275Figure 11.21. Distribution of sound from the speakers distributed on a trellis systemoverhead of the audience at the Pritzker Pavilion. (Courtesy of TALASKE, graphics byDOXA.)LARES TM (Lexicon Acoustic Reinforcement and Enhancement System), deliverslingering, enveloping sound characteristics through supplemental loudspeakers. Ituses a time varying technique to maintain stability by shifting the output in timeenough to maintain stability but not enough to introduce tonal coloration. Thissystem simulates reflections and reverberations using specialized electronics anddigital processing. Time delay creates the impression that sound is coming fromthe stage rather than from the speakers. These two systems work together to deliversound throughout the seating area and the Great Lawn (cf. Figure 11.21).The system makes use of eight microphones deployed around the stage area forthe enhancement system and 30 or so microphones for reinforcement, 143 amplifiershoused in three separate equipment rooms, 90 CobraNet TM channels, and185 speakers. The beauty of this system is that the acoustic environment can beadjusted to work equally well for orchestral music, staged opera, blues, jazz, rock,and other music. The fixed seating area accommodates 4000 concertgoers, and theGreat Lawn has a capacity for a lawn crowd of up to 7000 stretching more than acity block behind the front area.Backstage facilities include warm-up rooms that are shared with the HarrisTheater for Music and Dance. A bridge near the concert area was configured tohelp mask the traffic noises at the pavilion.11.17 Subjective Preferences in Sound Fieldsof Listening SpacesBeyond the Sabine realm of architectural acoustics, which is based on the relativelysimple but effective formula T 60 = (0.161V )/ A i α i , other considerationscome into play in determining optimal configurations for listening spaces (Ando,1998). This involves combining the elements of psychoacoustics, modeling ofthe auditory-brain system, and mapping of subjective preferences. The physicalproperties of source signals and sound fields in a room are considered, in

276 11. Acoustics of Enclosed Spaces: Architectural Acousticsparticular the autocorrelation function (ACF) that contains the same informationas power density spectrum but it is adjusted to account for hearing sensitivity.Effective duration of the normalized ACF is defined by the delay τ e at which theenvelope of the normalized ACF becomes one-tenth of its maximal value. Theresponse of the ear includes the effects of time delay due not only to the room’sacoustical characteristics, but also the spatially incurred differences in the signalsreaching the right and the left ears. The difference in the sounds arriving at theear is measured by the “interaural cross-correlation function” or IACF, which isdefined byIACF(τ) =∫ t2t 1p L (t)p R (t + τ)√ ∫ t2∫ (11.25)p 2 t2L t 1p 2 R dtwhere L and R denotes entry to the left and right ears, respectively. The maximumpossible value of IACF is unity. The time t = 0 is the time of the arrival of thedirect sound from the impulse radiated by a source. Integration from 0 to t 2 msincludes the energy of the direct sound and any early reflections and reverberantsounds occurring during the t 2 interval. Because there is a time lapse of about 1 msfor a sound wave to impinge the other side of the ear after impinging one side, itis customary to vary τ over the range from –1 ms to +1 ms. In order to obtain asingle number that measures the maximum similarity of all waves arriving at thetwo ears with the time integration limits and the range of τ, it is customary to choosethe maximum value of IACF, which is then called the interaural cross-correlationcoefficient (IACC), i.e.t 1IACC = |IACF(t)| maxDifferent integration periods are used for IACC. The standard ones includeIACC A (t 1 = 0 to t 2 = 1000 ms), and IACC E(arly) (0–80 m), IACC L(ate) (80–1000 m). The early IACC is a measure of the apparent source width ASW and thelate IACC is a measure of the listener envelopment LEV. IACC is generally measuredby recording on a digital tape recorder the outputs of two tiny microphoneslocated at the entrances to the ear canals of a person or a dummy head, and quantifyingthe two ear differences with a computer program that performs the operationof Equation (11.25). IACC A is determined with a frequency bandwidth of 100 Hz–8 kHz and for a time period of 0 to about 1 s.Subjective attributes for a sound field in a room have been developed experimentallywith actual listeners. The simplest sound field is considered first, a situationwhich consists of the direct sound and a single reflection acting in lieu of a setof reflections. The data obtained are based on tests in anechoic chambers (whichallowed for simulation of different concerts halls) with normal hearing subjectslistening to different musical motifs. From these subjective tests the optimum designobjectives are established, namely the listening level, preferred delay time,

References 277preferred subsequent reverberation time (after the early reflections), and dissimilarityof signals reaching both ears (involving IACC).These factors are each assigned scalar values and then combined to yield asubjective preference that can vary from seat to seat in a concert hall. Some ratherinteresting results of investigations include the fact that the right hemisphere ofthe brain is dominant for “the continuous speech.” while the left hemisphere isdominant when variation occurs in the delay time of acoustic reflection. The lefthemisphere is usually associated with speech and time-sequential identifications,while the right hemisphere is allied with nonverbal and spatial identifications. Aproposed model for the auditory-brain system was developed (Ando, 1998) thatincorporates a subjective analyzer for spatial and temporal criteria and entailsthe participation of the left and the right hemispheres of the brain. The powerdensity spectra in the neural processes in the right and left auditory pathways yieldsufficient information to establish autocorrelation functions.It is obvious that different individuals are likely to have different subjectivepreferences with respect to the same musical program, so seating requirementscan differ, with respect to the preferred listening level and to the initial time delay,and even lighting. For example, evaluations were conducted for a performance ofHandel’s Water Music with 106 listeners providing the input on their preferenceswith respect to listening level, reverberation time, and IACC. The information thusobtained can provide insight into how the acoustic design of a concert hall anda multipurpose auditorium can be accomplished. Procedures for designing soundfields include consideration of temporal factors, spatial factors, the effect of soundfield on musicians, the conductor, stage performers, listener, and the archetypalproblem of fusing acoustical design with architecture. Multipurpose auditoriumspresent bigger challenges, some of which have been met very well and many whichhave not. In the design procedure, a number of factors other than acoustical includemeasurable quantities such as temperature, lighting levels, and so on, and otherless tangible determinants that can be aesthetically evocative.ReferencesAmerican Society for Testing and Materials. 1981. A test method for sound absorption andsound absorption coefficients by the reverberation room. Method. ASTM C423-81a.Ando, Yoichi. 1985. Concert Hall Acoustics. Berlin: Springer-Verlag.Ando, Yoichi. 1998. Architectural Acoustics: Blending Sound Sources, Sound Fields, andListeners. New York: AIP Press and Springer-Verlag. (A followup to Yoichi’s previoustext. The author, arguably Japan’s most prominent architectural acoustician, introduces atheory of subjective preferences, based on a model of the auditory cognitive functioningof the brain.)Bies. D. A. and Hansen, C. H. 1996. Engineering Noise Control: Theory and Practice,2nd ed. London: E. and F. N. Spon.Beranek, Leo L. 1962. Music, Acoustics, and Architecture. New York: John Wiley & Sons.Beranek, Leo L. July 1992. Concert Hall Acoustics—1992. Journal of the AcousticalSociety of America 92, 1: 1–39.

278 11. Acoustics of Enclosed Spaces: Architectural AcousticsBeranek, Leo. 2004. Concert and Opera Houses: Music, Acoustics, and Architecture, 2nded. New York: Springer-Verlag. (A tremendous compendium of the acoustical characteristicsof musical auditoria and how they were achieved. This volume is a must-havetext for architectural acousticians and aficionados of architectural acoustics.)Cremer, L., and Muller, H. A. 1982. Principles and Applications of Room Acoustics, vols. 1and 2. English translation with additions by T. J. Schultz. New York: Applied SciencePublishers.Crocker, Malcolm J. (ed.). 1997. Encyclopedia of Acoustics, vol. 3. New York: JohnWiley & Sons: Chapters 90–93: 1095–1128.Eyring, C. F. 1930. Reverberation time in “dead rooms.”Journal of the Acoustical Societyof America 1: 217–241.Harris, Cyril M. 1966. Absorption of sound in air vs. humidity and temperature.Journalof the Acoustical Society of America, 40: 148–49.Harris, Cyril M. January 1967. Absorption of sound in air vs. humidity and temperature,NASA contractor report NASA CR-647. Springfield, VA 22151: Clearinghouse forFederal Scientific and Technical Information.Harris, Cyril M. (ed.). 1991. Handbook of Acoustical Measurements and Noise Control,3rd ed. New York: McGraw-Hill.International Standardization Organization. 1963. Measurement of sound absorption in areverberation room. ISO R354-1963.Jordan, V. 1974. Presented at the 47th Audio Engineering Society Convention,Copenhagen.Kinzey, B. Y., Jr., and Howard, S. 1963. Environmental Technologies in Architecture.Englewood Cliffs, NJ: Prentice Hall.Knudsen, Vern O., and Harris, Cyril M. 1978 (reissue). Acoustical Designing in Architecture.Woodbury, NY: Acoustical Society of America. (A classic text by two of the giantsin field of architectural acoustics.)Kutruff, H. 1979. Room Acoustics, 2nd ed. London: Applied Science.Kuttruff, H. 1991. Room Acoustics, 3rd ed. London: Elsevier Applied Science.Lambert, Mel. August 2004. Deus ex machine: Canopy and line array fly at the HollywoodBowl. Systems Contractor: 56.Millington, G. 1932. A modified formula for reverberation. Journal of the AcousticalSociety of America, 4: 69–82.Morse, Philip M., and Bolt, Richard H. 1944. Sound waves in rooms. Review of ModernPhysics, 16: 65–150.MVI Technologies. 1997. Mediacoustic: Teaching acoustics by computer. Lyon, France:01 dB Stell. (A multinational compact disk conceived and prepared by acousticiansat l”Ecole Polytechnique de Lausanne, l’Ecole Nationale es Travaux Publics de Lyon,Liverpool University, and the companies Acouphen, Estudi Acustic, and o.1 dB. Itcontains excellent modules on basic physics, noise and man. room acoustics, and noisecontrol.)Sabine, Wallace Clement. Reissued 1993. Collected Papers on Acoustics. Los Altos, CA:Peninsula Publishing. (Included are the original papers that laid the foundation of modernarchitectural acoustics.)Sette, W. J. 1933. A new reverberation time formula. Journal of the. Acoustical Society ofAmerica, 4:193–210.Seattle Times. June 21, 1998. Benaroy Hall: orchestrating the sound of music. SectionA: 9.

Problems for Chapter 11 279Siebein, Gary W. June 1994. Acoustics in buildings: a tutorial on architectural acoustics,presented at the 127th Annual Meeting of the Acoustical Society of America, Cambridge,MA.Problems for Chapter 111. The amplitude of the reflected wave is one-half that of the incident wave fora certain angle and frequency. What is the reflection coefficient? What is thecorresponding sound absorption coefficient?2. Find the average sound coefficient of 125 Hz and 2 kHz for a wall constructedof different materials as follows:Area (m 2 )Material85 Painted brick45 Gypsum board on studs35 Plywood paneling4 Glass window3. Find the time constant of a room that is 8 × 10 × 2.5 m and an average soundabsorption coefficient of 0.34.4. A room has dimensions 3.5 m high × 30 m length × 10 m width. Two of thelonger walls consist of plywood paneling; the rear wall is painted brick and thefront has gypsum boards mounted on studs. The ceiling is of acoustic panelingand the floor is carpeted. For 250 Hz and 125 people in the audience:(a) Determine the reverberation time from the Sabine equation.(b) Compute the room constant R.(c) Comment on the suitability of the room for use as an auditorium.5. An auditorium has dimensions 6.0 m high × 22.0 m length × 15.5 m width. Thefloor is carpeted and one of the longer walls has gypsum boards mounted onstuds the entire length, while the other three walls and the ceiling are constructedof plaster. For 500 Hz:(a) determine the reverberation time from the Sabine equation.(b) compute the room constant R.6. In Problem 5 there is a set of two swinging doors 2.5 m in height and 2.5 m intotal width. However, there are open cracks along the bottom and between thedoors 1 cm wide. In addition there is an open window on the gypsum-boardedwall that is 0.8 × 1 m. Find the effect of these openings on the reverberationtime of the auditorium.7. A room has dimension 4.0 m high × 18.0 m length × 9.5 m width. The flooris carpeted and one of the longer walls has gypsum boards mounted on studsalong the entire length, while the other three walls are constructed of paintedbrick. The ceiling is plastered. For 500 Hz:

280 11. Acoustics of Enclosed Spaces: Architectural Acoustics(a) determine the reverberation time from the Sabine equation.(b) compare the reverberation time obtained from the Eyring formula with thatobtained in part (a) above.(c) compute the room constant R.8. An isotropic source emitting sound power level of 106 dB(A) re 1 pW is operatingin the room of Problem 4. What sound pressure (on the A-weighted scale)will be registered by a meter 5.2 m from the source? If 125 people are not seatedin the room, how will it affect the reading? Assume that no one blocks the pathof direction propagation between the meter and the signal source.9. Why does an anechoic chamber provide what we call a free field?

12Walls, Enclosures, and Barriers12.1 IntroductionAcoustics constitutes an important factor in building design and in layouts ofresidences, plants, offices, and institutional facilities. A building not only protectsagainst inclement weather; it must also provide adequate insulation against outsidenoises from transportation and other sources. Interior walls and partitions need tobe designed to prevent the intrusion of sound from one room into another. Exposureof workers to excessive occupational noises can be decreased by construction ofappropriate barriers or enclosures around noisy machinery. A shell constructed ofthe densest materials may be the most effective barrier against noise transmission,but such an enclosure can lose a major portion of its effectiveness if there are weaklinks that tend to promote sound transmission rather than hinder it. For example,a large proportion of sound energy enters a building through its windows, evenclosed ones, and many cracks and crevices inevitable in real structures permitsound to enter the structure’s interior.12.2 Transmission Loss and Transmission CoefficientsSound absorption materials tend to be light and porous. Sound isolation materials,however, generally are massive and airtight, thereby forming effective soundinsulation structures between the noise source and the receiver. When airbornesound impinges on a wall, some of the sound energy is reflected, some energy isabsorbed within the wall structure, and some energy is transmitted through thewall. Sound pressure against one side of the wall may cause the wall to vibrateand transmit sound to the other side. The amount of incident energy transmittedto the wall depends on the impedance of the wall relative to the air. The amountof the sound transmitted through the wall that is finally transmitted to the air onthe receiver side also depends on the impedance of the wall relative to air. Adouble wall construction that incorporates airspace between panels repeats theprocess between the two panels, which results in even more insulation againstsound.281

282 12. Walls, Enclosures, and BarriersMost construction materials transmit only a small amount of acoustic energy,with the major portion of the energy undergoing reflection or conversion intoheat due to impedance mismatch, absorption within the material, and damping.Heavy walls, usually of masonry, allow very little of the sound to pass through,owing to their high mass per unit area. A wall of gypsum boards mounted onboth sides of a stud frame provides effective insulation against sound penetrationdue to energy losses resulting during the passage of sound from air to solid toair between the two panels to solid and thereon to air on the other side of thewall.A measure of sound insulation provided by a wall or some other structural barrieris given by the transmission loss TL, given in units of dB by( )W1TL = 10 log(12.1)W 2where W 1 represents the sound power incident on the wall and W 2 the sound powertransmitted through the wall. Because the frequency loss varies with the frequencyof the sound, it is usually listed for each octave band or one-third octave band. Thefraction of sound power transmitted through a wall or barrier constitutes the soundtransmission coefficient, τ, written asτ = W 2(12.2)W 1Combining Equations (12.1) and (12.2) we obtain( ) 1TL = 10 log(12.3)τorτ = 10 −TL/10 . (12.4)In Figure 12.1 we consider the case of a plane wave approaching a panel inthe y–z plane (i.e., the plane is normal to the x-axis) at an angle of incidence θ.Subscripts I , R, and T are used to identify, respectively, the incident wave, thereflected wave, and the transmitted wave. Arrows indicate the directions of thewave propagation. The wave equation for a plane wavebears the solutions:∂ 2 p∂t 2= c2 ∇ 2 pik(ct−x cos θ−y sin θ)p I = P I ep R = P R e ik(ct+x cos θ−y sin θ) . (12.5)ik(ct−x cos θ−y sin θ)p T = P T eHere the real part of p denotes the sound pressure, P, the (complex) pressureamplitude, and k, the wave number.

12.3 Mass Control Case 283Figure 12.1. Transmission loss through a panel in the y–z plane.12.3 Mass Control CaseConsider the panel as being quite thin, i.e., its thickness is considerably smallerthan one wavelength of the sound in air; and let us also neglect the material stiffnessand damping in the panel. We can stipulate the conditions of (a) the continuity ofvelocities normal to the panel and (b) a force balance occurs inclusive of the inertialforce. From application of the first condition we express the particle velocity u atthe panel asu panel = u T cos θ = u I cos θ − u R cos θ. (12.6)When the sound pressure and the particle velocity are in-phase they are related byZ = ρ u= ρc (12.7)where Z denotes the characteristic resistance, which represents a special case ofspecific acoustic resistance.Setting x = 0 at the panel, we apply Equation (12.7) and Equation set (12.5) toEquation (12.6) to obtainP T = P I − P R . (12.8)The second condition applied over a unit surface area of the panel givesp I + p I − p T − ma panel = 0 (12.9)where m represents the mass density per unit area of the panel and a panel theacceleration normal to the surface. Through the use of Equation (12.7) we can

284 12. Walls, Enclosures, and Barrierscorrelate the panel velocity with the transmitted particle velocity:cos θu panel = p Tρc .Using the last of the Equation set (12.5) in the preceding expression and differentiatingwith respect to time, we now obtainik(ct−x cos θ−y sin θ) cos θa panel = ikcP T eρc= iωP T cos θ.ρcInserting the above expression into Equation (12.9) and applying Equations (12.5)at the panel, where x = 0, the amplitudes of the pressures are related to each otherbyP I + P R − P T = imωP Tcos θ. (12.10)ρcThe pressure term P R in Equation (12.10) may now be eliminated through the useof Equation (12.8). The ratio of transmitted pressure amplitude to incident pressureamplitude may now be obtained asP TP I=1.imω cos θ1 +2ρcWe now express the sound transmission coefficient asτ = p2 rms(T )p 2 rms(I )= |P T | 2|P I | 2 = 1( ) imω cos θ 2.1 +2ρcFrom the definition of Equation (12.3), we obtain the mass law transmission lossequation:( ) [ ( ) ]1 mω cos θ2TL = 10 log = 10 log 1 +. (12.11)r 2ρcSetting the angle of incidence equal to zero, Equation (12.11) reduces to[ ( ) ]mω2TL 0 = 10 log 1 +(12.12)2ρcwhich constitutes the statement for the normal incidence mass law for approximatingtransmission loss of panels with sound at 0 ◦ angle of incidence.12.4 Field Incidence Mass LawIn the situation of transmission loss between two adjoining rooms, the sound sourcein one room may produce a reverberant field. The incident sound emanating from

12.5 Effect of Frequencies on Sound Transmission through Panels 285the source may strike the wall at all angles between 0 and 90 ◦ . For field incidenceit is customary to assume that the angle of incidence lies between 0 and 72˚, whichresults in a field incidence transmission loss of approximatelyTL = TL 0 − 5dB.We can modify the mass law equation by converting the angular frequency ofsound (rad/s) into cyclic frequency (Hz), i.e. ω/2π = f , by setting the acousticimpedance for air ρc = 407 rayls, and assuming mω/(2ρc) ≫ 1. This gives thefield incidence mass law equationTL = 20 log( fm) − 47 dB (12.13)where m denotes the mass per unit area (kg/m 2 ).Equation (12.13) indicates for the mass-controlled frequency region that transmissionloss of a panel increases by 6 dB per octave, Doubling either the panelthickness or the density of the panel will also engender an additional 6 dB loss for agiven frequency. While Equation (12.13) is useful for the prediction of a material’sacoustical behavior, laboratory or field testing should be conducted to measure thetransmission loss of actual structures under real environmental conditions.Example Problem 1A wall is considered to have its sound transmission mass-controlled. Plot thetransmission loss as a function of the product of frequency and mass per unit area.SolutionUsing Equation (12.13), which expresses the field incidence mass law for typicalconditions, we can writeTL = 20 log( fm) − 47 dB.For the normal incidence mass law:TL = 20 log( fm) − 42 dB.Figure 12.2 shows the semilog plot of both relationships as two straight lines. Itshould be noted here that these values tend to be considerably higher than thoseof the actual transmission losses.12.5 Effect of Frequencies on Sound Transmissionthrough PanelsPanel bending stiffness constitutes the governing factor in low-frequency soundtransmission. The panel resonances play the principal role in determining thenature of transmission of higher frequency sounds. The panel may be consideredmass-controlled in the frequency range from twice the lowest resonant frequencyto below the critical frequency (discussed below).

286 12. Walls, Enclosures, and BarriersFigure 12.2. Semi-log plot of transmission loss versus product of frequency and surfacemass.Ver and Holmer (1971) develop the sound transmission coefficient equation forpanels manifesting significant bending stiffness and damping, which is given asfollows:τ =[1 + η( mω cos θ2ρc1)( Bω 2 sin 4 θmc 4 )] 2+[( mω cos θ2ρc)(1 − Bω2 sin 4 )] 2.θmc 4(12.14)The panel thickness is assumed to be small compared with the wavelength of theincident sound, B = panel bending stiffness (N m), η = composite loss factor(dimensionless), and m = panel surface density (kg/m 2 ).12.6 Coincidence Effect and Critical FrequencyIn propagating though panels and other structural elements, sound can occur aslongitudinal, transverse and bending waves. Bending waves give rise to the coincidenceeffect. In Figure 12.3 a panel is shown with an airborne sound wave ofwavelength λ incident at angle θ. A bending wave of wavelength λ b is excited in thepanel. The propagation velocity of bending waves depends on the frequency, withhigher velocities occurring at higher frequencies. The coincidence effect occurswhenθ = θ ∗ = arcsin(λ/λ b ) (12.15)

12.6 Coincidence Effect and Critical Frequency 287Figure 12.3. The coincidence effect affected by bending waves.orsin θ ∗ = λ λ b.With the asterisk indicating the occurrence of coincidence, θ ∗ denotes the coincidenceangle. Under such circumstances, the sound pressure on the surface of thepanel falls into phase with bending displacement. This results in a highly efficienttransfer of acoustic energy from the airborne sound waves on the source side ofthe panel to bending waves in the panel, and thence to airborne sound waves inthe receiving room on the other side of the panel. This is a highly undesirablesituation if the panel is meant to prevent the transmission of sound from one roomto another as a noise control measure.Figure 12.4 shows an idealized plot of transmission loss for a panel as a functionof frequency, with stiffness-controlled, resonance-controlled, mass-controlled, andcoincidence-controlled regions. The transmission loss curve drops considerably inthe region beyond the critical frequency owing to the coincidence effect.From Equation (12.15) it can be deduced that the coincidence effect cannotoccur if the wavelength λ of the airborne sound is greater than the bending modewavelength λ b in the panel. The minimum coincidence frequency, or the criticalfrequency, exists at the critical airborne sound wavelengthλ = λ ∗ = λ b

288 12. Walls, Enclosures, and BarriersFigure 12.4. Idealized plot of transmission loss versus frequency.where the critical frequency f ∗ is given byf ∗ = c λ ∗which corresponds to the grazing incidence θ = 90 ◦ . Also, there is a criticalangle θ ∗ , at which coincidence will occur, for any frequency above the criticalfrequency.Figure 12.5 contains a plot of critical frequencies against thickness for a numberof common construction materials. It must be realized that the critical frequencyFigure 12.5. Thickness versus critical frequency for a number of construction materials.(Source: Brüel and Kjær, 1980)

12.7 The Double-Panel Partition 289often falls in the range of speech frequencies, rendering some partitions nearlyuseless for providing privacy and preventing speech interference.From Equation (12.14), it is seen that the coincidence effect depends on characteristicsof the plate or panel and of the airborne sound wave. At coincidenceBω 2 sin 4 θ ∗= 1mc 4and inserting the above condition into Equation (12.14) yields the sound transmissioncoefficient for the coincidence conditionτ = τ ∗ 1=∗]mωη cos θ 2.[1 +2ρcThe corresponding transmission loss for the coincidence condition is( ) 1TL = TL ∗ ∗)fmη cos θ= 10 log = 20 log(1 + .τ ∗ ρcFrom the above equations, it would appear that τ ∗ = 1 and TL ∗ = 0 for undampedpanels (in which the loss factor η = 0). The above transmission loss equation ispremised on the theoretical behavior of an infinite plate, and the finite boundariesof actual structures such as windows and walls and the presence of damping inreal construction materials will produce a different response to sound waves.12.7 The Double-Panel PartitionA single-panel wall can exhibit resonant frequencies that fall below the rangeof speech frequencies. Most walls are constructed of two panels with an airspacebetween them, and they may yield low-frequency resonances in the speech range. Atypical interior partition consists of two gypsum board panels (ranging in thicknessfrom 1.3 cm to 1.9 cm, or from 0.5 in. to 0.75 in.) separated 9 cm (3.5 in.) by 5 cm ×10 cm (2 × 4) 1 wood or metal studs. In Figure 12.6 a double-panel configuration isshown; the two panels of mass per unit area m 1 and m 2 , respectively, are separatedby an airspace h. Because the air entrapped between the two panels behaves asa spring, a spring-mass analogy, shown in Figure 12.7, can be applied, with krepresenting the spring constant between two masses. The wall response modecan be depicted by two masses vibrating at the same frequency. A node (i.e. a“motionless” point) on the spring exists, thus effectively resolving the spring intotwo springs with spring constants (or spring rates) k 1 and k 2 , The natural frequencyof the dual-mass system is given byf n =√k1/m 1=2π√k2/m 2. (12.16)2π1 Lumber sizes are given by figures that are almost always nominal rather than representative of actualvalues. A two-by-four stud generally measures 1.75 × 3.5 in.

290 12. Walls, Enclosures, and BarriersFigure 12.6. A double-panel partition.The individual spring constants are related to the spring constant of the compositespring by1k = 1 + 1 . (12.17)k 1 k 2Eliminating k 2 between Equations (12.16) and (12.17) results ink 1 = k m 1 + m 2.m 2Inserting the last expression into the first portion of Equation (12.16) yields thenatural frequency f n of the system:√k m 1 + m 2mf n = 1 m 2. (12.18)2πIn order to establish the effective spring constant of the air between the panels,it will be assumed that an isentropic process constitutes the action of sound waves,since the pressure changes occur too rapidly for an isothermal process to take place.From thermodynamics, for an isentropic process the pressure p of air (consideredFigure 12.7. Spring-mass analogy of the double-panel partition of Figure 12.6.

an ideal gas) varies in the following manner:12.7 The Double-Panel Partition 291pv γ = constant (12.19)where v represents the specific volume (equal to the reciprocal of density) and γ ,the ratio of specific heats. Differentiating Equation (12.19) givesdpdv =−γ p v . (12.20)Sound waves cause the ambient pressure and specific volume of air to vary onlyslightly from the quiescent values of p 0 and v 0 , and thus the instantaneous valuesp and v differ very little from p 0 and v 0 in Equation (12.20). Because the massof air entrapped between the panels remains constant, the ratio of panel displacement(arising from the sound pressure pushing on the panels) dh to the original(quiescent) spacing h 0 should equal the ratio of the change of specific volume tothe quiescent value of the specific volume:dh= dv(12.21)h 0 v 0But the spring constant represents the force per unit displacement, or for a unitarea of the wallk =− dpdh . (12.22)Combining Equations (12.21) and (12.22) and dropping the subscripts (sinceh 0 ≈ h, p 0 ≈ p), we now havek =− γ phwhich, upon inserting into Equation (12.19), gives√γ p m 1 + m 2hmf n =1 m 2.2πSetting γ = 1.4 and p = 101.325 kPa (the standard atmosphere), the lowfrequencyresonance of the double-panel wall may now be found from√m1 + mf n = 602. (12.23)hm 1 m 2The surface masses m 1 and m 2 are given in kg/m 2 and the panel spacing h inmeters.Example Problem 2Predict the transmission loss for an 8-in. wall of poured concrete for 800-Hz sound.The concrete has a density of approximately 150 lb/ft 3 (2406 kg/m 3 ).

292 12. Walls, Enclosures, and BarriersSolutionSurface density m = 8 in. × 1m/(39.37 in.) × 2406 kg/m 3 = 488.9 kg/m 2Using field incidence law Equation (12.13)TL − 20 log(fm) − 47 = 20 log(800 × 488.9) − 47 = 65 dBLaboratory tests indicate a TL closer to 50 dB. Actual installations may yield evenlower values because of flanking noise transmission paths.Example Problem 3Find the resonant frequency of a double-panel wall constructed of a 15-kg/m 2 anda 20-kg/m 2 gypsum board with 9-cm airspace.SolutionApplying Equation (12.23):√ √m1 + m 2 15 + 20f n = 60 = 60= 68 Hz.hm 1 m 2 0.09(15)(20)12.8 Measuring Transmission LossRecasting Equation (12.1) for transmission loss TL in terms of vector sound intensity,we write( )IITL = 10 logI Twhere I I = incident vector sound intensity and I T = transmitted vector soundintensity, For sound pressure level and particle velocity in phase, sound intensityis given byI = p2 rmsρcwith the result that transmission loss could be restated as( p2)TL = 10 log rmsI.p 2 rmsTBut the above equation is not usable for measurement purposes, since distinctioncannot be made between root-mean-square (rms) pressures attributable to incidentsound, transmitted sound and reflected sound.In order to determine the transmission loss, it is required to set up a sourcechamber and a receiving chamber, with a panel of the specimen material installedin a window between the two chambers. Such a setup is shown in Figure 12.8. Thechambers must be designed so as to minimize the sound transmission paths other

12.8 Measuring Transmission Loss 293Figure 12.8. Setup for a transmission loss measurement.than that through the evaluation specimen. Sound level is measured in the sourcechamber and the receiving chamber with a microphone system, a measuring amplifier,and filter. The measured sound is filtered to mitigate the effect of backgroundnoise on the measurement procedures, with the filter settings being identical for thegenerated sound and the measuring sound filters. The sound level may be averagedby rotating the microphone about its base on the mounting tripod. The transmissionloss is found by applying the measured results in the following equation:( )AmTL = L S − L R + 10 logA RwhereL S = average sound level in the source chamberL R = average sound level in the receiving chamberA m = area of the material under investigationA R = equivalent absorption area of the receiving room.Another measurement setup is given in Figure 12.9. A pink noise generator,filters, audio power amplifier, and speaker constitute the sound source system.A computer-controlled analyzer incorporates a random noise generator and

294 12. Walls, Enclosures, and BarriersFigure 12.9. An alternative to the setup of Figure 12.8, which provides for automaticevaluation of transmission loss. The sound source can consist of a pink noise generator,filters, power amplifier, and speaker. The analyzer, which can generate random noise, canautomatically position a microphone boom for spatial and temporal averaging.provisions for automatic remote control of the microphone boom for spatial andtemporal averaging. Such an analyzer can measure in one-third octave bands thefollowing parameters: source chamber level, receiving chamber background SPL,receiving chamber SPL, and receiving chamber reverberation time.12.9 FlankingFlanking occurs when noise from a sound source is transmitted through pathsother than direct transmission through a wall between the source chamber and thereceiver on the opposite side of the wall. Among the causes of flanking are acousticalleaks through cracks around doors, windows and electrical outlets, passageof the sound through suspended ceilings with resulting reflection into adjacentrooms, HVAC (heating/ventilation/air-conditioning) ducts, floor and ceiling joists.The insulation effect of a partitioning wall is effectively decreased by flanking andacoustic leaks, examples of which are shown in Figure 12.10.12.10 Combined Sound Transmission CoefficientA wall may contain a number of elements such as windows, doors, openings,and cracks. The effective or combined sound transmission coefficient depends on

12.10 Combined Sound Transmission Coefficient 295Figure 12.10. Examples of flanking.the values of the sound transmission coefficients of the individual elements andtheir respective areas. Assuming the incident sound to be fairly uniform over theindividual elements, we can compute the combined sound transmission coefficientfromn∑A i τ ii=1τ combined =(12.24)n∑A iwhere i designates the i-th element of the subdivided wall area A and the correspondingvalue of τ. For cracks and open areas, τ = 1, which is to say the soundtransmission through such crevices may be considered to be virtually unimpeded.The corresponding transmission loss of the composite wall then becomes⎛ n∑⎞( )A1iTL combined = 10 log= 10 log ⎜i=1⎟τ combined⎝ n∑⎠ . (12.25)A i 10 −TL i /10Example Problem 4A 6.0 m × 2.5 m wall is for the most part constructed of 8-cm thick dense pouredconcrete with a transmission loss of 52 dB at 500 Hz. An opening in the wall isprovided for a 2.2 m × 1.0 m wood door with a transmission loss of 25 dB at500 Hz. There is a crack across the width of the door that is 1.0 cm high. Estimatethe effective transmission loss of the composite wall at 500 Hz.SolutionThe total area A of the wall is 6.0 × 2.5 = 15.0 m 2 , but the concreteportion of the wall is 15.0 − 2.2 × 1.0 − 0.01 × 2.2 = 12.778 m 2 . Applyingi=1i=1

296 12. Walls, Enclosures, and BarriersEquation (12.25)15.0TL combined = 10 log12.778 × 10 −55/10 + 2.2 × 10 −25/10 + 0.022 × 10 −0/10= 27.1dBSuppose that a door with TL = 45 dB was installed instead; what would be thenew value of the composite TL for the wall?Solution15.0TL combined = 10 log12.778 × 10 −55/10 + 2.2 × 10 −45/10 + 0.022 × 10 −0/10= 28.3dBThis example shows that negligible improvement will occur because the crackconstitutes the principal means of negating the combined effectiveness of theacoustic insulation of the concrete wall and the door. The use of an acoustical doorwhich effectively seals the area will result in an appreciably improved value of thetransmission loss.12.11 Noise Insulation RatingsTwo principal methods of rating sound insulation are discussed in this section,namely Sound Transmission Class (STC) and Shell Isolation Rating (SIR). Theformer designation constitutes a single-number description of noise insulationeffectiveness of a structural element and it is widely used to describe the characteristicsof interior partitions or walls with respect to noise occurring in the rangeof speech and music frequency. The SIR methodology was developed by Pallettet al. (1978) at the U.S. National Bureau of Standards (now National Instituteof Science and Technology) in order to establish a simple system to predict thecapacity of building shells to attenuate transportation noise.Sound Transmission Class (STC)The values of STC are computed from transmission loss values measured in onethirdoctave bands in the 125 Hz–4 kHz range. Figure 12.11 illustrates the standardSTC contour. This contour begins at 125 Hz, sloping upward at 3 dB per one-thirdoctave (i.e., 9 dB per octave) until it reaches 400 Hz. The slope of the contour thenchanges at 400 Hz to 1 dB per one-third octave (3 dB per octave), and it remainsconstant at 1.25 kHz. In the range from 1.25 to 4 kHz the slope is zero. The STCsingle-number classification of a specific wall is designated by the value of TLat 500 Hz. If the values of TL for the wall at the 16 one-third octave points are

12.11 Noise Insulation Ratings 297Figure 12.11. The standard STC contour.known for the region from 125 Hz to 4 kHz and plotted, the STC for the wall isestablished by superimposing the contour of Figure 12.11 upon the TL curve sothat (a) there is not more than 8-dB deficiency between the TL and the STC contourat any one-third octave frequency and (b) the total deficiency between the STCcontour (i.e., the value of the STC contour less the value of the TL curve summedat all one-third octave frequencies from 125 Hz to 4 kHz) must be ≤32 dB. Whenthe curve is adjusted to meet these two criteria, the STC value of the wall is takento be equal to the value of the TL of that contour at 500 Hz.Example Problem 5Find the STC value of the TL curve of Figure 12.12.SolutionThe standard STC contour is overlaid on the TL plot and positioned to satisfy thetwo criteria described above. From the value of the contour at 500 Hz, it is seenthat the STC rating of the wall is approximately 48 dB.Table 12.1 lists the STC ratings of a number of structural elements.

298 12. Walls, Enclosures, and BarriersFigure 12.12. STC overlay on a TL versus frequency plot.Shell Isolation Rating (SIR)The SIR method stems from intensive studies conducted by investigators at theNational Institute of Standards and Technology in the effort to develop a simplesystem to predict the attenuation of A-weighted transportation noise by buildingTable 12.1. Sound Transmission Class (STC) Ratings of Common ConstructionMaterialsDescriptionSTC4-in.- (100-mm) thick brick wall with airtight joints 404-in.- (100-mm) thick brick wall with gypsum board on one side 45–508-in.- (200-mm) thick brick wall with gypsum board on one side 50–604-in.- (100-mm) thick hollow concrete block with airtight joints 36–414-in.- (100-mm) thick hollow concrete block with gypsum board on one side 42–488-in.- (200-mm) thick hollow concrete block with airtight joints 46–508-in.- (200-mm) thick hollow concrete block with gypsum board on both sides 50–552 × 4-in.- (nominal) wood or metal stud wall with gypsum board on 33–43both sides, spackled at joints, floor, and ceiling 20–30Single-glazed window 26–44Double-glazed window 20–27Hollow core wood or steel door 38–55Specially mounted acoustical door 70Double-walled soundproof room, 12-in.- (300-mm) thick walls including airspace 21–29Quilted fiberglass mounted on vinyl or lead (limp mass) barrier septrum

12.11 Noise Insulation Ratings 299shells. This method is based on statistical studies, and +3 dB per octave was chosenas the standard SIR contour. The STC contour method is generally preferred fordescribing the noise attenuation effectiveness of building structures in the presenceof non-transportation sounds, viz. speech, television, and radio. The SIR systemis favored for the evaluation of noise reduction provided by building shells againsttransportation noise.An element SIR (or member SIR) is an estimate of the difference in the A-weighted sound levels when a structural element or member is placed between thetransportation noise source and the receiver. It is assumed that all noise transmissionoccurs through the subject element. A room SIR (or composite SIR) refers to theestimate of the A-weighted sound level difference caused by the presence of allmembers that act as noise transmission paths between the source and the receiver.An estimate of the member SIR may be obtained from a set of transmission lossmeasurements in one-third octave or octave bands in the following manner:1. The transmission loss is plotted against frequency.2. The +3 db per octave SIR reference contour is plotted on an overlay sheet,which is then superimposed on the TL versus f plot in the highest position sothat the sum of the deficiencies is less than twice the number of test frequencies.A deficiency is defined as the number of decibels by which the SIR referencevalue exceeds the measured TL of the member.3. The value of TL at 500 Hz on the SIR reference curve constitutes the value ofthe SIR of the member.The statistical quality of SIR prediction is improved by using a larger numberof TL laboratory measurements, and the one-third octave band measurements willgenerally yield more reliable results than those based on octave bands. Manufacturers’catalogues frequently report the one-third octave TL measurements usedto evaluate STC. An example of an SIR determination is given in Figure 12.13for the TL plot of Figure 12.12. Much of the tedium of SIR contour fitting can beeased through the use of computer plotting.The field incidence mass law, expressed by Equation (12.13), can be utilized toextrapolate data. If the transmission loss is known for all applicable frequenciesfor a structural element 1, and it is desired to find the TL of a similar element witha different surface mass, then( )m2TL 2 = TL 1 + 20 log .m 1If the relationship holds in the applicable frequency range, the values of SIR aresimilarly related.Example Problem 6A shell constructed of 5-in. thick concrete with 55 lb/ft 2 surface mass carries arating of SIR 43. Estimate the SIR rating for an 8-in. thick concrete with a surfacemass of 92 lb/ft 2.

300 12. Walls, Enclosures, and BarriersFigure 12.13. SIR determination for TL versus f plot of Figure 12.12.Solution( )m8in.SIR 8in. = SIR 5in. + 20 logm 4in.( ) 92= 45 + 20 log = 49.5 dB55Table 12.2 gives the SIR values of a number of structural elements.12.12 Noise Reduction of a WallLet us consider two rooms that are acoustically separated by a partition of area S win Figure 12.14. L p1 and L p2 denote the spatially averaged values of the soundpressure levels in room 1 and room 2, respectively. We assume the noise sourcein room 1 produces a purely reverberant field near the partition. This occurs if thesound pressure level can be described by Equation (11.23):( ) 4L p1 = L w1 + 10 logR 1where 4/R 1 ≫ Q/(4πr 2 ), R 1 is the room constant of room 1, and r the distancefrom the source in room 1 to the position of measured L p1 . This assumption permitsus to simplify the analysis by assuming a constant sound pressure over the entirearea of the dividing wall.

12.12 Noise Reduction of a Wall 301Table 12.2. Shell Isolation Rating (SIR).Description Weight (lb/ft 2 ) SIR ratingDense poured concrete or solid block walls4 in. thick 50 416 in. thick 73 438 in. thick 95 4612 in. thick 145 4916 in. thick 190 51Hollow concrete block walls6 in. thick 21 418 in. thick 30 43Brick veneered frame walls — 48–53Stuccoed frame walls — 34–52Frame walls with wood siding — 33–40Metal walls, curtain walls — 25–39Shingled wood roof with attic 10 40Steel roofs — 36–51Fixed, single-glazed windows (the higher SIR — 22–39values apply to windows with special mountingsand laminated glass)Fixed double-glazed windows (the higher SIR — 22–48values apply to windows with large spacesbetween the glass and special mountings)Double-hung windowsSingle-glazed — 20–24Double-glazed — 20–29Casement windows, single-glazed — 19–29Horizontal sliding windows 16–24Glass doors 2.3–3 24Wood doors (the higher SIR values apply toweatherstripped doors)Hollow core 1.2–5 14–26Solid core 4–5 16–30Steel doors (the higher SIR values apply toacoustical doors) 4–23 21–50Figure 12.14. Two rooms separated by a dividing wall of area S w .

302 12. Walls, Enclosures, and BarriersThe partition absorbs a certain amount of power W α from the reverberant fieldin room 1, namely( )Sw α wW α = W r (12.26)S 1 α 1whereW α = power absorbed by the dividing wallW r = power in the reverberant fieldS w = area of the wallα w = absorption coefficient of the wallS 1 = total surface area of room 1α 1 = average absorption coefficient.Let us assume that all the power incident upon the wall will be absorbed, i.e.,α w = 1. The portion of the power W 1 of the source in room 1 that becomes thepower in the reverberant field is given byW r = (1 − ᾱ)W 1Substituting the above into Equation (12.26) givesW α = (1 − ᾱ)W 1S wS 1. (12.27)The power W α absorbed by the wall can be expressed in terms of the room constantR 1 , so Equation (12.27) becomes1 − ᾱ 1W α = W 1 S w = W 1S wSᾱ 1 R 1which can now be combined with Equation (12.2) to yieldW 2 = W 1S w τ(12.28)R 1where W 2 represents the power transmitted into room 2 and τ the transmissioncoefficient of the wall.We can consider the direct field in the region near the partition in room 2 to bea uniform plane wave progressing outward from the radiating wall. The energy inthe direct field equals the product of the power transmitted into the room multipliedby the time required for the plane wave to transverse the room. This time is givenby t = L/c, where L denotes the length of room 2. The energy density δ d2 ofthe direct field is given by the directed field energy divided by the room volumeV = S w L:δ d2 = W 2 LV c . (12.29)From Equation (11.21), the reverberant energy is given byδ r2 = 4W 2cR 2. (12.30)

12.12 Noise Reduction of a Wall 303Combining the last two equations gives the total energy δ 2 density near the wall:( 1δ 2 = + 4 )W2S w R 2 c . (12.31)Equation (12.28) can now be used to eliminate W 2 in Equation (12.31) to yieldδ 2 = 4 (W 1 1R 1 c 4 + S )wτ. (12.32)R 2The mean-square pressure in room 2 is given in terms of energy density byp2 2 = ρ 0c 2 δ 2which is then inserted into Equation (12.32) to yieldp2 2 = 4W (1 1ρ 0 cτR 1 4 + S )w. (12.33)R 2From the use of the definitions( ) 2 ( )p2W1L p2 = 10 logand L W 1 = 10 log20 μPa10 pWEquation (12.33) becomes( ) ( ) ( 4 1 1L p2 = L w1 + 10 log − 10 log + 10 logR 1 τ 4 + S )w(12.34)R 2where we have assumed that ρ o c = 407 rayls.In inspecting Equation (12.34) we observe that the first two terms in the righthandside of the equation represents L p1 under the conditions of a reverberant fieldnear the wall in room 1. We also invoke Equation (12.3) which expresses TL interms of transmission coefficient τ, with the result that Equation (12.34) simplifiesto:( 1L p2 = L p1 − TL + 10 log4 + S )w. (12.35)R 2From Equation (12.35) one can estimate the sound pressure level L p2 near the wallin room 2, given the TL of the wall and the acoustic parameters of room 2. On theother hand, if we know the desired value of L p2 , (which is the usual case) we canrearrange Equation (12.35) to find the necessary transmission loss of the wall asfollows:( 1TL = L p1 − L p2 + 10 log4 + S )w. (12.36)R 2The above equation is valid in both English and metric units.The term L p1 − L p2 is referred to as the noise reduction denoted by the termNR. Equation (12.36) becomes( 1NR=TL− 10 log4 + S wR 2). (12.37)

304 12. Walls, Enclosures, and BarriersWe note if room 2 is entirely absorbent, i.e., it supports no reverberation field or ifthe wall is an outside wall that it radiates outdoors, the value of R 2 approaches aninfinite value and Equation (12.37) revises toNR = TL + 6dB. (12.38)Example Problem 7A20ft× 8 ft wall with a transmission loss of 32 dB separates two rooms. Room 1contains a noise source that produces a reverberant field with a SPL = 110 dBnear the wall. Room 2 has a room constant R 2 of 1350 ft 2 . Determine the soundpressure level near the wall in room 2.SolutionWe apply Equation (12.32) to find the value of L p2 :( )1 20 × 8ft2L p2 = 110 − 32 + 10 log +4 1350 ft 2 dB= 73.7dB12.13 Sound Pressure Level at a Distance from the WallFor the most part we wish to predict the sound pressure level at some distancesfrom the wall. For appreciable distances the reverberant field will dominate overthe direct field. In Figure 12.14, it is desired to find L p3 at a point located somewhatfar from the wall, which results from the sound pressure level L p1 in room 1. Butit should be noted that the difference (L p1 − L p3 ) does not represent noise reduction,because region 3 in the chamber of room 2 is not directly contiguous to thewall.An expression for L p3 will be derived on the basis that it consists almost entirelyof the energy density in the reverberant field, i.e., the energy density is givenbybut since p 2 = ρ 0 c 2 δ, Equation (12.39) becomesδ r2 = 4W 2R 2 c = δ 3 (12.39)p 2 3 = 4W 2ρ 0 cR 2. (12.40)Here p3 2 represents the mean-square pressure in those regions of room 2 where thedirect field emerging directly from the wall as a plane wave has already dispersed

12.14 Enclosures 305to the extent that the field can be considered a superposition of randomly reflectedcomponents, i.e., it is essentially a reverberant field. Equation (12.28) can be usedto eliminate W 2 in favor of W 1 in Equation (12.40) which then becomesp3 2 = ρ 0 c 4W 1 S wτR 1 R 2and because τ = W 2 /W 1 , L p3 = 10 log (p 3 /20 μPa), we will obtain( )SwL p3 = L p1 + 10 log − TL (12.41)R 2where ρ o c = 407 rayls, and L p3 denotes the sound pressure level at a distancesufficiently far from the wall that the direct field is negligible in comparison withthe reverberant field.Example Problem 8Find the sound pressure level of room 2 at a large distance from the wall in theExample Problem 7.SolutionApplying Equation (12.41) yields12.14 EnclosuresL p3 = L p1 + 10 log= 110 + 10 log(Sw)− TLR 2( ) 20 × 8− 32 = 68.7 dB1350Enclosures may be categorized as being either full enclosures or partial enclosures.These structures of varying sizes may enclose people or noise generatingmachinery. It is advisable to never enclose more volume than necessary. For example,an entire machine should not be enclosed if only one of its components(such as a gear box, motor, etc.) constitutes the principal noise source. In somecases, a partial enclosure may suffice to shield a worker from excessive exposureto noise. The walls of an enclosure should be constructed of materials that willprovide isolation, absorption, and damping—all necessary for effective noise reduction.Moreover, any presence of cracks or leaks can radically reduce the noisereduction of an enclosure, so all mechanical, electrical, utility, and piping outletsmust be thoroughly sealed. Access panels should fit tightly, and viewing windowsshould be constructed of double panes and be impervious to acoustical leakages.The interior of the enclosures should be lined with highly absorbing material so

306 12. Walls, Enclosures, and BarriersFigure 12.15. Plan view of a hood with energy densities δ 1 and δ 2 just inside and outsidethe hood.that the sound level does not build up from reflections, thereby decreasing the wallvibration and the resultant radiation of the noise.Hoods constitute a special category of acoustical full enclosures designed specificallyfor the purpose of containing and absorbing excessive noise from a machine.A hood is designed to minimize leakages at its physical input and output, and accessmust be provided to allow periodic servicing. Usually hoods are sized so thatthe enclosed machine component does not occupy more than a third of the internalvolume. The effectiveness of a hood is denoted by the amount of noise reductionand insertion loss.In the equilibrium condition, the total power W 1 radiated by a source is absorbedby the interior walls of the hood. In Figure 12.15 δ 1 and δ 2 , respectively, denotethe energy densities of the interior and the exterior of the hood in the immediatelyvicinity of the enclosure wall. We assume the wall to be thin enough so thatvolumes V 1 and V 2 are nearly equal and denoted by V . As the time t required for anacoustic wave to travel distance L is given by L/c, the energy contained in volumeV 1 may be approximated by E 1 = W 1 L/c, with the energy density now givenbyδ 1 = E 1= W 1LV 1 Vc

12.14 Enclosures 307and similarly for the external side of the wallδ 2 = E 2= W 2LV 2 Vcand because the transmission coefficient τ = W 2 /W 1δ 2 = τ W 1L= τδ 1 .VcUsing the relationship p1 2 = ρ 0c 2 δ 1 and p2 2 = ρ 0c 2 δ 2 , and the fact L p =10 log (p/p ref ) 2 we readily obtain(L p1 = 10 logρ 0 c 2 δ 1p 2 refNoise reduction NR is given by)andL p2 = 10 logNR = L p1 − L p2 = 10 log(1/τ)(ρ 0 c 2 δ 2p 2 refwhich we note is also the definition for transmission loss TL. Hence,NR = TLwhich indicates that the noise reduction from just within the hood (located ina virtual free field) to a region very near the external hood wall is equal to thetransmission loss of the wall alone. It should be realized here the value of L p1 issupposed to be the value obtained with the hood in place, not the original lowervalue of L p1 measured in the vicinity of the noise source before the hood is placedover it. L p2 will be correspondingly higher, as the result of the hood insertion. Asa result, it is more useful to know the insertion loss IL of the hood rather than thenoise reduction. The insertion loss IL is given by).IL = L p0 − L p2 (12.42)where L p0 denotes the sound pressure level at a selected location without the hood;and L p2 , the sound pressure level at the same point with the hood enshrouding thesound source.In deriving an approximate expression for insertion loss, we shall assume thatthe room is considerably larger than the hood. The sound pressure level L p0 at anylocation in the room may be found from Equation (11.21)(Q0L p0 = L w0 + 10 log4πr + 4 ). (12.43)2 RIn a similar fashion we can express SPL at the same location with the hood in placeby(Q2L p2 = L w2 + 10 log4πr + 4 ). (12.44)2 RWe now need to determine the total power W 2 emitted by the noise source andits hood acting as a single unit. Since the combination of noise source (usually

308 12. Walls, Enclosures, and Barriersa machine) and hood maintains a fairly equilibrium condition, the space averageof the time-average energy density remains a constant value under the hood. Theentire acoustic power emitted by the source is absorbed by the hood as losses oris radiated through the walls and from thence outside the hood. That amount ofacoustic energy which passes through the hood is W 2 , which can be approximatedby Equations (12.26) and (12.28):whereW 2 = W 0(Sh α hᾱW 2 = power radiated into the room by the hoodW 0 = acoustic power of the sourceS h = area of the walls and ceiling of the hoodα h = absorption coefficient of the walls and ceilingS = total area under the hood)¯τ (12.45)ᾱ = average value of sound absorption coefficient under the hood¯τ = average value of transmission coefficient for the hood,excluding the floor.We shall now simplify Equation (12.45) by assuming that the total surface S =S h + S f , where S f is the floor area, is such that S f ≪ S h so that S ≈ S h . Withthe system in equilibrium, we can set α h = 1, which means that all of the energyimpinging on the walls of the hood becomes absorbed one way or the other.Equations (12.45) becomes¯τW 2 = W 0 (12.46)ᾱwhere ¯τ ≤ ᾱ ≤ 1. The limits are established so that when ᾱ approaches unity, thedefining expression for τ is satisfied. But when ᾱ approaches ¯τ, nearly the entireacoustic power of the source is radiated outside the hood.Setting Q 0 = Q 2 = Q in Equations (12.43) and (12.44) (i.e. the Q of the hoodsourcecombination equals that of the source alone) and substituting into Equation(12.42), we obtain( )W0IL = L w0 − L w2 = 10 log . (12.47)W 2Inserting Equation (12.46) into Equation (12.47) yields(ᾱ )IL = 10 log(12.48)¯τwhere ¯τ ≤ ᾱ ≤ 1.In actuality the average absorption coefficient ᾱ has a lower nonzero limit as theresult of air absorption inside the enclosure, viscous losses of waves near grazingincidence in the acoustical boundary layer on inside of the hood, and the changefrom adiabatic to isothermal compression in the immediate vicinity of the inner

12.15 Small Enclosures 309walls. According to Crèmer (1961), the latter two effects may be taken into accountin terms of sound frequency f by the expressionα f = 1.8 × 10 −4√ f .The average excess air absorption coefficient ᾱ ex due to air absorption in a roomor enclosure isα ex = k 4V Swhere k is an experimentally determined constant, V is the enclosed volume, and Sthe room or enclosure area. Combining the last two expressions, we get a minimumᾱ min expressed byᾱ min = α ex + α f = 4kVS + 1.8 × 10−4√ f (12.49)which is also valid for reverberation rooms. Equation (12.49) helps to establishthe upper limit of the reverberation time that can be achieved in an echo chamber.The two limiting cases for Equation (12.48) areIL = 0 dB for ᾱ = ¯τ, IL = 10 log(1/τ) = TL dB for ᾱ = 1.The first case of IL = 0 obviously represents the worst case, and the second casewhere IL = TL represents the best case for the insertion loss of the hood. Thisconnotes that ᾱ should be near unity as possible and ¯τ be made much less thanunity for the most effective noise attenuation by a hood.The above analysis is premised on the presence of high frequencies with a diffusefield inside the enclosure. For a more complete treatment that includes the effectsof low frequencies, the reader is referred to Ver (1973).Example Problem 9Determine the insertion loss of a hood with an average absorption coefficient of0.30 and a transmission coefficient of 0.002.SolutionApplying Equation (12.48)(ᾱ )IL = 10 log¯τ12.15 Small Enclosures( ) 0.30= 10 log = 21.8 ≈ 22 dB0.002A small enclosure is one that fits closely around the noise source. If the noisesource’s geometry contains flat planes that are parallel to the walls of the enclosure,standing wave resonances can occur at frequencies that are integer multiples of

310 12. Walls, Enclosures, and Barriersthe half-wavelengths of the generated noise. This can render the enclosure uselessfrom the viewpoint of noise attenuation unless the inside of the enclosure is linedwith sound-absorption material at least one-quarter wavelength thick. If there isno sound absorption material inside the enclosure, situations can occur where thenoise generated at resonant frequencies may actually be louder outside than wouldbe the case without the enclosure!12.16 Acoustic BarriersThe term acoustic barrier (or noise barrier) refers to an obstacle that interruptsthe line of sight between a noise source and receiver but does not enclose eithersource or receiver. An acoustic barrier may be in the form of a fence, a wall, a berm(a mound of earth), dense foliage, or a building between the noise source and thereceiver. Noise attenuation occurs from the fact that noise transmission through thebarrier is negligible in comparison with refracted noise, particularly if the barrieris solid, without holes or openings, and it is of sufficient mass. Because the soundreaches the receiver by an indirect path over the top of the barrier, the sound levelwill be less than the case would be if the sound had traveled the (shorter) directpath. The refraction phenomenon is highly dependent upon the frequency contentof the sound. The calculations for barrier attenuation are based in part on Fresnel’swork in optics.In Figure 12.16 consider a room prior to inserting a barrier in the position shown.The mean-square pressure at the sound receiver’s location is given by( Qp0 2 = Wρ 0c4πr + 4 )2 Rwhere p0 2 denotes the mean-square pressure without the barrier. The sound pressurelevel is expressed by( QL p0 = L w + 10 log4πr + 4 )2 RwhereL p0 = SPL without the barrierL w = power level of the sourceQ = directivity of the sourceR = room constant without the barrierr = distance between the source and the receiver.Now let us insert the barrier as shown in Figure 12.16. The mean-square pressurep2 2 , at the receiver with the barrier installed is given byp2 2 = p2 r2 + p2 d2 (12.50)

12.16 Acoustic Barriers 311Figure 12.16. Schematic of a room with a barrier in position. Region 1 and region 2 areseparated by a plane defined by ABB ′ A ′ .where the first term on the right of Equation (12.50) is the mean-square pressureat the receiver due to the reverberant field and the second term represents themean-square pressure due to the diffracted field around the edges of the barrier.Recognizing that L p2 = 10 log(p2 2/p2 ref), we can express the sound pressure levelL p2 at the receiver in terms of the reverberant and diffracted fields:( p2L p2 = 10 log r2+ pb22p 2 ref). (12.51)The barrier insertion loss IL is given in terms of SPL by( p2)IL = L p0 − L p2 = 10 log 0. (12.52)For the room of Figure 12.16, we shall assume an equilibrium condition and thatthe area of the barrier is considerably smaller than the planar cross section of theroom, i.e.LH ≫ L b h. (12.53)p 2 2

312 12. Walls, Enclosures, and BarriersIn this situation the reverberant field in the shadow zone of the barrier may beassumed to be the same with or without the barrier’s presence. This reverberantfield represents the minimum SPL in the shadow zone. The space average δ r of thetime-average reverberant energy density in the room without the barrier is givenbyδ r = 4W Rc = p2 rρ 0 c . (12.54)For the condition stated by Equation (12.53) δ r = δ r1 = δ r2 , where the numericalsubscripts denote the regions on each side of the barrier in Figure 12.15. Equation(12.54) can now be rewritten aspr2 2 = 4Wρ 0cR . (12.55)Here pr2 2 denotes the mean-square pressure in the reverberant field of the shadowzone of the barrier. Our next step is to include the mean-square pressure pb2 2 inarea 2 at the location of the receiver due to the diffracted field from the edges ofthe barrier, and this is given by (see Moreland and Musa, 1972):n∑pb2 2 = 1p2 d2= pd2 2 3 + 10N D (12.56)ii=1where pd2 2 represents the mean-square pressure attributable to the direct field withoutthe barrier, the Fresnel number N i is defined bywhereN i ≡ 2d iλ(12.57)d i = difference in direct path and diffracted path between the source and receiveλ = wavelength of the soundand D is the diffraction constant defined by 2n∑ 1D ≡. (12.58)3 + 10Ni=1iIn the case of Figure 12.16, the following path differences exist:d 1 = (r 1 + r 2 ) − (r 3 + r 4 )d 2 = (r 5 + r 6 ) − (r 3 + r 4 ) (12.59)d 3 = (r 7 + r 8 ) − (r 3 + r 4 )2 In some literature the alternative definition for the diffraction coefficient is given byn∑ 1D =3 + 20Ni=1i

12.16 Acoustic Barriers 313In the case of rectangular barriers, three values of d i ,orn = 3, will usuallysuffice to describe the shadow zone of the barrier. Through Equation (11.20) wecan write for the mean-square pressure pd2 2 due to the direct field as follows:pd2 2 = QWρ 0c. (12.60)4πr 2with r constituting the direct length from the source to the receiver. CombiningEquations (12.56) and (12.60) yieldsp 2 b2 = WQDρ 0c4πr 2= WQ B4πr 2 ρ 0c. (12.61)where Q B ≡ Q D is the effective directivity of the source toward the direction ofthe shadow zone. Inserting Equations (12.61) and (12.55) into Equation (12.50)results in( Qp2 2 = Wρ B0c4πr + 4 )2 Rand the corresponding sound pressure iswhereL p2 = L w + 10 logQ B = Qn∑i=1( Q B4πr + 4 )2 R(12.62)λ3λ + 20d i. (12.63)In English units, where distances are expressed in feet instead of meters, Equation(12.62) becomes( Q BL p2 = L w + 10 log4πr + 4 )+ 10. (12.64)2 RFor a rectangular barrier n = 3 in Equation (12.63), and the required path differencesare given by Equation (12.59).From Equations (12.52) and (12.62) we obtain the barrier insertion loss IL:⎛Q⎜IL = 10 log4πr + 4 ⎞⎝2 R ⎟Q B4πr + 4 ⎠ . (12.65)2 RThe above equation applies to either the metric system or English system. It is interestingto note that if the barrier is located in an extremely reverberant environment,such as an echo chamber, the acoustic field reaches the receiver by rebounding unabatedfrom the surfaces of the room to the degree that the effectiveness of the barrierin blocking the direct field becomes insignificant. Because 4/R ≫ Q/(4πr 2 )and 4/R ≫ Q B /(4πr 2 ) for a reverberant room, Equation (12.65) will give a valueof IL = 0 dB.

314 12. Walls, Enclosures, and Barriers12.17 Barrier in a Free FieldThe case of a barrier located outdoors or in an extremely acoustically absorbentroom is of special interest. In a free field the room constant R ≈∞, with the resultthat Equation (12.65) becomes( ) ( ) ( )Q Q1IL = 10 log = 10 log = 10 log . (12.66)Q B QDDwhereD =n∑i=1λ3λ + 20d i. (12.67)In treating the rectangular barrier in a free field or an acoustically high absorptivitychamber we expand Equation (12.67) for n = 3 and insert the result into therightmost term in Equation (12.66) to yield[ ()]111IL =−10 log λ+ +. (12.68)3λ + 20d 1 3λ + 20d 2 3λ + 20d 3If the barrier is of semi-infinite length, i.e., L B ≈∞in Figure 12.16, Equation(12.68) reduces to( )λIL =−10 log. (12.69)3λ + 20d 1Example Problem 10Consider the room of Figure 12.16 to be an anechoic chamber with dimensions20 ft long × 15 ft wide × 12 ft high. A sound source is located in the room1.5 ft above the floor, 8 ft away from one of the shorter walls, in the center and2 ft directly behind a 6-ft-wide × 5-ft-high rectangular barrier. The barrier itselfhas a very high transmission loss rating, and a receiver in the form of a SPL meteris located also in the center 2 ft above the floor and 6 ft away from the source.Determine the insertion loss of the barrier for the 1000-Hz octave band.SolutionThe path differences as given by Equation (12.59) assume the following values:d 1 = (r 1 + r 2 ) − (r 3 + r 4 ) = (4.03 + 4.98) − 6 = 3.01 ftd 2 = (r 5 + r 6 ) − (r 3 + r 4 ) = (3.61 + 4.98) − 6 = 2.59 ftd 3 = (r 7 + r 8 ) − (r 3 + r 4 ) = (3.61 + 4.98) − 6 = 2.59 ft.The wavelength λ at 1000 Hz is found fromλ = c f= 1128 ft/s ÷ 1000 Hz = 1.128 ft

Equation (12.67) becomes(11D = λ+ +3λ + 20d 1 3λ + 20d 2[= 1.128= 0.05912.18 Approximations for Barrier Insertion Loss 315)13λ + 20d 313(1.128)+20(3.01) + 13(1.128)+20(2.59) + 13(1.128)+20(2.59)The insertion loss is then calculated:IL = 10 log(1/D) = 10 log(1/0.059) = 12 dB]12.18 Approximations for Barrier Insertion LossConsider the barrier of Figure 12.17. The Fresnel number N, given by Equation(12.57) can be restated asN =2(A + B − C)λ=2(A + B − C) fcA number of researchers in the field developed and verified analytic models,with a view to apply the results to highway barriers. For a point source locatedbehind an infinitely long solid wall or berm, the attenuation A ′ provided by abarrier are given by the following equations, where the arguments of tan and tanhare given in radians:A ′ = 0 for N < −0.1916 − 0.0635b( ′√ )−2π NA ′ = 5(1 + 0.6b ′ ) + 20 logtan √ for − 0.1916 − 0.0635b ′ ≤ N ≤ 0−2π N( √ )2π NA ′ = 5(1 + 0.6b ′ ) + 20 logtanh √ for 0 < N < 5.032π NFigure 12.17. The geometry of a barrier used in the calculation of the Fresnel number.

316 12. Walls, Enclosures, and Barrierswhere b ′ = 0 for a wall and b ′ = 1 for a berm. The correction factor b ′ allowsfor the experimentally determined fact that berms tend to produce 3 dB moreattenuation than walls of the same height. The above set of equations apply onlyto barriers that are perpendicular to a line between the source and the receiver. Adetailed discussion of attenuation through scattering and diffraction is given byPierce (1981).ReferencesBarry, T. M., and Reagan. J. A. December 1978. FHWA Highway Noise Prediction Model.U.S. Department of Transportation Report FHWA-RD-77-108.Beranek, Leo L. 1971. Noise and Vibration Control. New York: McGraw-Hill: 566–568.Brüel & Kjær. 1980. Measurements in Building Acoustics. Nærum, Denmark: 19.Crèmer, L. 1961. Statistische Raumakustik. Chapter 29. Stuttgart: Hitzel Verlag:.Moreland, J. B., and Musa, R. S. 1972. International Conference on Noise Control Engineering,October 4–6, 1972, Proceedings: 95–104.Pallett, D. Wehrli, Kilmer, R. R., and Quindry, T. 1978. Design Guide for Reducing TransportationNoise in and around Buildings. Washington, DC: U.S. National Bureau ofStandards.Pierce, A. D. 1981. Acoustics, An Introduction to its Physical Principles and Applications.New York: McGraw-Hill.Reynolds, D. D. 1981. Engineering Principles of Acoustics. Boston: Allyn and Bacon.Ver, I. L. 1973. Reduction of noise by acoustic enclosure. Isolation of Mechanical Vibration,Impact, and Noise (J. C. Snowdon, and E. E. Ungar, eds), AMD-Vol. 1, ASMEDesign Engineering Conference,: 192–220.Ver, I. L. and Homer, C. I. 1971. Interaction of sound waves with solid structures. NoiseControl (Beranek, L. L., ed.). New York: McGraw-Hill, 270–361.Problems for Chapter 121. A room is subdivided in its middle by a concrete wall with a transmission lossof 52 dB. The room is 4 m high, and the division occurs across the entire widthof 12.5 m. In the middle of the dividing wall there is a door 2.5 m high × 1.0 mwide. The door’s rated transmission loss is 28 dB. There is a crack underneaththe door that extends across the door’s width and it is 2.5 cm high.(a) Determine the effective transmission loss of this structure.(b) If you could reduce the crack to 0.30 cm (by lengthening the door) whatwill be the change in the overall transmission loss?2. A 5.8 ft 2 sample of a building material is being tested by being placed in awindow between a receiving room and a source room. There is no appreciablesound transmission except that through the sample. Average sound levels inthe 1-kHz octave band are 93 dB in the source room and 68 dB in the receivingroom. The equivalent absorption area of the receiving rooms is 18 ft 2 . Estimatethe transmission loss (TL) and the transmission coefficient (TC) for the sampleat 1 kHz.

Problems for Chapter 12 3173. A 0.75 m 2 sample of building material is placed in a window between areceiving room and a source room. Sound transmission occurs only through thesample. The sound level in the source room is 89 dB and that for the receivingroom, with 3.0 m 2 equivalent absorption, is 69 dB. Find the transmission lossand the transmission coefficient for this material sample.4. A wall consists of the following: a 22-dB (transmission loss at 500 Hz) wooddoor which takes up 25% of the exposed area, 2.2% airspace, and the remainderof the exposed wall area is a solid wall with 52 dB TL. Determine the TL ofthe composite wall at 500 Hz.5. Repeat the above problem but with the airspace reduced to 0.2%.6. Repeat Problem 4, using a 36-dB TL door. Also determine the benefit ofreducing the airspace from 2.2% to 0.2%.7. A room is subdivided in its middle by a concrete wall with a transmissionloss of 62 dB. The room is 4.5 m high, and the division occurs across theentire width of 11.8 m. In the middle of the dividing wall there is a door 2.5 mhigh × 1.0 m wide. The door’s rated transmission loss is 29 dB. There is acrack underneath the door that extends across the door’s width and it is 2.6 cmhigh.(a) Determine the effective transmission loss of this structure.(b) If you could reduce the crack to 0.30 cm (by lengthening the door) whatwillbe the change in the overall transmission loss?8. Develop an equation for transmission loss versus frequency for 4-mm glass inthe mass-controlled region. Glass may be assumed to have a specific gravityof 2.6.9. It is desired to increase the transmission loss of a panel in the mass-controlledregion by 5 dB Find the necessary change in thickness.10. Find the transmission loss and transmission coefficient for 8-mm glass at thecritical frequency. Assume a loss factor of 0.06.11. Redo Problem 9 for 14-mm glass with a loss factor of 0.1.12. Consider a double wall panel with airspace h (given in mm). One panel has70% the surface mass of the other, and the sum of the surface masses is m t (inkg/m 2 ). Plot the resonant frequency versus the product of m t and h.13. Determine the resonant frequency of a double-paneled partition constructedof 4 lb/ft 2 and 6 lb/ft 2 panels with 5.2-in. airspace.14. A wall was measured for its transmission loss in one-third octaves beginningat 125 Hz. The values were: 25, 24, 30, 32, 39, 41, 41, 46. 47, 49, 47, 46, 48,49, 45, and 46. Find the sound transmission class (STC).15. Redo Problem 14 with the TL values in the four highest third-octaves being39, 40, 38, and 39.16. Find the SIR for the wall of Problem 14.17. A 6 m × 2.5 m wall with a transmission loss of 35 dB separates two rooms.A noise source in one room yields a reverberant field with a sound pressurelevel of 120 dB near the wall. The other room has a room constantR = 130 m 2 . Predict the sound level pressure near the wall in the latterroom.

318 12. Walls, Enclosures, and Barriers18. Predict the insertion loss of a hood that has an average absorption coefficientof 0.4 and a transmission coefficient of 0.003.19. Determine the insertion loss at 1000 Hz of an outdoor noise barrier with anoise source 1.2 ft above the ground, 12 ft away from the wall that is 8 ft talland 20 ft wide. The SPL meter is located 3 ft above ground and is 6 ft awayfrom the wall. Both source and receiver are equally far from the ends of thewall.

13Criteria and Regulations forNoise Control13.1 IntroductionLoss of hearing constitutes only one of the effects of sustained exposure to excessivenoise levels. Noise can interfere with speech and sleep, and cause annoyanceand other nonauditory effects. Annoyance is quite subjective in nature, rendering itdifficult to quantify. Loud steady noise can be quite unpleasant but impulse noisecan be even more so. Moreover, impulse noise involves a greater risk of hearingdamage. Community response, which can range anywhere from simple telephonecalls to municipal authorities to massive public demonstrations of outrage, can beused as a measure of annoyance to citizens.In this chapter we discuss these effects and describe the criteria and regulationsthat have been implemented over nearly three decades for the purpose of controllingenvironmental noise. In a broader sense, the term environmental noise pertainsto noise in the workplace and in the community. This pertains to noise sources arisingfrom operation and use of industrial machinery, construction activities, surfaceand air transportation. We must also consider recreational noise which arise fromthe use of snowmobiles, drag strip racers, highly amplified stereo systems in thehome and in motor vehicles, and so on. A considerable number of regulations existin a number of nations, for the purpose of protecting hearing from overexposureto loud noise, providing salutary working conditions, shielding communities fromexcessive environmental noise arising from the presence of manufacturing plantsor construction activities and nearby surface and air transportation. The enactmentand enforcement of these regulations motivates the control of noise levelsin the workplace and in the community, not only for the purpose of promoting aproper auditory environment but also to avoid penalties that can be levied for uncorrectedviolations. The principal U.S. laws germane to the issue of noise controlare: National Environmental Policy Act of 1969 Noise Pollution and Abatement Act of 1970 Occupational Safety and Health Act of 1970 Noise Control Act of 1972.319

320 13. Criteria and Regulations for Noise ControlAs the corollary to these acts, the Environmental Protection Agency (EPA)(1973), the Department of Labor (DOL), and the Department of Transportation(DOT) have been designated as the principal federal agencies to issue and enforcenoise control regulations.Under the National Environmental Policy Act of 1969, any federally fundedconstruction project requires the preparation and submission of an EnvironmentalImpact Statement (EIS), which assesses the public impact of the noise thatwill be generated by the erection and operation of the completed project. Thecreation of the Office of Noise Abatement and Control (ONAC) as a branch ofEPA was sanctioned by the Noise Pollution and Abatement Act of 1970. Thisoffice carries total responsibility for investigating the effect of noise on publichealth and welfare. The Occupational Safety and Health Act of 1970 set up themechanics of enforcing safe working conditions, of which noise exposure criteriaconstitute a part. The Noise Control Act of 1972 probably contains the mostimportant piece of federal legislation in regard to noisy environments. While EPAis given the primary responsibility for monitoring sound levels in the community,this legislation provides for a division of powers among the federal, state, and localgovernments.13.2 Noise Control Act of 1972With the passage of the Noise Control Act of 1972, the U.S. Congress establishedthe national policy of promoting an environment that is free of excessivenoise that would be deleterious to health, safety, and welfare. EPA was chargedwith the responsibility of coordinating federal efforts in noise control research,identification of noise sources and the promulgation of noise criteria and controltechnology, the establishment of noise emission standards for commercial products,and the development of low-noise emission products. This agency also bearsthe responsibility of evaluating the adequacy of Federal Aviation Administration(FAA) flight and operational noise controls and the adequacy of noise standards onnew and existing aircraft (including recommendations on retrofitting and phaseoutof existing aircraft). However, under the provisions of Section 611 of the FederalAviation Act of 1958, FAA retained the right to prescribe and amend standardsfor the measurement of aircraft noise and sonic boom, but EPA can raise objectionsto FAA standards that, in the opinion of EPA, do not protect public healthand welfare. Irreconcilable differences may be resolved by filing of a petition forreview of action of the Administrator of EPA or FAA only by filing in the UnitedStates Court of Appeals for the District of Columbia Circuit. EPA may subpoenawitnesses and serve relevant papers to obtain information necessary to carry outthe act.Section 17 of the Noise Control Act also assigns the task of developing noiseemission regulations for surface carriers engaged in interstate commerce by railroads,in consultation with the Secretary of Transportation. States and local

13.3 The Occupational Safety and Health Act of 1970 321governments are enjoined from adopting different standards except when renderednecessary by special local conditions and then only with the permission ofEPA and the Secretary of Transportation. The provisions of Section 18 are nearlyidentical to those of Section 17 except they apply to motor carriers engaged ininterstate commerce.13.3 The Occupational Safety and Health Act of 1970Under the directive of the Walsh–Healey Public Contract Act which was passed byCongress in 1969, the U.S. Department of Labor developed the first occupationalnoise exposure standard. In 1970, the Occupational Safety and Health Act waspassed to apply the requirements of the standard to cover all workers engaged ininterstate commerce. The Occupational Safety and Health Administration (OSHA)exists as an arm of the Department of Labor. The assigned task of this agency is toestablish safety regulations (including those pertaining to noise levels) and enforcethem.According to OSHA regulations, the daily noise dose D in percent is given interms of slow-response time-average A-weighted sound levels byD = 100n∑i=1C iT i(13.1)where C i is the duration of exposure to a specific sound pressure level (SPL) and T iis the allowable daily duration for exposure to noise at that value of SPL. Table 13.1lists the A-weighted slow-response noise levels and their corresponding maximumdaily exposure times T i . Under OSHA regulations noise dosage exceeding 100%is not permitted. It will be noted in Table 13.1 that each 5 dB(A) increase in soundlevel cuts in half the allowable exposure time. For a time-average A-weightedaverage of 90 dB(A), the permitted exposure time is 8 h. Elevation of this noiseTable 13.1. OSHA Noise ExposureLimits (OSHA, 1978).SPL, dB(A)(slow response)Permissible Exposure,h/day90 892 695 497 3100 2102 1.5105 1110 0.5115 0.25 or less

322 13. Criteria and Regulations for Noise Controllevel by 5 dB(A) cuts the allowable exposure time to 4 h. Exposure to noise levelsabove 115 dB(A) is not permitted. Moreover, for exposure to noise having a slowresponsetime-average A-weighted sound level of any value, the instantaneouspeak sound pressure level may not exceed 140 dB. It should also be noted thatthe dosage levels are considered to be attributable to noise that actually reach aworker’s station at the ear level. If a worker moves from place to place in thecourse of his or her occupational assignment, the dosage must be based on thenoise exposure and the time spent at each station during the work period.Example Problem 1Monitoring of a factory noise environment at a worker’s station yielded the followingtime samplings over the course of a normal 8 h: Compute the dosage andcomment on the result.Exposure Level, dB(A)Exposure Period, h85 1.590 3.092 2.595 0.597 0.3100 0.2SolutionApplying Equation (13.1) and using Table 13.1, we obtain( 3D = 100 ×8 + 2.56 + 0.54 + 0.33 + 0.2 )= 112%2This dosage of 112% violates OSHA regulations. However, if this 8-h workdayis cut down by 100/117 to 6.8 h or less (assuming the noise level distribution isfairly consistent throughout the day), the daily dosage will not exceed 100%. It canalso be arranged to move the worker to a region, where the noise level averages86 dB(A) or less (i.e., below the OSHA noise level range), for 1.16 h in order toround out 8 h daily on the job.Annual audiology tests of all workers exposed to this environment are mandatedif the noise level equals or exceeds 85 dB(A) over the course of a work day.Corrective measures must be taken when the dosage D exceeds unity. Aside fromshortening time exposures to the offending noise, noise control measures describedin the next chapter must be taken to decrease the sound level pressure of noisesources. Hearing protection devices such as earplugs, earmuffs, and special helmetsare considered to be temporary stopgaps, and these do not necessarily remove theobligation of the work facility management to cut down on noise levels. OSHA

13.4 Perception of Noise 323inspectors have the discretion to levy fines for violations that are not removed afteran initial visit to the facility.13.4 Perception of NoiseLoudness is not the only characteristic of noise that determines the degree ofannoyance. There are other factors, acoustical as well as nonacoustical, that areimportant. In a classic series of laboratory studies conducted by Kryter and hisassociates (1959, 1963), human subjects rated sounds of equal duration accordingto their noisiness, annoyance, or unacceptability. Through the use of octave bandsof noise Kryter with others established equal noisiness contours, which resemblethose for equal loudness, but less acoustic energy is required at higher frequenciesto produce equal noisiness and more is needed at low frequencies.The unit of noisiness index is the noy N (this term apparently derives from thesecond syllable of the word annoy). Figure 13.1 displays the equal loudness indexcurves. In order to determine the logarithmic measure of the perceived noise level(PNL), the following standardized procedural steps need to be taken:1. Obtain and tabulate the octave band (or 1/3 octave band) sound pressure levelsproduced by the noise.2. Use Figure 13.1 to calculate the noisiness index for each octave (or 1/3 octave)band. Then determine the total noisiness index from either one of the followingtwo expressions:( )∑N t = N max + 0.3 N i − N max (13.2a)iN t = N max + 0.15( ∑iN i − N max)(13.2b)Equation (13.2a) applies to one-octave bands and (13.2b) to one-third octavebands. For each octave (or 1/3 octave) band, N max represents the maximumvalue of N i found within that band.3. The total perceived noisiness index N t , which is summed over all frequencybands, is converted to the perceived noise level PNL (or L PN ) through therelationshipL PN = 40 + (33.22) log N t (13.3)There have been some questions raised regarding the validity of this procedurebecause the listeners in the laboratory trials do not always seem to distinguishthe difference between loudness and noisiness and annoyance. But this procedureis now being used to evaluate single-event aircraft noise. The U.S. FederalAviation Administration (FAA) adopted the PNL method to certify newaircraft.

324 13. Criteria and Regulations for Noise ControlFigure 13.1. Equal loudness index curves.13.5 Effective Perceived Noise LevelNoise of longer duration is obviously more annoying than noise of short duration.Moreover, if pure tones are present in the broadband noise system, the noise isjudged to be even noisier than noise without such tones. In order to take into

13.6 Indoor Noise Criteria 325account the factors of duration and the presence of pure tones, the effective noiseperception level (EPNL) has been defined asL EPN = L PN + C + D. (13.4)Here C is the correction factor for pure tones and D is the duration correction. Thetone correction varies from 0 dB up to a maximum of 6.7 dB. The estimation of Centails a complex procedure (Edge and Cawthorn, 1977) that involves examinationof the band spectra to detect any band whose level exceeds the level of the bands toeither side. The duration correction D, expressed in decibels, which accounts forduration of the noise, may be either positive or negative but it is usually negative.It is calculated from( d/0.5)∑D = 10 log 10 antilog L PNT(k)− 13 − L PNTmax (13.5)10k=0Here d represents the total length of the time elapsed when the noise begins toexceed the background level to the moment when it falls back to the level ofimperceptibility. The number 0.5 represents the increment index, i.e., if the totalduration d of the detectable sound is 5 s, then ten intervals are being consideredin the summation of Equation (13.5). L PNT is the tone-corrected value of L PN (i.e.L PNT = L PN + C).13.6 Indoor Noise CriteriaIn order to render communication possible in both indoor and outdoor areas at work,it is necessary to minimize the speech interference arising from the background.The A-weighted sound level can be utilized to determine the acceptability of indoornoise, but it cannot give an indication as to which part of the frequency spectrumis responsible for interference. A number of noise evaluation curves are availablefor rating the acceptability of noise in indoor situations. The most frequentlyused families of curves are the noise criterion (NC) curves, noise rating (NR)curves, room criterion (RC) curves, and balanced noise criterion (NCB) curves.These curves were developed in order to provide criteria to either determine theacceptable noise levels in buildings or to specify the acceptable noise in buildings.Noise Criterion CurvesThe NC curves of Figure 13.2 were the result of an exhaustive series of interviewswith people in offices, industrial spaces, and public areas. It was found that theprinciple concern is the interference of noise with speech communication andlistening to radio, television, and music. In order to find the NC rating of a particulararea, the octave-band sound pressure levels of the noise are measured and plottedon the family of the NC curves of Figure 13.2. The highest curve penetrated byany octave band and pressure level of the measured noise defines the NC value forthe spectrum.

326 13. Criteria and Regulations for Noise ControlFigure 13.2. Noise criterion (NC) curves.Example Problem 2A noise generator in a room was found to yield 50 dB straight across the octaveband from center frequencies 63–8000 Hz. What is the RC rating for that room?SolutionDraw a horizontal line at the 50 dB level. The highest NC curve penetrated is 50.The NC value is therefore NC 50.Noise Rating CurvesThe noise rating (NR) curves of Figure 13.3 are very similar to the NC curves.Their original purpose was to determine whether noise heard from industrial plantsis acceptable at nearby apartments and houses. As with the NC curves, the noisespectrum is also measured at the affected locations and plotted in Figure 13.3. TheNR curves differ in that they include corrections for time of day, fraction of thetime the noise is heard, and the type of neighborhood. In the range between 20 and50 for NR or NC, there is little difference between results obtained from the twosets of calculated procedures.

13.6 Indoor Noise Criteria 327Room Criterion CurvesFigure 13.3. Noise rating (NR) curves.The difficulty with NC curves is that they are not defined in the low frequencyrange—i.e, the 160 and 31.5 Hz octave bands—and complaints have been registeredthat they allow too much noise above the 2 kHz range. Accordingly, Blazierdeveloped a set of room criterion curves on the basis of an extensive study conductedfor ASHRAE of generally acceptable background spectra in 68 unoccupiedoffices. Most of the A-weighted sound pressure levels lie in the range of 40–50 dB.Blazier determined that the curve he derived from the measure data had a slope ofapproximately –5 dB/octave, and he developed a family of straight lines with thisslope (cf. Figure 13.4). It was also established that intense low-frequency noise75 dB or more in region A of Figure 13.5 is apt to cause mechanical vibrationsand rattles in lightweight structures. Noise in region B has a low probability togenerate such vibration. The RC value of a measured spectrum is defined as thearithmetic average of the sound levels at 500, 1000, and 2000 Hz. These curves arebased on measurements made with air conditioning noise only, so they are mainlyused in rating the noise of HVAC systems.Balanced Criterion CurveBeranek (1989a,b) modified the NC curves by adding the 16- and 31.5-Hz octavebands and modifying the slope of the curves so that it became –3.33 dB/octavebetween 500 and 8000 Hz. He also incorporated regions A and B of the RC curves

328 13. Criteria and Regulations for Noise ControlFigure 13.4. Room criterion (RC) curves.of Figure 13.4. The rating number of the resulting balanced noise criterion (NCB)curve, shown in Figure 13.5, represents the arithmetic average of the octave-bandlevels with center frequencies of 500, 1000, 2000, and 4000 Hz. Figure 13.5 isuseful for rating air-conditioning noise in buildings. Table 13.2 (Beranek andVer, 1992) lists the recommended categories of NCB curves for different uses ofbuilding interior space.Example Problem 3In Figure 13.5, the background noise spectrum from air conditioning is plotted as adashed line. Find the NCB rating and comment on the noisiness of the equipment.SolutionWe note that this plot is tangent to the NCB-40 curve and no octave-band levelexceeds this curve. This is a spectrum that might be acceptable for a business officebut is barely acceptable in a bedroom or a residential living room. According toTable 13.2, the NCB-40 rating is not at all acceptable in a concert hall, theater, ora house of worship.

13.7 Equivalent Sound Level, Day–Night Equivalent Sound Level 329Figure 13.5. Balance noise criterion (NCB) curves.13.7 Equivalent Sound Level, Day–Night Equivalent SoundLevel, and Equivalent Day–Evening–Night Sound LevelFrom Chapter 3, we repeat here the definition of Equation (3.29) for the equivalentsound pressure level L eq , which is the A-weighted sound pressure level averagedover a measurement period T .( 1L eq = 10 logT∫ T0) (10 L/10 1dt = 10 logNN∑10 L n/10n=1)(13.6)

330 13. Criteria and Regulations for Noise ControlTable 13.2. Recommended NCB Curve Categories on the Basis of Interior Use andSuggested Noise Criteria Range for Steady Background Noise.ApproximateType of Space (and Acoustical Requirements) NCB Curve a L A ,dBBroadcast and recording studios (distant microphone pickupused)Concert halls, opera houses, and recital halls (for listening tofaint musical sounds)Large auditoriums, large drama theaters, and large churches(for very good listening conditions)Broadcast, television, and recording studios(close microphone pickup used only)Small auditoriums, small theaters, small churches, musicrehearsal rooms, large meeting and conference rooms(for very good listening), or executive offices andconference rooms for 50 people (no amplification)Bedrooms, sleeping quarters, hospitals, residences,apartments, hotels, motels, etc. (for sleeping, resting,relaxing)Private or semiprivate offices, small conference rooms,classrooms, libraries, etc. (for good listening conditions)Living rooms and drawing rooms in dwellings (for conversingor listening to radio and television)Large offices, reception areas, retail shops and stores,cafeterias, restaurants, etc. (for moderately good listeningconditions)Lobbies, laboratory work spaces, drafting and engineeringrooms, general secretarial areas (for fair listeningconditions)Light maintenance shops, industrial plant control rooms, officeand computer equipment rooms, kitchens and laundries(for moderately fair listening conditions)Shops, garages, etc. (for just acceptable speech and telephonecommunication). Levels above NC or NCB 60 are notrecommended for any office or communication situation.For work spaces where speech or telephone communication isnot required, but where there must be no risk of hearingdamage.10 1810–15 18–23Not to exceed 20 28Not to exceed 25 33Not to exceed 30 3825–40 38–4830–40 38–4830–40 38–4835–45 43–5340–50 48–5845–55 53–6350–60 58–6855–70 63–78a See Figure 13.5.This averaging time T can be chosen to be anywhere from a few seconds tohours. L eq can be readily measured through the use of an integrating sound levelmeter. Because it takes into account both magnitude and duration, the equivalentsound level has proven to be a viable parameter for evaluating environmental noisefrom industry, railroads, and traffic. L eq is found to correlate very well with thepsychological effects of noise.In order to account for different response of people to noise at night, the U.S.Environmental Agency developed the day–night equivalent level (DNL), as defined

13.7 Equivalent Sound Level, Day–Night Equivalent Sound Level 331by Equation (3.30), which we repeat here in modified form:L dn = 10 log 15 ( 10 L d /10 ) + 9 ( 10 (L n+10)/10 )24(13.7)where L d represents the 15-h daytime A-weighted equivalent sound level (from7:00 a.m. to 10:00 p.m.) and L n is the 9-h nighttime equivalent A-weighted soundlevel (from 10 p.m. to 7:00 a.m.). The nighttime value carries a penalty of 10 dBbecause noise at night is deemed to be much more disturbing than noise generatedduring the day. The use of the day–night equivalent level as a parameter is increasingin the United States and some nations for evaluating community noise andsome cases of airport noise. The U. S. Federal Interagency Committee on UrbanNoise (FICON) adopted DNL as the descriptor of environmental noise that affectsresidences.In 2002, the European Union Parliament and Council issued Directive2002/49/EC in an effort to establish common assessment methods for rating environmentalnoise in terms of harmonized indicators for the determination ofnoise levels. The concrete figures of any limit values are to be determined bythe Member States of the European Union. The common noise indicators adoptedare day–evening–night equivalent level L den to assess annoyance and the nightequivalent level L night to assess sleep disturbance. L den and L night are defined asfollows:L den = 10 log 1)(12 × 10 L day10 + 4 × 10 L evening +510 + 8 × 10 L night +101024whereL day = A-weighted average sound level determined over 12-h day periodsof the yearL evening = A-weighted average sound level determined over 4-h evening periodsof the yearL night = A-weighted average sound level determined over 8-h night periodsof the year.We note that +5 dB(A) and +10 dB(A) penalties have been imposed to reflect theneed for quieter periods of evenings and nights, respectively. While the day spans12 h, the evening 4 h, and night, 8 h, the Member States may shorten the eveningperiod by 1 or 2 h and lengthen the day and/or the night period, correspondingly,provided the measurements are consistent for all sources and the reporting Statesprovide the European Commision with information on any systematic differencefrom the default option. The default values of (local) time are 0700–1900 h forday, 1900–2300 h for evening, and 2300–0700 h for night. Use of L den will likelyincrease in the future for airport noise and other environmental assessments inEuropean Union.

332 13. Criteria and Regulations for Noise ControlFigure 13.6. Example of a cumulative distribution and its percentile sound levels.13.8 Percentile Sound LevelsMuch of environmental noise tends to have a great deal of fluctuations in soundlevels. There have been indications that unsteady noise, which occurs from noisesources such as passing aircraft or ground vehicles, is more disturbing than steadynoise. In order to consider fluctuations in noise levels and the intermittent characteristicsof some noises, percentile sound levels are used internationally asdescriptors of traffic and community noise. The level L n represents the soundlevel exceeded n% of the time. For example, L 20 represents the sound level exceeded20% of the time. In Figure 13.6 a cumulative distribution is given withexamples of L 10 , L 50 , and L 90 levels. In this example, L 10 = 85 dB(A) denotesthat 85 dB(A) is the level exceeded 10% of the time. L 50 = 75 dB(A),and this is termed the median noise level, because half the time the fluctuatingnoise level is greater than this value and the other half of the time the noise isless.In Japan the median noise level is used to describe road traffic noise. Levelssuch as L 1 or L 10 are used to describe the more intense short-duration noises. InAustralia and in the United Kingdom, L 10 is applied (over an 18-h period, from0600 to 2400 h) as target values for new roads and for insulation regulations fornew roads. High percentage levels such as L 90 or L 99 are usually used to denotethe minimum noise level or residual or background noise level.13.9 Rating of Aircraft NoiseAircraft noise is a cause of increasing concern in many nations. A considerableamount of effort has gone into developing the means for predicting and evaluatingthe annoyance caused by aircraft noise in communities. A 1995 survey (Gottlob,1995) of rating measures revealed that 11 measures are in use in 16 countriessurveyed. We discuss a few of these measures below.

Composite Noise Rating13.9 Rating of Aircraft Noise 333The composite noise rating CNR traces its history as far back as the early 1950s.The CNR, originally used as a basic parameter, measures the level rank on thebasis of a seta set of curves placed about 5 dB apart in the mid-frequency range,in nearly the same fashion as the NC and NR curves. The level rank was obtainedby superposing the noise spectrum on the curves and determining the highest zoneinto which the spectrum protruded. The rank thus found was then modified toinclude algebraic corrections to reflect the spectrum characteristics, peak factor,repetitive nature, background noise level, time of day, adjustment to exposure,and even public relations factors. The realized value of CNR was associated with arange of community annoyance categories established by case histories, which canrange from no annoyance through mild annoyance, varying degrees of complaints,threats of legal action, to downright vigorous community response.The CNR had to be modified to meet the advent of jet aircraft (first militaryand then commercial) in the late 1950s. Instead of being assigned a level rank, themilitary aircraft noise was converted into an equivalent sound pressure level (SPL)in the 300–600 Hz range on the basis of a set of curves similar to the level-rankcurves. The time-varying SPL was averaged and modified by corrections similarto those described in the preceding paragraph. When commercial jetcraft arrivedon the scene, the CNR was modified to use the perceived noise level (PNL) insteadof SPL. The final version of CNR is of the form:CNR = PNL max + N + Kwhere PNL max denotes the average maximum perceived noise level for individualaircraft flyover events (either landing or take-off) for a 24-h period, N is a correctionfactor for the number of aircraft flyovers, and K is an arbitrary constant. The CNRmethodology has proven to be useful for predicting community response to aircraftnoise, ranging on a scale from no reaction to vigorous community response. Eachground point can be represented byCNR ij = PNL ij + 10 log(N d,ij + 16.7N n,ij ) − 12 (13.8)where N d,ij and N n,ij represent the number of daytime and nighttime events foreach aircraft class iand flight path j. A penalty of 10 dB is imposed on the nighttimeflights, and the factor of 16.7 takes into account that there are fewer nighttime hoursthan daytime. The following expression gives the total CNR for the ground point:CNR = 10 log ∑ i∑jantilog CNR ij10(13.9)But Equation (13.9), the final version of CNR, does not correct for backgroundnoise, prior history, public relations, or other factors such as the occurrence ofpure tones. We discussed CNR here even though it is no longer used, because thisparameter serves as the precursor of currently used noise measures and descriptorssuch as NEF and NNI, which are described below.

334 13. Criteria and Regulations for Noise ControlNoise Exposure ForecastThe noise exposure forecast (NEF) resembles CNR, but it uses the effective perceivednoise level EPNL instead of PNL, thus automatically incorporating theannoying effects of pure tones and the duration of flight events. NEF can be expressedasNEF = EPNL + N + Kwhere EPNL designates the average effective perceived noise level for individualaircraft flyovers over a 24-h period, N provides the correction for the number offlyover events, and K is an arbitrary constant. As was done for CNR in Equation(13.8). NEF can be calculated for NEF ij for a certain ground point:NEF ij = EPNL ij + 10 log(N d,ij + 16.7 N n,ij ) − 88 (13.10)Here EPNL ij denotes the EPNL for aircraft class i and flight path j; and N d,ijand N n,ij are the number of daytime and nighttime events, respectively, for eachaircraft of class i and flight path j. In order to distinguish NEF values from CNRvalues, the constant 88 was selected for Equation (3.10). ThenNEF = 10 log ∑ ∑antilog NEF ij(13.11)10i jHowever, the use of NEF has been replaced by the day–night level L dn in the UnitedStates. NEF is still being used in Canada and in a modified form in Australia.Noise and Number IndexThe noise and number index (NNI) is a rather subjective method for rating aircraftnoise annoyance, developed and used in the United Kingdom. A survey conductedin 1961 of noise in the residential areas within a 10-mile radius of the LondonHeathrow Airport resulted in the creation of NNI, which is defined byNNI = (PNL) N + 15 log 10 N − 80 (13.12)where (PNL) N comprises the average peak noise level of all aircraft operatingduring a day and is given by((PNL) N = 10 log1NN∑n=1)antilog PNL10(13.13)PNL denotes the peak perceived noise level generated by a single aircraft during theday and N is the number of aircraft events over a 24-h period. Since no annoyanceapparently occurs at levels less than 80 PNdB a constant of 80 was introduced intoEquation (13.12) so that a zero value of NNI corresponds to no annoyance.Another survey conducted in 1967 revealed that for the same noise exposurethe reported annoyance was less than in 1961. This may be attributable to thefact that noise-sensitive people may have left the affected area, noise-insensitivepeople moving into the area, or the residents got used to their environment or

13.10 Evaluation of Traffic Noise 335became apathetic to the noise, or the background noise arising from surface traffic,construction or operation of industrial facilities had increased sufficiently to maskmore of the aircraft noise.The use of NNI was superseded in the United Kingdom in 1988 in favor ofmeasurements based on the A-weighted L eq (cf., Equation (13.6)). This parameteris averaged over the period from 0700 to 2300 h (7 a.m. to 11 p.m.). The measureL dn of Equation (13.7) is not applied because nighttime flights are severely limitedin the United Kingdom. Both Switzerland and Ireland still make use of NNI.Equivalent Sound LevelWhile some nations continue to use NEF or NNI as descriptors, it has been establishedthat L eq and L dn are much simpler to measure and to evaluate since theyseem to correlate well with subjective response. In addition to the United Kingdomwhich uses only L eq over an 18-h period (since nighttime flights are restricted),both Germany and Luxembourg have adopted the L dn method with (0600–2200 h)day and (2200–0600 h) night classifications. In the United States, as the result ofthe publication of EPA Report 550/9-74-004 in 1974 and similar documents, theuse of CNR and NEF has been well superseded by the day–night equivalent levelDNL for rating of potential impacts of noise and for planning purposes and landusage near military and civilian airports. In fact, a directive (Part 256 of Title 32of the Code of Federal Regulations) was issued in 1977 by the U.S. Office of theSecretary of Defense that the day–night average sound level DNL must be usedas the basis for evaluating the impact of noise by air installations and that neitherCNR nor NEF may be utilized.In 2002, the European Parliament and Council issued Directive 2002/30/ECon the establishment of rules and procedures with regard to the introduction ofnoise-related operating restrictions at airports. The objectives of this Directive are(a) to lay down rules for the Community to facilitate the introduction of operatingrestrictions in a consistent manner at airport administrative level so as to limit orreduce the number of people significantly affected by the harmful effects of noise,(b) to provide a framework which safeguards internal market requirements, (c) topromote development of airport capacity in harmony with the environment, (d)to facilitate the achievement of specific noise abatement objectives at the levelof individual airports, and (e) to enable measures to be chosen with the aim ofachieving maximum environmental benefit in the most cost-effective manner.13.10 Evaluation of Traffic NoiseHighway traffic noise probably impacts more people than any other source ofoutdoor noise. Consequently, many national, state, and local governments set requirementsto assess the existing or potential noise impact of highways. In theUnited States, federal agencies are required by law to provide environmental impactstatements (EIS) for proposed new roads and for any reconstruction of existing

336 13. Criteria and Regulations for Noise Controlroads, where environmental impacts are likely to occur and federal funding is usedto finance all or part of the project. The Federal Highway Administration (FHWA;not to be confused with FHA, the Federal Housing Administration) delegates theresponsibility for preparation of environmental impact statements to the individualdepartments of transportation of the affected states. Moreover, the U.S. Departmentof Housing and Urban Development (HUD) also requires that when a developerplans residential property with the aid of construction funds or a guarantee of suchfunds from HUD, the developer must assess the potential impact of transportationnoise upon that development.Assessments of traffic noise are usually made in terms of the overall A-weightedsound levels. Octave-band and one-third octave-band levels are generally used onlyfor the purpose of developing vehicle noise abatement measures such as acousticbarriers. Traffic noise tends to vary greatly over time, so methods are required todeal with this variation and its resultant impact on people.Time PeriodThe time period used as a basis by FHWA is the 1-h period when the trafficis at its heaviest. This period is called the worst noise hour. The Departmentof Housing and Urban Development and certain states, California for example,require assessment over a 24-h period. Two types of descriptors are in general use:statistical descriptors and time-averaging descriptors.Statistical Descriptors. The 10-percentile-exceeded level L 10 encompasses theacoustic magnitude of individual traffic noise events, such as passages of heavytrucks, and also the number of such events. Originally the FHWA noise abatementcriteria specified only in terms of a 1-h L 10 . This descriptor is clearly inadequatefor situations where (1) the hourly traffic rates are low, (2) vehicles are not evenlyspaced along a road, and (3) the values of L 10 could not be combined mathematicallyon the basis of calculations for separate events.Time-Average Descriptors. Time-average descriptors are now widely used forassessing traffic. The most common such descriptor is the 1-h average sound level(abbreviated 1HL), which is essentially the A-weighted equivalent continuoussound level L eq taken over a 1-h period. This carries the advantage of (1) easycomputation through the use of integrating meters, (2) assumptions regardingvehicle spacing are rendered unnecessary, and (3) the average levels for separatecategories of sources may readily be combined. The principal disadvantage is thatit can be extremely sensitive to isolated events having a high sound level, but whichdo not necessarily provoke a correspondingly high human response.FHWA Assessment ProceduresThe Federal Highway Administration (FHWA, 1976) requires that expected trafficimpacts be determined and analyzed. Highway projects are classified byFHWA into two categories: Type I, which is a proposed federal or federal-aided

13.10 Evaluation of Traffic Noise 337project to construct a new highway or to make major physical alterations to anexisting road, and Type II, which is a project for noise abatement procedures thatare added to an existing highway with no major alterations of the highway itself.All Type I projects are subject to FHWA regulations. Development and implementationof Type II projects is not mandatory, but a traffic noise analysis is mandatedfor eligibility of noise abatement measures for federal funding.In the procedure for traffic noise analysis, the following steps are specified inFHWA regulations:1. Identify activities and land uses that may be affected by the traffic noise. Thiscan involve the use of aerial photographs, land-use maps, and highway plans, inaddition to the results of field reconnaissance. Also determine regions of humanactivity and identify major sources of noise.2. Predict future traffic noise levels through a method consistent with the FHWATraffic Noise Prediction Model, employing noise emission levels that are eitherpublished in the FHWA regulations or determined by the agency using specifiedprocedures.3. Determine existing noise levels by actual measurements at sites that will beaffected by the proposed highway. These ambient levels provide the basis forassessing impact and evaluating abatement feasibility.4. Determine existing traffic noise impact. The predicted levels must be comparedwith the existing levels and with criteria based on land use. Table 13.3 lists thenoise levels for different categories of land use. FHWA regulations describe twotypes of impacts on the land. The first type occurs when the predicted futurelevels “substantially exceed” the existing levels (but “substantially exceed” isnot defined by FHWA, so different state transportation departments interpret theterm differently, with minimum increases in A-weighted sound level rangingfrom5to15dB(A), with 10 dB(A) being a typical value). The second type ofimpact occurs when the predicted future levels “approach or exceed” the noiseabatement criteria of Table 13.3. The criteria of Table 13.3 are listed accordingto the land-use activity and are in terms of 1-h average sound levels (L 1h )or1-h 10-percentile-exceeded levels.5. Evaluate abatement measures. In situations where severe impacts are identified,a state transportation agency need to examine means of reducing substantiallyor eliminating the impact. FHWA regulations specify that primary considerationshould be given to exterior impacts where frequent human use occurs and wherea decreased level would be of benefit. The state agencies must also (a) considerthe opinions of affected residents, (b) identify in the environmental reports theabatement measures likely to be incorporated into the project and impacts whereno solutions are apparent, and (c) include the abatement measure in the projectplans and specifications to be approved. Some abatement measures includetraffic management (e.g., prohibition of certain vehicle types, time restrictionsfor certain vehicle type, speed limits, traffic control devices, etc.) and acquisitionof property to serve as buffer zones to preempt development that would beadversely affected by traffic noise.

338 13. Criteria and Regulations for Noise ControlTable 13.3. Yearly Day–Night Average Sound Levels for Land-Use Compatability(Source: Federal Aviation Administration, 1985) ∗Yearly day–night average sound level (L dn ), dBLand Use Below 65 65–70 70–75 75–80 80–85 Over 85Residential:Residential other than mobile homes Y N(1) N(1) N N Nand transient lodgingsMobile home parks Y N N N N NTransient lodgings Y N(1) N(1) N(1) N NPublic use:Schools Y N(1) N(1) N N NHospital and nursing homes Y 25 30 N N NChurches, auditoriums, and concert halls Y 25 30 N N NGovernmental services Y Y 25 30 N NTransportation Y Y Y(2) Y(3) Y(4) Y(4)Parking Y Y Y(2) Y(3) Y(4) NCommercial use:Offices, business, and professional Y Y 25 30 N NWholesale and retail—building materials, Y Y Y(2) Y(3) Y(4) Nhardware, and farm equipmentRetail trade—general Y Y 25 30 N NUtilities Y Y Y(2) Y(3) Y(4) NCommunication Y Y 25 30 N NManufacturing and production:Manufacturing, general Y Y Y(2) Y(3) Y(4) NPhotographic and optical Y Y 25 30 N NAgriculture (except livestock) and forestry Y Y(6) Y(7) Y(8) Y(8) Y(8)Livestock farming and breeding Y Y(6) Y(7) N N NMining and fishing, resource production Y Y Y Y Y Yand extractionRecreational:Outdoor sports arenas and spectator sports Y Y(5) Y(5) N N NOutdoor music shells, amphitheaters Y N N N N NNature exhibits and zoos Y Y N N N NAmusements, parks, resorts, and camps Y Y Y N N NGolf courses, riding stables, and water Y Y 25 30 N NrecreationNumbers in parentheses refer to notes.∗ The designations contained in this table do not constitute a federal determination that any useof land covered by a program is acceptable or unacceptable under federal, state, or local law. Theresponsibility for determining the acceptable and permissible land uses and the relationship betweenspecific noise contours rests with the local authorities. FAA determinations under Part 150 are notintended to substitute federally determined land uses for those determined to be appropriate by localauthorities in response to locally determined needs and values in achieving noise-compatible land uses.Key: Y(yes): Land use and related structures compatible without restrictions. N(no): Land use andrelated structures are not compatible and should be prohibited. NRL noise level reduction (outdoor toindoor) to be achieved through incorporation of noise attenuation into the design and construction ofthe structure. Land use and related structures are generally compatible; measures to achieve an NLRof 25, 30, or 35 dB must be incorporated into the design and construction of the structure.

Highway Construction Noise13.10 Evaluation of Traffic Noise 339No specific quantitative rules or guidelines for limiting highway construction noiseare provided by the FHWA, but some state agencies use the criteria of Table 13.3as guidelines for assessing the impact of construction. However, a computer model(HICNOM) is available from FHWA to be used to predict construction noise andasses abatement measures.Vehicle NoiseMajor contributors to vehicle noise include the engine exhaust and air intake, engineradiation, fans and auxiliary equipment, and tires. To a lesser degree othernoise sources are the transmission, driving axles, and aerodynamic noise due tothe passage of the vehicle through air. The relative importance of each componentdepends on the vehicle type and condition, vehicle load (passenger and cargo),speed, acceleration, and highway grade, and road surface condition. In order toaid in prediction of highway noise, FHWA conducted an exhaustive series of measurementsof noise emission from, automobiles, trucks, buses, and motorcycles.The procedure for determining the noise emission levels entails the followingsteps:1. A level open space free of large reflecting surfaces within 30 m (100 ft) of thevehicle path or microphone is identified. A 150 ◦ clear line-of-sight arc from themicrophone position is required.2. The surface of the ground should be free of snow and may be hard or soft.The roadway should be relatively level, smooth, dry concrete or asphalt. Thereshould be no gravel.3. The background level from all sources except the vehicle in question should beat least 10 dB(A) lower than the level of the vehicle in question.4. The microphone is situated 15 m (50 ft) from the centerline of the lane of travel.5. The microphone is mounted 1.5 m (5 ft) above the roadway surface and notless than 1 m (3.5 ft) above the surface upon which the microphone stands. Itshould be oriented according to the manufacturer’s specifications.6. The vehicle in question should be traveling at steady speed without accelerationor deceleration.Vehicles are grouped by FHWA into three classes:1. Automobiles (A): All vehicles with two axles and four wheels, including automobilesdesigned for transportation of nine passengers or fewer, and lighttrucks and SUVs. Generally, the gross vehicle weight (GVW) is less than 4500kg (10,000 lb).2. Medium Trucks (MT): All vehicles having two axles and six wheels, generallyin the weight class 4500 kg (10,000 lb) GVW 12,000 kg (26,000 lb).

340 13. Criteria and Regulations for Noise ControlTable 13.4. Typical Octave Band Sound Pressure Levels of Automobiles and HeavyTrucks, Measured at 1.2 m Above the Ground at a Distance of 15.2 m.Octave-band center frequency, HzA-weighted sound125 250 500 1000 2000 4000 level, dB(A)Automobile speed56 km/h (35 mph) 65 61 62 61 57 53 6588 km/h (55 mph) 71 68 66 68 66 60 72Heavy truck speed56 km/h (35 mph) 87 84.5 81.5 78 74.5 70.5 83.588 km/h (55 mph) 87.5 85 87.5 82.5 77 73.5 87.53. Heavy Trucks (HT): All vehicles having three or more axles, including threeaxlebuses and three-axle tractors with and without trailers. Generally GVW >12,000 kg (26,000 lb).Table 13.4 shows the octave-band sound pressure levels typical of automobilesand trucks at two different speeds.Measurements are made by vehicle type for each selected speed ±5 km/h. Fora given category of vehicles at a given speed, the reference energy emission levelis given byL 0 = L 0mean + 0.115L 2 0SDwhere L 0mean is the arithmetic average emission level for the specific categoryand speed and L 0SD is the standard deviation of that emission level.Vehicle Noise Prediction According to the FHWA ModelOn the basis of many measurements, FHWA developed a model for predictinghighway noise (Barry and Reagan, 1978). We first compute the reference meanlevel [(L 0 ) E ] i for each vehicle type i. This level represents a speed-dependent valueof the average (energywise) of the maximum passby levels measured at a referencedistance of 15.2 m (50 ft) for a given vehicle type. The U.S. Federal reference meanemission levels are computed from:Automobile: (L 0 ) E = 38.1 log(S) − 2.4 dB(A)Medium trucks: (L 0 ) E = 33.9 log(S) + 16.4 dB(A)Heavy trucks: (L 0 ) E = 24.6 log(S) + 38.5 dB(A)where S is the average operating speed in km/h.Adjustments for Traffic Conditions(13.14a)(13.14b)(13.14c)A number of adjustments have to be made to the reference mean emission levelscomputed through the use of Equations (13.14). These adjustments are for (a) the

13.10 Evaluation of Traffic Noise 341traffic density, (b) distances to areas of human activity, (c) finiteness of roadwaysegments, (d) presence of road gradients, (e) shielding by buildings, (f) shieldingby rows of trees, and (g) barrier attenuation.The traffic flow adjustment ( traffic ) i is rendered to account for the hourly flowof vehicle type i and to average the level over a 1-h time period as follows:( traffic ) i = 10 log (N i d 0 /S i ) − 25 dB(A) (13.15)whereN i = hourly flow rate of vehicles of type i, vehicles/hd 0 = the reference distance of 15.2 mS i = speed of the ith vehicle type, km/hDistance adjustment takes into consideration the type of ground cover (e.g.,sound will attenuate more readily over a soft surface than over a hard parking lotpavement). Field data gathered by FHWA have shown an average rate of attenuationto be approximately 4.5 dB(A) per doubling of distance over a grassy surface andan attenuation of 3.0 dB(A) per doubling of distance over a paved surface. Thesetwo rates are used by FHWA to establish a ground cover parameter called analpha factor (α). For hard sites, the alpha factor has a value of zero; for soft sites, itassumes the value 0.5. The distance adjustment distance can be generally expressedas distance = 10(1 + α) log(d 0 /d)where d is the perpendicular distance from the receiver to the center of thetravel lane. Table 13.5 provides guidance on when to use propagation rates of 3.0or 4.5.The methodology described above assumes an infinitely long road. In analyticalpractice, highways are subdivided into a series of straight segments of finite length.Table 13.5. Guidelines for Choosing Sound Propagation Rates.Situation1. All situations in which the source or the receiver are located 3 mabove the ground or whenever the line of sight a averages morethan 3 m above the ground2. All situations involving propagation over the top of a barrier3 m or more in height3. Where the height of the line of sight is less than 3manda. there is a clear (unobstructed) view of the highway, theground is hard, and there are no intervening structures, orb. the view of the roadway is interrupted by isolated buildings,clumps of bushes, or scattered trees, or the interveningground is soft or covered with vegetationPropagation rate,dB(A)3(α = 0)3(α = 0)3(α = 0)4.5 (α = 1/2)a The line of sight (L/S) is a direct line between the noise source and the observer.

342 13. Criteria and Regulations for Noise ControlFigure 13.7. Subdivision of a road for noise analysis purposes with segment ends Aand B defining the angles with respect to the normal from the observer to the road. In(a) an angle clockwise to the observer is positive and an angle counterclockwise is negative.In (b) the angles are measured from the normal line to the extension of the roadsegment.The 1-h average sound levels are computed separately for each segment, and thenthey are combined in the end. Such subdivision of the roadway for analyticalpurposes should be executed (a) where the traffic volumes or speeds change (e.g.,at an exit or entry ramp or a road fork), (b) where the ground cover changessignificantly, (c) where a curved road is being analyzed as a series of straightsegments, and (d) where the vertical gradient of the road changes. With referenceto Figure 13.7, the finite roadway segment adjustment (which depends on theground cover) is defined as∫ φ 21 segment = 10 logπ (cos φ)a dφ (13.16)φ 1where φ 1 and φ 2 are angles in radians at the receiver, as shown in Figure 13.6;φ 1 is the angle to the left end of the segment and the angle φ 2 to the right end ofthe segment. If an angle is measured counterclockwise from the normal line, it isassigned a negative value; if measured clockwise, a positive value. For a hard siteEquation (3.16) reduces to segment = 10 log φ 2 − φ 1π(13.17)In the case of a soft site, Equation (13.16) needs to be evaluated by numerical integration.For the general case of a “soft” site and an infinitely long roadway (whereφ 1 =−π/2 radians and φ 2 =+π/2 radians), the adjustment is −1.2 dB(A).

13.10 Evaluation of Traffic Noise 343It is commonly observed that large trucks become noisier as they travel uphill inlower gears. The FHWA recommends that the 1-h average levels for heavy trucksbe increased as a function of roadway gradient 1 in the following manner:1. 0–2%: grade = 0 dB(A)2. 3–4%: grade =+2 dB(A)3. 5–6%: grade =+3 dB(A)4. Over 7%: grade =+5 dB(A).If there are one or more rows of buildings present between an observation pointand a road, an adjustment for the shielding, shielding , by these buildings must beestimated. A rule of thumb can be used in the following manner: if 40–65% ofthe length of the first row of buildings is occupied by the building themselves,then subtract 3 dB(A) from the average sound level. If 65–90% of the length ofthe row is occupied by the buildings, subtract another 3 dB(A) from the averagesound level; and if the percentage exceeds 90%, the buildings may be treated asnoise barriers. Each successive row of buildings adds an additional 1.5 dB(A) tothis adjustment, up to a maximum reduction of 10 dB(A). Any excess groundattenuation stops at the first row.A second type of shielding, also referred to as shielding , can occur from thepresence of trees between the road and the receiver. The tress may be dense enoughto virtually disallow a direct view of the road. For a 30-m (100 ft) belt width, FHWAsuggests an adjustment of –5 dB(A). For an additional 30-m belt width there is anadditional –5 dB(A) adjustment for a total of –10 dB(A). When this adjustment isused, both distance adjustment and a segment adjustment should be used on thebasis of the propagation rate of 3 dB(A) per doubling of distance from the linesource.Barrier attenuation adjustment barrier can be computed using the path-differenceprocedure described in the last chapter. According to the FHWA model, this shouldbe computed by using an incoherent line source model separately for each vehicletype. The attenuation is computed for a series of paths, defined by the angleswith the perpendicular to the source-to-receiver line, and the results are combinedvia numerical integration. This calculation is usually achieved with the use of acomputer program such as the FWHA STAMINA program.The total 1-h average sound levels for each vehicle type can be obtained bysumming up the various adjustments:(L eq ) i = [(L 0 ) E ] i + ( traffic ) i + ( distance ) i+ ( grade ) i + ( shielding ) i + ( barrier ) i (13.18)The 1-h average levels for automobiles, medium trucks, and heavy trucks may nowbe combined to yield the total 1-h average level.1 When a road changes its elevation n meters (or feet) for every 100 meters (or feet) horizontal travel,the road is said to have an n percent gradient.

344 13. Criteria and Regulations for Noise ControlComputer ProgramsThe FHWA noise prediction methodology has been incorporated into the followingcomputer programs:1. SNAP 1.1, intended for relatively simple geometries, can handle up to twelveroadways and one simple barrier. This program provides a detailed output andcan provide separate results according to vehicle type.2. STAMINA 2.0 and its derivative programs are used by most states, along witha noise barrier program called OPTIMA. Complex highway sites with manyroadways, barriers, and receivers can be analyzed with this program. Additionalvehicle types can be specified in addition to the three classes of automobiles,medium trucks, and heavy trucks. The output of the program can include the 1-haverage sound level and 10-percentile-exceeded level for each receiver for theinitial barrier segment elevations, the average sound levels for each individualroad segment, and matrices of sound energy contributions from each barriersegment at each elevation for each receiver.3. OPTIMA is used to design noise barriers on the basis of input information suchas the type of material being used for each barrier. This program can be used tooptimize the design on a cost effective basis.13.11 Evaluation of Community NoiseTransportation noise, both surface and air, constitutes the dominant source of noiseexposure in residential neighborhoods. Other sources can include noise emanatingfrom industrial and commercial enterprises, rowdiness of individuals carousingin the streets, passing “boom cars,” operation of lawnmowers or snowblowers,chain saws or other gear, and local construction. Concern about noise pollutionbegan to intensify when commercial jet began to appear in the sky in the late1950s. There was little in the way of standardization of community response surveymethods, questionnaire items, and even noise measurements and analyticaltechniques throughout the 1960s and 1970s. But research was carried on duringthese years on annoyance, speech, and sleep interfering properties of noise. Researchershave come up with a number of ways to assess community response. Inaddition to frequency-weighting schemes, some of the properties of noise thoughtto have relevance include tonality, impulsiveness, rise time, onset time, periodicity,time of day, and temporal variability. Dozens of physical measure of sound havebeen considered as predictors of annoyance caused by noise exposure (Pearsonsand Bennett, 1974; Schultz, 1982). Even today there is no single purely physicalmetric that can function as a definite predictor of annoyance with noise exposure,and it may not even be possible to develop such a predictor.Social SurveysPerhaps the least ambiguous procedure to evaluate the prevalence of noiseinducedannoyance in a community is through the means of a social survey.

13.11 Evaluation of Community Noise 345Survey techniques, however, can range widely in the degree of their respectivesophistication.Empirical Dosage Response Relationship. Schultz (1978) documented quantitativedosage-response relationships through meta-analysis, which constituted amajor step toward the setting up of a standard method for predication of transportationnoise effects. Schultz executed a “best-fit” third order polynomial to a dataset relating the day–night average sound level (DNL) to the degree of annoyancein communities. A simple quadratic-fit equation provides a purely empirical basisfor predicting the prevalence of annoyance in communities:% highly annoyed = 0.036L 2 dn − 3.27L dn + 79.14 (13.19)However, Equation (13.19) produces meaningless predictions when evaluated outsidethe range 45 dB < L dn < 85 dB. The U.S. Federal Interagency Committeeon Noise prefers a different logistic fit to a subset of data reported by Fidell et al.(1991), which resulted from disregarding the results of certain studies in whichrelatively low levels of noise were associated with a high degree of reported annoyance:100% highly annoyed =(13.20)1 + e 11.13−0.141L dnEquation (13.20) predicts somewhat lower levels of annoyance at lower noiseexposure levels than the quadratic fit of Equation (13.19). For example, at L dn =65 dB. Equation (13.20) yields a prevalence rate of 12.3% in contrast to 18.8%obtained through the use of (13.19).Equal Energy Hypothesis. DNL has been adopted by many federal agencies asa convenient descriptor of long-term environmental noise descriptor, which soonenough became a predictor of annoyance. The DNL index is a time-weightedaverage (in effect, the average acoustic energy per second with arbitrary nighttimeweighting), which is sensitive to the duration and magnitude of individual noiseevents and directly proportional on an energy (10 log N) basis to number of events.Reliance on such an integrated energy metric is based on the “equal energy”hypothesis, which states the notion that the number, level, and duration of noiseevents are fully interchangeable determinants of annoyance as long as their product(energy summation) remains equal. This quantification of noise exposure in termsof DNL for the purpose of predicting annoyance carries the implication that aperson would be annoyed to the same degree by small numbers of very highlevel noise events as by large numbers of lower level noise and/or longer durationnoise events. The equal energy hypothesis has provided an adequate account fordata on the prevalence of annoyance to sporadic (e.g., urban) noise in the range55 < L dn < 75 dB, but the validity of the hypothesis falls off in extreme cases. Forexample, no community is likely to tolerate even infrequent operation of a noisesource powerful enough to damage hearing or a very occasional shock wave frompassing supersonic jet plane from a nearby military base.

346 13. Criteria and Regulations for Noise Control13.12 Guidelines and Regulations in the United States,Canada, Europe, and JapanThe most well-established of noise effect guidelines in the United States are thosepromulgated by the Federal Interagency Committee on Noise (FICON). FICONis composed of a number of federal agencies with interest in environmental noise[e.g., FAA and Department of Defense (DoD)]. The guidelines and recommendationsof FICON for “land-use compatibility” are couched in ranges of DNL.FICON considers noise exposure levels lower than L dn = 65 dB to “be compatiblewith most residential land uses.” However, FICON does confess its realization thatthis limitation may be somewhat too high in highly rural areas where it would bemore appropriate to characterize the effects of noise pollution not in acoustic termsbut rather in terms of annoyance.On the international scene, a major degree of consensus has evolved over theyears as to what constitutes unacceptable levels of noise exposure. In the mid1980s the Organization for Economic Cooperation and Development (OECD) 2suggested a standard guideline value for average outdoor noise levels of 55 dB(A),which pertains to normal daytime in order to prevent significant interference withnormal activities of local communities. According to OECD:1. At 55–60 dB(A) noise creates annoyance.2. At 60–65 dB(A) annoyance increases considerably.3. Above 65 dB(A) constrained behavior patterns occur, symptomatic of seriousdamage by noise.The World Health Organization (WHO) listed additional guidelines in 1996for noise exposure in dwellings, schools, hospitals, concert halls, and so on. Theexposure levels are given in terms of the time average A-weight sound level L eq .For example, a private bedroom should sustain a sound level no higher than L eq =30 dB(A) at night in order to promote undisturbed sleep; and the background noiselevel in a classroom should be no greater than L eq = 35 dB(A) to facilitate theteaching processes.Workplace Noise ExposureIn the United States the passage of the Noise Control Act of 1970 gave rise toOSHA regulations, which are described in Section 13.3. In Europe, the Council ofEuropean Communities issued a Council Directive 86/188/EEC on May 12, 1986,which sets the guidelines on the protection of workers from the risks related toexposure to noise at work. This directive does not prejudice the right of members ofEU to introduce or apply even stricter provisions that reduce the permissible levelsof noise. For an 8-h day, under the provisions of 86/188/EEC, a worker should2 While its name indicates its mission, the Organization for Economic Cooperation and Developmentalso concerns itself with environmental matters. Its membership includes industrial nations fromNorth America, Europe, and Asia.

13.12 Guidelines and Regulations in the United States 347not be exposed to more than A-weighted L eq = 85 dB per 8-h day. This value isdetermined from measurements made at the position occupied by a worker’s earsduring work, preferably in that person’s absence, using a technique that minimizesthe effect on the sound field. If the microphone has to be located close to theworker’s body, appropriate adjustments should be made to determine an equivalentundisturbed field pressure. The daily personal noise exposure does not take intoaccount the effect of any personal ear protectors used. Should the allowable dailyexposure be exceeded, the affected workers are to be notified, and proceduresmust be enacted to lower the exposure, which may include periodic checks ofthe work environment, provision of ear protectors, and audiology tests. If thedaily personal noise exposure exceeds 90 dB(A) or the maximum value of theunweighted instantaneous sound pressure is greater than 200 Pa, the employermust exert every effort to cut down this exposure.Vehicle Noise RegulationsExterior noise tests for road vehicles have been used in Japan since 1971. Thecurrent version of the tests is quite thorough and consists of three parts: (a) a fixedpassby test at 7 m, 60% rated engine speed or 65 km/h, (b) an acceleration passbytest at 7.5 m, and (c) a stationary test behind the exhaust outlet at 20 m.In these tests the microphone is placed 1.2 m above ground level. The noiselimits for new highway vehicles are given in Table 13.6 which went into effect in1998. Another 2 dB reduction in these noise limits has been in effect since 2002.Table 13.7 lists the mandatory exterior limits for new highway vehicles sold inthe European Union. The EU vehicle noise test procedure is prescribed in CouncilDirectives 92/97 EEC and 81/334/EEC, presented in the Official Journal ofthe European Communities. Amendments were added for automatic transmissionequippedand for high-powered vehicles. The test is based on ISO 362, which issubstantially equivalent to SAEJ140 (cf., Handbook of the Society of AutomotiveEngineers). A 1-dB(A) tolerance is permitted in this test. The test and its derivativesTable 13.6. Japanese Noise Limits for New Highway Vehicles.VehicleDescription GVWR (metric tons) Power (kW) Limit [dB (A)]Passenger car — — 76Light truck 3.5 >150 81Moped 50 cc < 125 cc 70>125 cc < 250 cc 73>250 cc 76

348 13. Criteria and Regulations for Noise ControlTable 13.7. European Union Highway Vehicle Noise Limits.Unloaded WeightNoise Limit,Vehicle Description (metric tons) Power (kW) dB(A)Passenger car — — 74Mini bus 9 seats >3.5 12 >150 80Motorcycles≤80 cc 75>80 ≤ 175 cc 77>175 cc 80are widely used throughout the world, including Japan (which does not have1-dB(A) tolerance). The test site configuration is used for all classes of vehicles,including motorcycles, with the measurement distance set at 7.5 m.Boom cars, sometimes referred to as “boomers,” are vehicles that have audioequipment installed specifically to generate excruciatingly loud sound levels. Becausethe market is so lucrative, particularly among automobile owners under30 years of age, manufacturers aggressively engage in the promotion of ultrahighpower amplifiers and rather hefty loudspeaker systems. Boomer systems areusually operated at levels that generate a high degree of annoyance in residentialneighborhoods, which constitute a problem of increasing proportions (Raichel,2000). Loudness competitions are held, and levels as high as 175 dB (two and ahalf times louder than the Boeing 747 jumbo jet) have been reported. Much ofthe equipment advertising emphasizes loudness; and it is apparent that fidelity ofsound reproduction rarely figures in the setup of a boom system.The problem of boom cars is not confined to the United States. It is also becominga problem in Latin America (Raichel and Miyara, 2001). The phenomenon of boomcars compares to tobacco addition—a danger to vehicle occupants and a nuisanceto “secondary listeners.” Countermeasures against such public nuisances include:(a) enactment and diligent enforcement of municipal laws specifying limitationson sound levels, (b) education in school systems on the dangers of excessive noiselevels, and (c) public criticism of manufacturers’ promotional efforts that stressextremely high-level sound outputs. In Chicago boomer cars that can be heard from22 m (75 ft) are subject to seizure and their owners may be fined $615. Buffalo,Cleveland, and Pittsburgh police are cracking down on boom cars. In Papillion,Nebraska, owners of car stereos that can be heard from 15 m (50 ft) away canbe sentenced to 3 months in jail. The police in Lorain, Ohio have been regularlyenforcing the law approved in 2002 that prohibits car steros from being audible at15 m or more from the car. A first offense brings a $300 fine. A second offense

13.12 Guidelines and Regulations in the United States 349costs $400 and the offender’s audio equipment gets confiscated as contraband.That equipment is then destroyed by the police (Lorain Morning Journal, 2003).Motorcycles and other recreational vehicles also can be troublesome to communitiesaround the world. In an informal survey of 33 motorcycles, audiologicalresearchers at the University of Florida found that nearly half of them producedsounds about 100 dB when throttled up—equivalent in intensity to a loud rock concertor a chainsaw. At the Institute for Sound and Vibration Research (ISVR) at theUniversity of Southampton, UK, it was found the noise levels under motorcyclehelmets can be quite high. Above 65 km/h (40 mph) the wind noise generated byairflow over a motorcycle and rider exceeds the noise level from the motorcycleitself. In Montana, Section 61-9-418 (2003) of the state code requires that all motorcyclesor quadricycles operating on the streets and highways must be equippedat all times with noise suppression devices, including an exhaust muffler, in goodworking order at all times. In addition, all motorcycles and quadricycles must meetthe following noise level limitations, on the basis of measurement of 50 ft fromthe closest point of the motorcycle or vehicle:(a) any cycle manufactured prior to 1970: 92 dB(A)(b) any cycle manufactured after 1969 but prior to 1973: 88 dB(A)(c) any cycle manufactured after 1972 but prior to 1975: 86 dB(A)(d) any cycle manufactured after 1974 but prior to 1978: 80 dB(A)(e) any cycle manufactured after 1977 but prior to 1988: 75 dB(A)(f) any cycle manufactured after 1987: 70 dB(A).In New Hampshire, a House Bill 326 was introduced in the legislature in 2005 toban modification of a motorcycle to amplify or increase the noise level beyondthat emitted by the original equipment installed by the manufacturer. No straightpipe exhaust system is allowed; and no motorcycle shall be operated, which has anoise level of more than 100 dB(A) when measured 10 ft (3 m) or further from theexhaust pipe or muffler. Noise measurement is to constitute a part of the mandatoryregular vehicular inspection.Railroad Noise RegulationsIn the United States, regulations for railroad noise are published in Section 40,Part 201, and Section 49, Part 210, of the Code of Federal Regulations. Testmeasurements are conducted within a cleared level area 30 m from the track centerline. Locomotives are tested either in motion or stationary status using remote loadcells. Noise limits have been specified for locomotives manufactured after 1979:Stationary, idle throttle setting: 70 dB(A)Stationary. All other throttle settings: 87 dB(A)In motion: 90 dB(A).For rail cars, the limits are:Moving at speeds 83 km/h (50 mph) or less: 88 dB(A)Moving at speeds greater than 83 km/h: 93 dB(A).

350 13. Criteria and Regulations for Noise ControlThe test is to be conducted during dry weather without the presence of dust orpowdery snow.The EU has no exterior noise standards for railroad equipment because of interminglingwith equipment from outside the EU and differences in rail gauges.Railroads have used, on their own, a number of noise standards such as the InternationalUnion of Railways ORE E82/RP4 and the ISO 3095: 1975.Highway Construction NoiseAn interesting development in EU is the use of sound power as a noise measure,rather than sound pressure at a single point in space, for earth-moving andother construction equipment. Sound power measures the total noise emanating inall directions from a source. The fundamentals of sound power measurement aregiven in Chapter 9, Sections 9.17 et seq.; and details are given in the EU directives95/27/EU, 89/514/EEC, 86/662/EEC, 84/533-536b/EEC, and 79/113/EEC.The distribution of measurement positions specified by 79/113/EE is given in ahypothetical hemisphere enclosing a stationary piece of earth-moving equipment.Sound pressure is measured at the prescribed measurement points, which is convertedinto sound intensity, applying the far-field approximation and integratedto yield sound power. The sound power measurement would be more accurate ifsound intensity measurements were used instead of sound pressure. In its currentform the test uses only half of the measurement points specified in 79/113EEC. Asimilar sound power test was developed for lawn mowers in 84/538/EEC.amendedby 88/181/EEC.New noise limits came into effect at the end of 1996 for earth-moving machineryof net installed power less than 500 kW. Machines with power exceeding 500 kWoperate in quarries and mines and so are considered to have a negligible effect oncommunity noise. For the period between 1996 and 2001, the permissible soundpower levels L WA ,inA-weighted decibels relative to 1 pW, are given by1. Tracked vehicles (except excavators):L WA = 87 + 11 log P for L WA ≥ 107 (13.21)2. Wheeled bulldozers, loaders, excavators loaders:3. Excavators:L WA = 85 + 11 log P for L WA ≥ 104 (13.22)L WA = 83 + 11 log P for L WA ≥ 96 (13.23)where P is the net installed power of the construction vehicle in kilowatts. Belowthe lower limit given above, the machine automatically passes the test. After2001, the numerical A-weighted decibel values in Equations (13.21–13.23), includingthe lower limits, are reduced by 3 dB(A). The coefficient 11 remainsunchanged.

13.12 Guidelines and Regulations in the United States 351Over the past few years, the European Commission of the EU has been laboringon a new directive regarding machinery used outdoors. Directive 200/14/ECrelating to the noise emission in the environment by equipment used outdoors wasadopted by the European Parliament and the Council and first published in May2000. This directive became effective January 3, 2003. Directive 2000/14/EC covers57 types of equipment for outdoor use, ranging from construction machineryto lawnmowers. It is intended to supersede similar directives that exist separatelyin the EU’s member nations. Directive 200/14/EC demands declarations frommanufacturers on the “guaranteed” sound power levels of their products beforethey can be marketed in the EU. Such products must bear a CE mark and anindication of their guaranteed sound output level and be accompanied by an EDdeclaration of conformity before they can be placed on the market. The guaranteedsound power level is that defined by ISO 3744-1995 with the addition of uncertainties(owing to production variations and measurement procedures), which themanufacturer confirms will not be exceeded. Therefore, in order to obtain estimatesof these uncertainties, a sampling procedure on the production line may beappropriate.The directive places sound power limits on 22 of the 57 types of machineryfor use outdoors. Of these 22, one half were already subject to noise limits laiddown in seven earlier directives (including Directive 98/0029 described in thefirst edition of this text). The remaining half of this group are subject to noiselimits for the first time. All manufacturers of products subject to noise limits mustfollow conformity assessment procedures under the supervision of organizationsappointed by the EU member states. Failure to comply with these regulations mayresult in nonconforming products being barred on the EU market.Over the past quarter century or so, a greater awareness of the impact of constructionnoise on the part of government agencies led to a series of codes andregulations for the control and mitigation of noise from construction sites. Theseacts generally cover (a) the erection, construction, alteration, repair, or maintenanceof buildings, structures, or roads; (b) the breaking up, opening, or boringunder any road or adjacent land in connection with the construction, inspection,maintenance, or removal of public or private works; (c) piling, demolition, ordredging works; or (d) any other work entailing engineering construction.A major cornerstone in the development of effective construction noise controlprograms may very well be the Construction Noise Control Specification 721.560developed by the Massachusetts Turnpike Authority for the Central Artery Tunnel(CA/T) Project, also known as “the Big Dig.” In the Boston area, at the close ofthe 20th century, this 12-year plus undertaking ranks to date as the largest infrastructureconstruction project in the United States. With this project’s completionthe notorious Boston traffic bottleneck on U.S. Interstate Highway I-93 to/fromLogan International Airport should be alleviated, thus freeing up the City of Bostonand the entire New England corridor to normal traffic flow. Apart from doublingBoston’s highway capacity, this project should lead to modernization of Boston’sunderground utilities and enable the city to achieve positive growth in the 21stcentury.

352 13. Criteria and Regulations for Noise ControlConstruction over the 11-km-long project occurred 24 h a day in various locationswithin the city. Construction equipment operated in close proximity tothousands of residential and commercial properties, in some cases as close as3 m away. Hundreds of construction machinery were operating at any one timethroughout the project. The range of equipment types used is wide, includingcranes, slurry trenching machines, hydromills, hoe rams, pile drivers, jackhammers,dump trucks, bulldozers, concrete and chain saws, and gas and pneumaticallypowered hand tools. Upon the completion of this project, more than 10 millioncubic meters of excavated materials will have been removed and nearly 3 millioncubic meters of concrete poured.In order to contain the acoustical impact of the project, while supporting andmaintaining the progress of construction, the CA/T Project people developed apolicy summary to establish an overall noise control program that includes (a) acommitment statement to minimize noise impact on neighboring residences whilesustaining construction progress; (b) a summary of the Project Noise ControlSpecification criteria and components; (c) an expression of willingness to developarea-specific noise mitigation strategies tailored to particular community needs andsensitivities; (d) an approach and criteria for judging the worthiness of mitigationmeasures; and (e) a commitment to provide qualified noise experts to oversee contractorcompliance in the field. Construction Noise Control Specification, 721.560adopted and enforced by the CA/T Project, ranks as the most comprehensive andstringent noise code of any pubic works project in the United States.In addition to the CA/T project, other outstanding sets of codes or regulationspertaining to construction noise (Raichel and Dallal, 1999a,b) include those by theHong Kong and Singapore governments as well as those developed for the I-15project in Utah. Germany’s “Blue Angel” seal program certifies noise-generatingequipment that operate below specified noise levels.ReferencesAnonymous. 1998. Low-Noise Cpnstruction has a Future (Leises Bauen hat Zukunft):Noise Reduction with the “Blue Angel” (Lämminderung mit dem “Blauen Engel”),1st ed. Kalsruhe, Germany, Fachinformationszentrum, Gesellschaft für Wissenschaftlishtechnische,Barry, T. M., and Reagan, J. R. 1978. FHWA Highway Traffic Noise Prediction Model.Report FHWA-RD-77-108. Washington, DC: Federal Highway Administration.Beranek, Leo L. 1989a. Balanced Noise Criterion (NCB) Curves. Journal of the AcousticalSociety of America 86: 650–664.Beranek, Leo L. 1989b. Applications of NCB noise criterion curves. Noise Control EngineeringJournal 33: 45–56.Blazier, W. E. 1981. Revised noise criteria for application in the acoustical design andrating of HVAC Systems. Noise Control Engineering16: 64–73.Bowlby, W. (ed.). 1981. Sound Procedures for Measuring Highway Noise: Final Report.Report FHWA-DP-45-1. Washington, DC: Federal Highway Administration.Bowlby, W., and Cohn, L. F. 1982. Highway Construction Noise: Environmental Assessmentand Abatement, Vol. 4: User’s Manual for FWHA Highway Construction Noise

References 353Computer Program, HICNOM. Vanderbilt University Report VTR 81-2. Washington,DC: Federal Highway Administration.Bowlby, W. 1980. SNAP 1.1—A Revised Program and User’s Manual for the FHWA Level 1Highway Noise Traffic Noise Prediction Computer Program. Report FHWA-DP-45-4.Arlington, VA: Federal Highway Administration.Bowlby, W., Higgins J., and Reagan, J. (ed.). 1982. Noise Barrier Cost Reduction Procedure,STAMINA 2.0/OPTIMA: User’s Manual. Report FHWA-DR-58-1. Washington,DC: Federal Highway Administration (based on Menge, C. W. 1981. User’s Manual:Barrier Cost Reduction Procedure, STAMINA 2.0 and OPTIMA, Report 4686.Cambridge, MA: Bolt, Beranek and Newman.Council of the European Communities. May 12, 1986. Council Directive 86/188/EE onthe Protection of Workers from the Risks Related to Exposure to Noise at Work.Department of Labor Occupational Noise Exposure Standard. May 29, 1971. Code ofFederal Regulations, Title 29, Chapter XVII, Part 1910, Subpart G, 36 FR 10466.Department of Labor Occupational Noise Exposure Standard. March 8, 1983. AmendedCode of Federal Regulations, Title 29, Chapter XVII. Part 1910, 48FR9776-9785.Edge, Jr., P. M., and Cawthorn, J. M. February 1977. Selected Methods for Quantificationof Community Exposure to Aircraft Noise, NASA TN D-7977.Environmental Protection Agency (EPA). July 1973. Public Health and Welfare Criteriafor Noise. Washington, DC: EPA.European Union Council. June 17, 1999. Proposal for a Directive of the European Paliamentand of the Council on the Approximation of the Laws of Member States Relating toNoise Emission by Equipment Used Outdoors. Interinstitutional File 09/0029. Brussels:European Council of EU.European Union Council. 2000. Directive 2000/14/EC of the European Parliament andof the Council of 8 May 2000 on the approximation of the laws of the Member Statesrelating to the noise emission in the environment by equipment for use outdoors (OfficialJounral L 162 of 03.07.2000).European Union Council. 2002. Directive 2002/30/EC of the European Parliament andof the Council of 26 March 2002 on the establishment of rules and procedures withregard to the introduction of noise-related operating restrictions at Community airports(Official Journal L 085, 29/03/2002: 0040-0046).European Union Council. 2002. Directive 2002/49/EC of the European Parliament and ofthe Council of 6 25 June 2002 relating to the assessment and management o environmentalnoise. (Official Journal L 189, 18/07/2002).Federal Aviation Administration (FAA). January 18, 1865 (revised). Part 150—Airportnoise compatibility planning. Federal Aviation Regulations. Washington, DC: FederalAviation Administration.Federal Highway Administration. July 8, 1982. Procedures for abatement of highway trafficnoise and construction noise. 23 C.F.R., Part 722. Federal Register vol. 47: 29653–29657.Federal Interagency Committee on Noise (FICON). 1992. Final Report: Airport NoiseAssessment Methodologies and Metrics. Washington DC: FICON.Federal Register, 40. May 28, 1975. p. 23105.Fidell, S., Barber, D., and Schultz, T. 1991. Updating a dosage-effect relationship for theprevalence of annoyance due to general transportation noise. Journal of the AcousticalSociety of America 89, 1: 221–233.Gottlob, D. 1995. Regulations for community noise. Noise/News International 3(4): 223–236.

354 13. Criteria and Regulations for Noise ControlSociety of Automotive Engineers (SAE). 1991. Handbook of the Society of AutomotiveEngineers. Warrendale, PA: SAE.Harris, Cyril M. (ed.). 1991. Handbook of Acoustical Measurements and Noise Control,3rd ed. New York: McGraw-Hill; Chapters 23–26, 46–50.Hong Kong Environmental Protection Department. 1999. A Concise Guide to theNoise Control Ordinance. Hong Kong: EPD. Internet address: http://www.info.gov.hk.epdinkh/noise/book2/book20.htlm.Jacques, Jean. June 2004. Noise and standardization, focusing on machinery and workplacedomains. Presented at the Joint Baltic-Nordic Acoustics Meeting, Mariehamn,Åland.Kryter, Karl D. 1970. The Effects of Noise on Man. New York: Academic Press.Lorain Morning Journal. October 9, 2003. Lorain more pleasant as law hammers down onboom car stereos. Lorain, Ohio.Lower, M. C., Hurst, D. W., Claughton and A. R., and Thomas, A. 1994. Sources andlevels of noise under motorcyclists helmets. Proceedings of the Institute of Acoustics16(2): 319–326.Lower, M. C., Hurst, D. W., and Thomas, A. 1996. Noise levels and noise reduction undermotorcycle helpmets. Proceedings of Internoise ’96, Book 2: 979–982.Massachusetts Turnpike Authority. Central Artery (I-93)/Tunnel (I-90) Project. RevisedJuly 28, 1998. Construction Noise Specification 721.560. Boston, MA: Commonwealthof Massacusetts.Montana State Code. 2003. Code 61-9-418. Motorcycle noise suppression devices.New Hampshire Transportation Committee. March 9, 2005. House Bill 326.Occupational Noise Exposure Standard. June 28, 1983. U.S. Department of Labor. OccupationalSafety and Health Administration (OSHA), Code of Federal Regulations, Title29, Part 1910, Section 1910.95 [29 CFR 1910.95], Federal Register 48: 29687–29698.Olson. N. 1970. Statistical Study of Traffic Noise, Report APS-476. Ottawa: NationalResearch Council of Canada, Division of Physics.Pearsons, K. S., and Bennett, R. 1974. Handbook of Noise Ratings. Washington, DC:NASA.Raichel, D. R., and Dallal, M. October 1999a. Prediction and attenuation of noise resultingfrom construction activities in major cities. Presented at the 138th Meeting of theAcoustical Society of America November 4, 1999. Journal of the Acoustical Society ofAmerica 106, 4, Pt. 2: 2261.Raichel, D. R., and Dallal, M. 1999b. Final Report: Construction Noise. Contract No.99F88151, Subagreement No. 001 with the New York City Department of Design andConstruction, Vols. 1 and 2. New York: The Cooper Union Research Foundation.Raichel, Daniel R. 2000. Boom cars: noise pollution at its worst. Presented December 7,2000 at the combined NoiseCon 2000/140th Meeting of the Acoustical Society ofAmerican, Newport Beach, CA.Raichel, Daniel R., and Miyara, Federico. October 22, 2001. “Autos baffle: La problemàticade los automóviles con refuero sonoro” presented at the Fourth International MultidisciplinaryConference on Acoustic Violence (Cuartes Jornados Interncionales MultiodiscoplinariasSobre Violencia Acústica), Centro Cultural “Bernardino Rivadavia”,Rosario, Argentina.Schultz, T. J. 1978. Synthesis of Social Surveys on Noise Annoyance. Journal of theAcoustical Society of America 64, 2: 277–405.Schultz, T. J. 1982. Community Noise Rating. New York: Elsevier.Singapore Government. 1990. The Environmental Public Health Act (Chapter 95) S466/90(Control of Noise from Construction Sites) Regulations.

Problems for Chapter 13 355Problems for Chapter 131. An outdoor survey of a neighborhood indicated averaged sound pressure levelsat 40 dB(A) from 7 a.m. to 9 a.m.;55dB(A) from 9 a.m. to 3 p.m.; 59.5 dB(A)from 3 p.m. to 10 p.m.; and 39.8 dB(A) from 10 p.m. to 7 a.m. What is theday–night sound level? Is the value obtained acceptable for a residential area?2. A factory environment was found to have an average of 88 dB for 2.5 h, 90 dBfor 1.6 h, 92 dB for 1.4 h, and 95 for 2.5 h. Is the 8-h exposure acceptableby OSHA standards? If not, what would be the maximum total daily workingtime, given the same histographic distribution?3. A lathe in a machine shop was found to yield the following octave-bandanalytical results:Center Frequency (Hz)Octave Band Loudness (dB)63 60125 70250 68500 751000 722000 754000 748000 58What is the RC rating for this machine shop?4. Redo problem 2 but find L eq on the basis of the information given. If the 8-hday is excessive exposure by the European Union standards, how much mustthat workday be cut down in order to meet the daily exposure guidelines?5. Examine the U.S. OSHA noise exposure regulations for workers and comparethem with the counterpart European regulations. Which set of regulations isbetter for the worker? State the reasons for your choice.6. Predict the mean emission levels for (a) an automobile going 135 km/h,(b) a medium truck moving at 120 km/h, and (c) a heavy truck travelingat 100 km/h.7. Find the traffic flow adjustment for 3600 cars per hour moving at an averageof 100 km/h for an observation point 15.2 m from the road.8. A number of noise sources are located at different distances from a measurementpoint on a property line. The data consisting of the measured sound levelsof each individual sources are given below:Number of Sources 1 2 3 1 2 4L at 25 ft 80 75 82 90 84 87Distance, ft 58 78 74 68 120 92(a) Determine the noise contribution of each source or group of sources.(b) Find the combined noise level at the measurement point on the propertyline.

356 13. Criteria and Regulations for Noise Control9. Use the data of Problem 8 to compute L eq24 and L dn if these sources runcontinuously at the same noise output levels from noon to 3 p.m.10. Find the permissible sound power levels of the following highway constructionvehicles under the European Union rules:(a) a nonexcavating tracking vehicle rated at 120 kW(b) a 200-kW bulldozer(c) a 100-kW excavatorWhat will be these corresponding values after the year 2001?11. Outline a noise code suitable for a suburban residential neighborhood. It shouldinclude the appropriate noise level limits, specification of how and wheremeasurements are to be taken.12. Develop a noise code for a rural jurisdiction.13. What are the problems that are applicable to urban areas? What are the specialneeds and goals with respect to noise levels in large cities?

14Machinery Noise Control14.1 IntroductionWorkplace noise at high levels is detrimental to the welfare of workers. Not onlyhigh sound levels can affect hearing and hinder oral communication, but alsothey detract the employee from performing at peak capacity. In spite of possibleeconomic drawbacks such as the cost of increased maintenance, the employerdoes have the moral obligation to provide a safe environment for both office andplant workers. Often when noise-attenuation measures are taken, some paybackmay accrue from the use of quieter machinery. For instance, when a diesel engineundergoes excessive vibration it becomes subject to severe stresses that can causeit to fail. Retuning of the engine so that it operates more smoothly lessens thestresses on its crankshaft or accessory parts and cuts down on its fuel consumptionas well as the noise output. In many highly industrialized nations, such as theUnited States, Germany, France, the Scandinavian nations, and Japan, there areregulations that limit the noise exposure levels. Some of these regulations werediscussed in Chapter 13.This chapter deals with industrial noise sources, predictions of their respectiveacoustic output, and means of attenuating noise in the workplace. Specific types ofnoise sources are considered, followed by descriptions of general methodologiesof noise control, which may or may not be machine specific.14.2 Noise Sources in the WorkplaceA considerable number of industrial machines and processes generate high levelsnoise that can cause physical and psychological stresses as well as considerablehearing loss. High-noise output machinery include blowers, air nozzles, riveters,pneumatic chisels and hammers, diesel generators, chipping hammers, rock crushers,die casting machines, drop hammers, metal presses, power saws, grinders, ballmills, stamping machines, and so on. In addition, building accouterments, such asfurnaces, air conditioning and ventilation (HVAC) systems and plumbing, as wellas office equipment can roil the office environment and add to the cacophony ofan industrial plant.357

358 14. Machinery Noise ControlIn setting up noise control measures, the first step is to identify the noise sourcesand to measure the sound power output. Ideally, it would be desirable to takea source into a well-defined environment such as a reverberation chamber andmeasure its sound power output, but many stationary sources cannot be moved.But the sound power output for immovable sources can still be estimated fromsound pressure measurements made on a hemispherical or rectangular envelopingsurface (cf. Chapter 9). The sound pressure level will be increased by the presenceof background noise and room reverberation, so a correction factor for either/bothbackground noise and reverberation must be applied in such cases. For reasonswhich we shall see in the sections following, a spectrum analysis of noise fromspecific machines will often prove useful in tracing malfunctioning machineryparts so that they can be realigned or replaced.Many noise problems in the workplace can be avoided by heeding the old adagethat an ounce of prevention is worth a pound of cure. Prior to any purchase ofmachinery, sound power output of each unit should be obtained beforehand, eitherdirectly from the vendor or by conducting actual measurements on an existinginstallation. Wise planning of the plant layout includes not only promoting productionefficacy and personnel safety; it should always involve prediction of noiseoutput of all equipment in normal operation. Adjustments can then be made atthis stage in the choice of quieter equipment and/or incorporating noise-reducingdevices so that the planned facility will operate within the sound exposure limitsmandated for workers.14.3 Estimation of Noise Source Sound PowerA nondirectional point source in a free field will radiate sound uniformly andradially in all directions. Such a source L W represents the true octave-band soundpower level, with units of decibels based on the reference power 1 pW (10 –12 W).Most pieces of machinery, however, are not point sources, nor do they radiate soundpower uniformly. In planning of facilities, it is generally necessary to estimate theexpected sound power for individual machines that will affect the environment.For certain machines, a sound power conversion factor F n can be used to determinethe output on the basis of the total power rating of the machine,whereP = F n × P m (14.1)P = sound power of the machine, WP m = machine rated power, WThe relationship of Equation (14.1) applies to both mechanical and electrical machinery.Estimated conversion factors for a number of common machinery arelisted in Table 14.1. It should be noted that the ranges are quite large for each typeof machine.

14.4 Fan or Blower Noise 359Table 14.1. Power Conversion Factors for Some Common Noise Sources.Conversion Factor (F n )Noise Source Low Mid-Range HighCompressors, air (1–100 hp) 3 × 10 −7 5.3 × 10 −7 1 × 10 −6Gear trains 1.5 × 10 −8 5 × 10 −7 1.4 × 10 −6Engines, diesels 2 × 10 −7 5 × 10 −7 2.5 × 10 −6Motors, electric (1200 rpm) 1 × 10 −8 1 × 10 −7 3 × 10 −7Pumps, >1600 rpm 3.5 × 10 −6 1.4 × 10 −5 5 × 10 −5Pumps,

360 14. Machinery Noise ControlFigure 14.1. Exploded view of a centrifugal fan. (Harris, 1991, p. 41.2.).centrifugal fans. The selection of the fan type, size, and speed depends firston the performance necessary to move a given amount of air against a specifiedpressure, and the noise characteristics are then established on a secondarybasis. Figures 14.1 and 14.2 show the general construction of a centrifugalFigure 14.2. Components of an axial flow fan. (Harris, 1991, p. 41.2.).

14.4 Fan or Blower Noise 361Figure 14.3. Three types of centrifugal fans. (Harris, 1991, p. 41.3.).fan and an axial-flow fan, respectively, along with listing of commonly usednomenclature.Centrifugal fans come in with a variety of blades. Three types are shown inFigure 14.3. Axial-flow fans divide into three main categories (vaneaxial, tubeaxial,and propeller), illustrated in Figure 14.4. In addition to flow and pressure requirements,fans are selected to meet environmental conditions, withstand corrosion,allow ease of maintenance, budget limits, and so on. The noise characteristics ofvarious types of fans are fairly predictable, as they are not significantly altered byminor changes in the fan geometry.Fan Noise CharacteristicsTable 14.2 lists the broadband noise characteristics of typical fan designs. Theaverage specific sound power levels in eight octave bands are given for welldesignedfans installed in well-designed systems. This data can be utilized toestimate fan noise at the design stage. In the selection process during the designstage, actual noise data should be obtained from the manufacturer. Octave-bandnoise levels should be used in calculations. Single-number ratings for fan noiseshould be avoided.When a blade passes over a given point, the air receives an impulse. The repetitionrate of this impulse, termed the blade frequency, determines the fundamentaltone that is produced by the blade. It can be predicted fromf B = nN(14.2)60wheref B = blade frequency, Hzn = fan speed, number of revolutions per minute (rpm)N = number of blades in the fan rotorAccording to Equation (11.23), the sound pressure level L p from a specific sourcein a room depends on room conditions as well as on the sound power level L Wof that source. In order to predict the contribution of the fan noise to the room

362 14. Machinery Noise ControlFigure 14.4. Three types of axial flow fans. (Harris, 1991, p. 41.3.)sound level, the data on sound power levels of the fan models under considerationshould be provided by the manufacturer. However, the manufacturer has no controlover the system design for the room, nor that of the acoustical nature of the roomand so cannot be responsible for the resultant noise level. If lower sound pressurelevels are required than that generated by even properly designed fans, then it maybe necessary to provide acoustic attenuators installed as an integral part of thefan assembly. Figure 14.5 shows a centrifugal fan with sound attenuators on boththe inlet and the outlet, which is used in the supply system of a central stationventilating system. A sound attenuator is fitted at the outlet to lessen the flow ofacoustic energy from the discharge of the air to the supply air ductwork.Specific Sound Power LevelSpecific sound power level is defined as the sound power level generated by aparticular fan operating at an air flow rate of 1 m 3 /s (2120 cfm) and at a pressure of

14.4 Fan or Blower Noise 363Table 14.2. Relative Sound Power Generated by Different Types of Fans.Octave-Band Center Frequency, HzFan Type Wheel Size 63 125 250 500 1000 2000 4000 8000 BFI aCentrifugal fansAirfoil or backward-curved Over 0.75 m 85 85 84 79 75 68 64 62 3or backward-inclined Under 0.75 m 90 90 88 84 79 73 69 64 3Radial fans:Low pressure (4–10 in. Over 1 m 101 92 88 84 82 77 74 71 7static pressure) Under 1 m 112 104 98 88 87 84 79 76 7Mediam pressure (10–20 in. Over 1 m 103 99 90 87 83 78 74 71 8static pressure) Under 1 m 113 108 96 93 91 86 82 79 8Highpressure (20–60 in. Over 1 m 106 103 98 93 91 89 86 83 8static pressure) Under 1 m 116 112 104 99 99 97 94 91 8Forward-curved All 98 98 88 81 81 76 71 66 2Axial fansVaneaxialHub ratio 0.3–0.4 All 94 88 88 93 92 90 83 79 6Hub ratio 0.4–0.6 All 94 88 91 88 86 81 75 73 6Hub ratio 0.6–0.8 All 98 97 96 96 94 92 88 85 6Tubeaxial Over 1 m 96 91 92 94 92 91 84 82 7Under 1 m 93 92 94 98 97 96 88 85 7Propeller All 93 96 103 101 100 97 91 87 5a BFI = blade frequency increment.Note: The data listed here are given in terms of specific sound power in dB re 1 (10) −12 W based on avolume flow of 1 m 3 /s and a total pressure of 1 kPa. Equation (14.3) must be used to adjust for actualpressures and volume flow rates. To convert these values into English units, subtract 45 dB in all bands.The base for the English units is a total presure of 1 in. water gauge and a volume flow rate of 1 ft 3 /min.These values are those for total sound power radiated from the fan. To obtain the power levels at eitherthe inlet or the outlet, subtract 3 dB from all bands. No change in BFI is to be made. In performingthe calculations for the sound power levels from this table, do not use a total pressure less than 0.125kPa. From applications where the total pressure is lower than 0.125 kPa, simply use the value of0.125 kPa.Figure 14.5. Centrifugal fan with sound attenuators at inlet and outlet. (Harris, 1991.P. 41.13).

364 14. Machinery Noise Control1 kPa (4.0 in. water gauge). Table 14.2 lists the relative sound power for a varietyof fan types. In reducing all fan noise data to this common base, the concept ofspecific sound power level allows direct comparison to be made between the octaveband levels of different types of fans. A blade frequency increment (BFI) is alsolisted in Table 14.2; this represents the number of decibels that must be added tothe level of the octave band, which includes the blade frequency in order to accountfor the presence of such a tone. Also, a means is provided for estimating the noiselevel of fans under actual operating conditions by a procedure that consists of thefollowing steps:1. Select the fan type and obtain the specific power levels in octave bands fromTable 14.2. These sound power levels are expressed in decibels re 1 pW.2. Adjust the octave-band levels for the volume flow rate and the operating pressureby adding to each octave band one of the following values:orwhere10 log Q + 20 log p t dB (for metric units) (14.3a)10 log Q + 20 log p t − 45 dB (for English units) (14.3b)Q = volume flow rate, m 3 /s or cfmp t = total pressure, kPa or inches water gauge3. Account for the blade frequency component of the fan by adding the BFI forthe fan type chosen to the octave-band level of that band which includes theblade frequency. The blade frequency is found from the use of Equation (14.2).4. The sum of the above equals the total sound power level of the radiation fromthe inlet and the outlet. Subtract 3 dB from each octave band to yield the soundpower level of radiation from the inlet or the outlet.Example Problem 2Consider a radial forward-curved fan with 24 blades, having a rotor diameter of0.8 m, and operating at 750 rpm with a volume flow rate of 18 m 3 /s and with atotal pressure of 1.5 kPa.Find the total sound output power at the inlet.SolutionWe use Table 14.2 and list the calculations at each step for each octave band levelin Table 14.3. For step 2, using Equation (14.3a)10 log Q + 20 log p t = 10 log 18 + 20 log 1.5 = 16

14.4 Fan or Blower Noise 365Table 14.3. Calculation Results for Sound Power Level.Octave-Band Center Frequencies, HzProcedural Step 63 125 250 500 1000 2000 4000 80001 98 98 88 81 81 76 71 662 16 16 16 16 16 16 16 163 0 0 2 0 0 0 0 04 −3 −3 −3 −3 −3 −3 −3 −3Radiation form inlet 111 111 103 94 94 89 84 79The blade frequency is from Equation (14.2)f B = nN60=750 × 2460= 300 Hzwhich lies in the octave band with the center frequency of 250 Hz. The BFI is 2 dB.Step 3 is now completed. Step 4 consists of subtracting 3 dB from each octaveband to obtain the noise from the inlet alone.Fan LawsFan laws can predict fan performance quite well over a wide range of size andspeed. These laws are as follows:( ) 3 ( )da naQ a = Q b (14.4)d b n bp ta = p tb(dad b) 2 (nan b) 2(14.5)where( ) 5 ( ) 3 da naP a = P b (14.6)d b n b( ) ( )danaL wa = L Wb + 70 log + 50 log(14.7)d b n bQ = volume flow rate, m 3 /sp t = total pressure, kPaP = fanpower,kWL W = sound power level, dB re 1 pWd = rotor diameter, mn = rotor speed, rpmSubscript a denotes the parameters for the base curve performance conditions, andsubscript b denotes the parameters for the desired performance conditions.

366 14. Machinery Noise ControlEquation (14.7) is less accurate than Equations (14.4)–(14.6) for predictingperformance characteristics but it is sufficiently accurate for estimating purposes.These fan laws state mathematically that when two fans have similar design configurations,their performance curves are similar, and at the equivalent point of ratingon each performance curve, the efficiencies should be equal. In order to apply thefan laws, it is necessary to have the test data for one fan in the same design series.The applicability of the fan laws is restricted to cases where all linear dimensionof the larger or smaller fan are proportional to the fan for which there are test data.14.5 Electric Motors and TransformersElectric motors convert electrical power into mechanical power. Some of the fundamentalnoises occurring in electric motors are caused by rotational unbalance,rotor/stator interaction, and slot harmonics. The electromagnetic force betweenthe armature and the field magnet, or rotor and stator, gives rise to vibration. Noiseis also generated by the excitation of natural frequencies of the motor structure,air resonance chambers, and the movement of air itself. The intensity of the noiseis typically a function of rotational speed and motor type.As a means of estimating the noise output of a motor, the following expressioncan be used to obtain the total sound power level in the bands 500, 1000, 2000,and 4000 Hz:whereL w = 20 log hp + 15 log n + K m dB (14.8)hp = rated horsepower (1–300 hp)n = rated speed of motor, rpmK m = motor constant = 13 dBOther more involved techniques have been developed, which entailed differentmotor constants for each of the octave bands of interest (Magrab, 1975; Webb,1976).Transformers exist for the purpose of stepping up or stepping down voltages.Their changing magnetic field causes deformation of the transformer coil, occurringat the alternating current (AC) frequency and higher harmonics, especially atthe twice the AC frequency. This results in the characteristic frequency of 120 Hzin the United States and 100 Hz wherever 50-Hz AC current is used.14.6 Pumps and Plumbing SystemsPump noise arises from both hydraulic and mechanical sources, namely, cavitation,pressure fluctuations in the fluid, impact of mechanical parts, imbalance, resonance,misalignment, and so on. The hydraulic causes are, however, the predominant noisegenerators. Pumps will generate even more noise if they are not operated at rated

14.6 Pumps and Plumbing Systems 367speed and discharge pressure, when the rate of compression is high or the inletpressure is below atmospheric, or if the temperature runs too high. An unfortunatesituation is created when the noise from the pump easily transmits through thefluid or piping to other system components.Pumps generate two types of noise: discrete tones and broadband noise. Thepump’s fundamental frequency f p is found fromwheref p = n × N c60n = pump rotational speed, rpmN c = number of pump chamber pressure cycles per revolution(14.9)In large pumps the noise emission is the loudest at this fundamental frequency.As the pump size decreases, the frequency at which the maximum noise emissionoccurs increases, often to a frequency that constitutes a third or fourth harmonic ofthe fundamental. Above 3 kHz, the noise becomes more broadband, approachingan essentially flat spectrum. This is due to phenomena such as high-velocity flowand cavitation.The total sound power level of pumps in the four octave bands of center frequencies400, 1000, 2000, and 4000 kHz can be estimated from the following:L w = 10 log hp + K p dB (14.10)where K p = pump constant which has the following values: 95 dB for the centrifugaltype, 100 dB for the screw type, and 105 dB for reciprocating type. Forrated speeds below 1600 rpm, 5 dB is to be subtracted for reciprocal pumps. Thesound power in each of the four bands may be considered to be 6 dB less than thetotal sound power L w computed from Equation (14.10).Controlling Noise in Plumbing SystemsTable 14.4 lists the sources of noise in a building’s plumbing system and theirlikelihood of being annoying. Flow can be either laminar flow, i.e., smoothlyflowing, or turbulent flow, in which occurs an irregular, random motion of the fluidparticles. The influencing factor that determines whether a flow will be turbulentor laminar is the Reynolds number Re, a dimensionless parameter defined byRe = ρ dvμwhere ρ is the density of the fluid, d is the internal pipe diameter, v is the flowvelocity, and μ is the absolute viscosity of the fluid. For Re < 2000, the flow islaminar. For transition region 2000 < Re < 4000, the flow may be either laminaror turbulent. For Re > 4000, the flow will be turbulent. Noise generated by laminarflow tends to be quite low in intensity and is usually of no concern.In most real plumbing systems, the velocities are sufficiently high to result inturbulent flow, which is a basic mechanism for noise generation within piping runs

368 14. Machinery Noise ControlTable 14.4. Different Type of Plumbing Noises. Their Means of Generation, andAnnoyance Potentials (from Harries, 1991, Ch. 44).Plumbing SystemComponent/Equipment Generation Mechanism Potential AnnoyancePiping runsCouplings Turbulence MinimalElbows Turbulence MinimalTees Turbulence MinimalFixturesBar sink Cavitation/turbulence/splash/waste flow MinimalBath tub Cavitation/turbulence/splash/waste flow Very significantBidet Cavitation/turbulence/splash/waste flow NominalFlushometer Cavitation/turbulence SignificantHose pipe valves Cavitation/turbulence NominalLaundry tubs Cavitation/turbulence/splash/waste flow NominalPressure regulator Cavitation/turbulence NominalShower Cavitation/turbulence/splash/waste flow Very significantSink Cavitation/turbulence/waste flow SignificantValves Cavitation/turbulence SignificantWater closet, tank stool Cavitation/turbulence/splash/waste flow Very significantUrinal Cavitation/turbulence/splash/waste flow NominalAppliancesDishwasher Vibration/cavitation/spray/water hammer Very significantDrinking fountain Cavitation/turbulence MinimalWashing machine Vibration/cavitation/impact/motor/ Very significantwater hammerWaste disposal Vibration/waste flow Very significantWater heater Cavitation/turbulence MinimalSupply and waste pumpsBooster Rotational flow/cavitation/motor SignificantRecirculation Rotational flow/cavitation/motor NormalSewage Rotational flow/cavitation/motor SignificantSump Rotational flow/cavitation/motor Significantand fixtures of the plumbing system. A potentially great cause of noise is cavitation,which is the formation and subsequent collapse of cavities (bubbles) within theflow of water through and past a restriction in the flow. For cavitation to occur,a localized restriction or a projection must exist within the piping system, whichensues in localized high velocities and low pressures. The formation and suddencollapse of these bubbles result in extreme local pressure fluctuations, which canbe detected as noise. Other water noises occur from splashing (i.e., impact of liquidstriking a surface) and waste water flow (i.e., flow into drainpipes).Much more serious is the sharp intense noise known as water hammer. It occurswhen a steady flow in a liquid flow system is suddenly interrupted, for example, bya quick-action valve. When the fluid is in motion throughout a piping system, evenat relatively low velocities, the momentum from this sudden interruption can bequite large. The sudden interruption of the flow creates an extremely sharp pressure

14.6 Pumps and Plumbing Systems 369rise that propagates as a shock wave upstream from the interrupting valve. Thesteep wavefront can be reflected numerous times back and forth throughout thevarious parts of the piping system until the energy is finally dissipated.Noise control factors which must be dealt with in order to effect noise controlinclude (a) water flow and piping characteristics, (b) radiation of sound to thebuilding structure, (c) selection and mounting of fixtures, (d) isolation of pumpsystems, and (e) water hammer noise control.Water pressure in a plumbing system influences the flow noise caused by piperuns and water supply valves. According to typical building codes, water pressureshould be maintained at least at 100 kPa (15 psi) but not more than 500 kPa (80 psi).For acceptable system performance, the supply pressure should be somewherebetween 230 and 370 kPa (35 and 55 psi), with a preference toward the lowerrange in order to minimize noise. Flow noise radiation from pipes can be lessenedby minimizing the number of pipe transitions (elbows, tees, y-connections, andthe like). This reduces the opportunity for the onset of turbulence and cavitation.Larger pipes are used in building design when noise control is given a higherpriority. In the U.S. 1 / 2 -inch diameter piping is generally used in domestic plumbingsystems but can be as high as 3 / 4 -inch diameter to cut down on noise by as muchas 3–5 dB.Noise resulting from water flow in pipes may be transmitted to the rooms throughwhich they pass, particularly if they are in direct contact with large radiation surfacessuch as walls, ceilings, and floors. Isolation of these pipes from the buildingstructure provides significant noise reduction. If the pipes are mounted with foamisolators, instead of being rigidly attached to the building structure, a considerablenoise reduction of 10–12 dB may be obtained. Whenever piping passes through astructure (block, stud, joist, or plate) or is in contact with a wall or masonry, resilientmaterials such as neoprene or fiberglass should be used to provide isolation.It is vital to seal with a resilient caulking around the perimeter of all pipes, faucets,and spouts that penetrate through floors, walls, and shower stalls.The impact of water hammer can rupture piping; thus, it will spring leaks, causeweakening of connections and produce damage to valves. Water-hammer pulsingassociated with the use of washing machines and dishwashers can be partiallydamped by connecting these machines to the water supply with extra-long flexiblehose. Figure 14.6 shows a schematic of a capped pipe that incorporates an airchamber for water-hammer suppression. The length of the pipe ranges from 30 to60 cm and may be of the same or larger diameter than the line it serves. The volumeof the air chamber, which serves as an air cushion, depends on the nominal pipediameter, the branch line length, and the supply pressure. But if the air chamberbecomes filled with water, it becomes ineffective. A petcock is provided alongwith a shut-off valve, so that the chamber may be drained of water and vented,thereby reactivating the unit.There are commercial devices, called water-hammer arresters, which are notsubject to the limitation of capped pipes, because a metal diaphragm separatesthe water from the air. These devices are best placed near quick-acting valves andshould also be installed at the termini of long pipe runs.

370 14. Machinery Noise ControlFigure 14.6. A capped pipe which serves to arrest water hammer.14.7 Air CompressorsAir compressors are widely used in industry, and they are identified as a majorsource of noise. Usually driven by a motor or a turbine, these devices are used toelevate the pressure of air or another gas. The noise emission characteristics are afunction of the type of unit. In portable air compressors the driving engine is themajor source of noise rather than the compressor itself.The second largest source is the cooling fan. Compressors are either rotaryor reciprocating. A reciprocating compressor typically generates a strong lowfrequencypulsating noise, the characteristics of which are dependent upon therotational speed and the number of cylinders. In centrifugal compressors the noisegenerated is a function of a number of parameters such as the interaction of the rotatingand stationary vanes, the radial distance between impeller blades and diffuservanes, the rotational speed, the number of stages, the inlet design, the horsepowerinput, turbulence, the molecular weight of the gas undergoing compression, andthe mass flow.As in the case of fans, the blade passage frequency constitutes an importancefrequency component in certain types of compressors. In the diffuser-type machines,the blade-rate component, arising from the movement of one set of bladespast another, is of primary importance. The frequency of this component is found

14.8 Gears 371fromwheref BRC = N r × N s60 K BRC× n Hz (14.11)f BRC = blade-rate component frequency, HzN r = number of rotating bladesN s = number of stationary bladesK RBC = greatest common factor of N r and N sn = rotational speed, rpmExample Problem 3Find the frequency of the blade rate component of a diffuser-type compressor withN r = 16 and N s = 24 and operating at a speed of 6000 rpm.SolutionUsing Equation (14.11) we find thatf BRC = N r × N s 16 × 24× n = × 6000 = 4800 Hz60 K BRC 60 × 8If the blade-rate component frequency falls within the audio range, we shouldexpect an increase of several decibels in the octave-band sound power level inwhich it occurs.The total sound power level in the four octave bands with center frequencies of500, 1000, 2000, and 4000 can be roughly estimated for both reciprocating andcentrifugal compressors from the expressionL w = 10 log hp + K c dB (14.12)where K c = air compressor constant = 86 dB for the 1–100 hp range. Equation(14.12) is very similar to Equation (14.10) for pumps, and either expression willyield a straight line in a semi-log plot. Also, it is not unreasonable to estimate thatthe sound power is equally divided among the four octave bands. Thus, each bandlevel is 6 dB below the total estimated from Equation (4.12).14.8 GearsInternal combustion engines and electric motors generally operate at speeds of oneto several thousand revolutions per minute. These high speeds help maximize thepower-to-weight and power-to-initial cost ratios. Gearing and other speed reducersare applied when the driven machinery requires high torque and low speeds.

372 14. Machinery Noise ControlMechanical power, shaft speed, and torque are related as follows:P kW = 10 −6 T ω (14.13)whereP kW = transmitted power, kWT = torque, N mmω = πn/30—angular velocity, rad/sn = shaft speed, rpmIn English units, Equation (14.13) becomesP hp =Tn(14.14)63,025whereP hp = transmitted power, hpT = torque, lb-inn = shaft speed, rpmIt was observed by Hand (1982) that gear noise increases with speed at the rate of6–8 dB per doubling of speed. It was also observed that an increase of 2.5–4 dB ingear noise occurs for each doubling of load. Thus, according to Equation (14.13)or (14.14) a reduction in speed results in an increase in torque, if the transmittedpower is to be sustained, and so the noise effect of speed reduction is somewhatoffset by the increase in the torque.Meshing FrequenciesThe profile of most gear teeth is that of an involute curve. Force transmits throughthe driving gear to the driven gear along the line-of-action, which is fixed in space(excepting planetary gear trains). If the gears are ideal, perfectly fabricated, rigid,and transmitting constant torque, the power should be transmitted smoothly andwithout vibration or noise. Real gears, however, have tooth errors in spacing andtooth profile, and in some cases, an appreciable shaft eccentricity. Gear teeth doact elastically and flex slightly under load. Consequently, the driving gear teeththat are not in contact deflect ahead of their theoretical rigid-body positions, whilethe driven teeth that are not in contact lag behind their theoretical positions. Thisresults in a rather abrupt transfer of force when each pair of teeth comes intocontact, instantaneously accelerating the driven gear and decelerating the drivinggear. The fundamental frequency of the noise and vibration is given byf = nN(14.15)60wheref = the fundamental tooth meshing frequency, Hzn = rotational speed of the gear in questionN = the number of teeth in the gear in question

14.8 Gears 373Figure 14.7. Gear noise spectrum.Harmonics—the noise and vibration at integer multiples of the tooth meshingfrequency—are usually present. The most significant contributions are usuallymade by the first two or three harmonics, f 2 = 2 f 1 , f 3 = 3 f 1 , and so on. A typicalgear noise spectrum is displayed in Figure 14.7.Tooth ErrorIf a single tooth is imperfectly cut, chipped, or damaged, then it will generate anoise impulse once every shaft revolution. The fundamental frequency of the noiseor vibration due to tooth error is given byf 1E = n 60(14.16)wheref 1E = fundamental frequency due to tooth error, Hzn = shaft speedHarmonics of tooth-error frequency may also occur. Moreover, if the shaft centerlineis not straight or if a gear or bearing is not concentric with the shaft centerline,noise and vibration at the tooth-error frequency may result. These can cause insideband frequencies f S that accompany the tooth meshing frequencies. Thesefrequencies are found as followsf S = f 1 ± f 1E (14.17)

374 14. Machinery Noise ControlGear TrainsConsider a pair of nonplanetary gears aand b in mesh. The tangential velocity V tat the mesh is the same for both teeth. But the V t of each tooth of gear i of radiusr i rotating at angular velocity ω i is given byV t = r i × ω (14.18)The number of teeth N i on a gear i is directly proportional to the radius r i ofthe gear, and ω i = 2πn i /60. Since r a × ω a = r b × ω b for gears a and b at theirmesh point, we obtainn a= N b(14.19)n b N aThus, the ratio of rotational speeds is inversely proportional to the ratio of numbersof teeth: If both gears are external spur gears, the speed ratio is negative, i.e., onegear will turn clockwise and the other counterclockwise. If one gear is an internalgear (i.e., a ring gear) the speed ratio is positive, meaning that both gears willrotate in the same angular direction. For a gear train comprised of several gears,the output to input speed ratio is given byn outputn input= ∏ i(NdrivingN driven)i(14.20)An idler gear serves as both a driving gear and a driven gear. Idler gears must beincluded in determining the direction of rotation of shafts in a gear train.Example Problem 4In Figure 14.8 showing a gear train, gear 1 on the input shaft has 40 teeth and rotatesat 3600 rpm. Gears 2, 3, and 4 have 90, 44, and 86 teeth, respectively. Narrow bandspectrum analysis of noise and vibration shows discrete tones and vibration energypeaks of 14, 27, 1227, 2400, 2455, and 4800. Establish the possible contributionsto the noise and vibration at these frequencies.SolutionGear speeds are determined fromfrom which we obtainn 2n 1= N 1N 2n 2 = n 3 = 3600 × 40 = 1600 rpm90

14.8 Gears 375Figure 14.8. Gear train for example problem 4.From Equation (14.20) we haveandn output 40 × 44=n input 90 × 86 = 0.227n output = 3600(0.227) = 818.7 rpmFundamental tooth-error frequencies are given byf 1E = n 60= 3600 = 60 Hz for gear 160= 160060= 818.760= 26.7 Hz for gears 2 and 3= 13.6 Hz for gear 4

376 14. Machinery Noise ControlFundamental tooth meshing frequencies are obtained fromf = nN60( ) 40= 3600 × = 2400 Hz for gear 160( ) 46= 1066 × = 1226.7 for gears 3 and 460If we compare the calculated results with the spectrum analysis, it seems thatthe 14- and 27-Hz discrete tones and vibration energy peaks correspond to thefundamental and the first harmonic of the tooth-error frequency for gear 4. Thiswould indicate that the shaft straightness and the teeth of gear 4 should be investigated.The discrete tones of 1227 and 2455 Hz correspond to the fundamentaland the first harmonic of the meshing frequencies of gears 3 and 4. Discrete tonesat 2400 and 4800 Hz correspond to the fundamental and the first harmonic of thetooth meshing frequency of gears 1 and 2.Contact Ratio (CR)The contact ratio CR for a pair of gears is defined as the average number of pairs ofteeth in contact. At least one pair must obviously be in contact at all times. Takinginto fact that tooth error, wear, shaft deflection, and machining tolerances affectthe way gears work, a contact ratio of 1.2 is selected as the practicable minimum.When the contact ratio is near minimum value, contact commences at the tips ofthe driven teeth, and impact loads due to teeth meshing are high. A contact ratioof 2 or more usually results in better tooth load distribution and quieter operation,since two or more teeth will be in contact at all times.In order to deal with contact ratio, we need to briefly review spur gear terminology:diametral pitch is defined by P d = N/d and d is the pitch (nominal) diameterof the gear, r is the pitch radius = d/2, φ is the pressure angle, and a is the addendumor extent of the tooth beyond the pitch radius, r a = r + a is the radius of theaddendum circle, and c = r 1 + r 2 (center distance for gears in mesh). It followsfrom Equation (14.19) for spur gears thatr 1= n 2r 2 n 1The pressure angle φ represents the angle of force transmission between gear teeth.Standard gears have pressure angles of 14.5 ◦ (a rather antiquated standard), 20 ◦ ,and 25 ◦ . The contact ratio between a pair of spur gears is given byCR =√r 2 a1 − r 2 1 cos2 φ +√r 2 a2 − r 2 2 cos2 φ − c sin φπ cos φP d(14.21)

14.8 Gears 377The subscripts 1 and 2 refer to the driver and driven gears, respectively. For fulldepth gears with standard pressure angles, the standard addendum is a = m =1/P d .For20 ◦ pressure angle stub teeth, the standard addendum is a = 0.8m =0.8/P d .Gears are selected to meet specified power and speed ratio requirements. Theserequirements affect the selection of gear material, tooth width, module or diametralpitch, number of teeth, and so on. If the number of teeth is increased for each gearin a train, the contact ratio will usually increase, with the result the tooth meshingfrequency and the accompanying noise level will decrease.Helical GearsWhen a pair of spur gear teeth begins to mesh, the contact occurs at once across theentire face width. In the case of a helical gear in which teeth entwine about an axialsurface rather than cut straight across (Figure 14.9), the contact occurs gradually,with the contact beginning at a point and then extending across the tooth face. Thisresults in reduced vibration and lower impact loads that produce tooth meshingfrequency noise. Hence, helical gears on parallel shafts may be substituted forspur gear trains to reduce noise; and the actual number of teeth in contact willalso be substantially increased. In order to ensure smooth and quiet operation, itis recommended that the thickness of the helical gear be 1.2 to 2 times the axialpitch (the distance between corresponding points on two adjacent teeth). Thrust(axial) loads must be taken into consideration when specifying helical gears in thedesign of shafts and bearings. Well-made helical gears cost more than spur gears,but the payback in terms of smoother and quieter operation can more than offsetthe greater expense.Figure 14.9. Helical gears. (Courtesy of Designatronics, Inc., New Hyde Park, NY.).

378 14. Machinery Noise ControlOther Aspects of Gear Noise ControlGear production methods involve die casting, milling, drawing, extruding, stamping,and production from sintered metal. These methods tend to produce gearswith significant tooth error to a more or lesser degree. Most gears produced bymilling cutters also tend to have their tooth forms only approximate the correctconfiguration. In order to produce a precise tooth form, each milling cutter shouldbe dedicated to only one diametral pitch or module and for a specific tooth number.In practice, a different milling cutter is used for each diametral pitch, but eachcutter is used for a range of tooth numbers. Inaccuracies in cast, drawn or extrudedgears result from shrinkage and other dimensional changes.Precision methods of generating gear teeth usually result in cutting down noisedue to tooth error. These include computer-controlled generating rack cutter, thegenerating gear shaper cutter, and the generating hob. Precision gears are usuallyfinished by grinding and shaving, and other finishing methods include burnishing,honing, and lapping.Large gears benefit by damping to absorb vibration energy. Constrained layerdamping, which sandwiches a layer of damping material between the gear web anda rigid steel plate, can reduce gear noise. If loads are low and the temperatures arenot excessive for the material, fiber, plastic, fiberglass reinforced, and compositegears are used in such applications. These materials have a low modulus of elasticityand high internal damping compared with steel. Shock loads from tooth meshingand tooth error are absorbed, thereby reducing noise.Gear enclosures or housing can and should be designed to control noise. Theenclosure should be isolated so as not to transmit vibration to adjacent structures;and provisions for adequate lubrication of the gears should be included. Resonancesof the enclosure should not correspond to the tooth meshing or any other excitationfrequency; otherwise the housing will radiate a large amount of noise energy. Thiscan be avoided by stiffening the enclosure structure to “tune out” its resonance toa higher frequency. The stiffeners themselves should be designed so that they havea low-radiation efficiency. Advantage should be taken of the directivity patterns ofnoisy gear trains in the orientation of machinery, so that personnel noise exposureis minimized.14.9 Journal BearingsA journal bearing is the simplest type of bearing. It consists of a portion of a shaftrotating inside a circular cylinder with a layer of lubrication separating the shaftand bearing surfaces. Hydrodynamic rotation depends on shaft rotation to pump afilm of lubricant between the shaft and the bearing. If the lubricant viscosity andshaft rotational speed are adequate for the load on the project bearing area, thenthick-film hydrodynamic lubrication will prevail. This ensures stable operationwithout metal-to-metal contact, as well as quiet operation. Starting, stopping, and

14.10 Ball and Roller Bearings 379load direction changing may result in temporary cessation of the lubricant film andin metal-to-metal contact. This calls for hydrostatic lubrication that introduces fluid(e.g., oil or air) to the bearing surface at a pressure sufficient to support the shaft,even when the bearing is stationary. A large increase in the noise level producedby a journal bearing is apt to indicate a failure of the lubrication system.14.10 Ball and Roller BearingsAn incipient failure of ball or roller bearing can be detected by noise and vibrationmeasurements. For the most part, bearing life can be predicted but premature failurecan and does occur. The occurrence of discrete tones in the noise spectrum for anoperating bearing may indicate manufacturing defects, and narrowband vibrationspectra can be utilized to detect wear and defects.A ball or roller bearing consists of the following elements: a set of balls orrollers are enclosed in a cage or separator sandwiched between an inner race(that is usually attached to the shaft) and the outer race (usually attached to thesupporting stationary structure). When the shaft turns, the balls (or rollers) rollalong the surface of the inner raceway, which allows for the rotational freedomof the shaft. The cage or separator follows the motion of the balls. The speedrelationships are as followsn O − n Cn I − n C=− D ID O(14.22a)andn B − n Cn I − n C=− D ID B(14.22b)wheren = rotational speed, rpmD = diameter, mm or in.and the subscripts denote the variables as follows:I = inner raceO = outer raceB = ball or rollerC = ball cage or separatorThe race diameters are measured at the point of contact with the balls or rollers.

380 14. Machinery Noise ControlBall or Roller Bearings with Stationary Outer RacesIn most bearing applications the outer race is stationary. Equation (14.22a)becomes0 − n C=− D I(14.23)n I − n C D Owhich yields the separator speedn C =n I D I(14.24)D O + D Iand the speed of the balls relative to the separator is given byn R B = n B − n C =− (n I − n C )D ID B=− n R I D ID B(14.25)Here n R Iis the speed of the inner race relative to separator speed. Equation (14.25)is obtained by eliminating n B through the use of relation (14.22b).Vibration and noise frequencies are related to the absolute speeds n, relativespeeds n R , and the number of balls or rollers ν B . For bearings with stationaryouter races, the following fundamental frequencies are likely to arise:1. Shaft imbalance:2. Outer race defect:f = n I60(14.26)3. Inner race defect:f O = n R O ν B60= n Cν B60(14.27)f I = n I Rν B60 = (n I − n C )ν B604. Defect or damage to one ball or roller:f B = 2n BR60 = n B − n C305. Imbalance or damage in the separator or cage:(14.28)(14.29)f C = n C(14.30)60With the appearance of the fundamentals as calculated above, harmonics 2 f ,3 f , ...are apt to occur.Example Problem 5Consider a ball bearing that uses twelve 8-mm-diameter balls. The inner andthe outer diameters are D I = 28 and D O = 44 mm, measured at the ball

14.10 Ball and Roller Bearings 381contact points. The outer race is stationary and the inner race rotates at speedn I = 4500 rpm clockwise. Determine (a) the rotational speeds and (b) the noiseand vibration frequencies which will occur if there are defects and imbalancepresent.SolutionIt should be obvious that D I + 2D B = D O.(a) Use Equation (14.24) to find the separator speedn C =n I D I= 4500 × 28 = 1750 rpmD O + D I 28 + 44and, from Equation (14.25), the speed of the balls relative to the separatorisn B R = n B − n C =− (n I − n C )D I=−(4500 − 1750) × 28D B 8=−9625 rpm (counterclockwise)(b) We find the fundamental frequencies as follows:1. Shaft imbalance:2. Outer race defect:f O = n O R ν B603. Inner race defect:f I = (n I − n C )ν B04. Defect or damage to one ball:f = n I60 = 4500 = 75 Hz60= n Cν B60=f B = 2n BR60 = n B − n C30=1750 × 1260(4500 − 1750) × 1260= 350 Hz= 550 Hz=− 962530 = 320.8HzNotice that the negative sign is ignored in this answer.5. Imbalance or damage in the separator or cage:f C = n C60 = 175060 = 29.2HzHarmonics of all of the above frequencies are likely to occur.Ball or Roller Bearings with Nonstationary Outer RacesFor some applications, the shaft attached to the inner race of a bearing remainsstationary while the outer race rotates. In this case, n I = 0 and we obtain the

382 14. Machinery Noise Controlfollowing speed relationships:with carrier speedn O − n C0 − n C=− D ID On C =n O D OD O + D Iand speed of the balls with respect to the carriern B R = n B − n C =− (n O − n C )D O=− n O R D OD BD BThe frequencies that may occur when the outer race is nonstationary are as follows:1. Shaft imbalance:f = n O602. Outer race defect:3. Inner race defect:f O = n R O ν B60= (n O − n C )ν B60f I = n I Rν B60 = n Cν B604. Defect or damage to one ball or roller:f B = 2n BR60 = n B − n C305. Imbalance or damage in the separator or cage:f C = n C60Harmonics will also constitute additional components of the spectrum.For any bearing fault, the noise and vibration spectra will be essentially tonal,i.e., characteristic peaks will exist. A great deal of the noise energy will be concentratedin the bands that include the fundamental frequency and its first and secondharmonics.14.11 Other Mechanical Drive ElementsChain DrivesFigure 14.10 shows a roller chain meshing with toothed sprockets. The roller chainis constructed of side plates and pin and bushing joints designed to mesh withsprockets. The flexibility of the chain aids in limiting shock and vibratory forces,but the initial contact between the chain and sprockets can be noisy at high speeds.

14.11 Other Mechanical Drive Elements 383Figure 14.10. Roller chain.For transmission of high loads at high speeds, inverted-tooth (or “silent”) chainsare often applied. Because the inverted-tooth chain engages with the sprocketswith less impact force, it generally operates more quietly than roller chain.The links are of finite length in either roller chain or inverted-tooth chain. Whilethe chain links engage a sprocket, the pitch line (i.e., the centerline of the link pins)bobbles up and down due to the chordal action. In Figure 14.10, the center of alink pin is located at R max from the center of the sprocket over which it moves.R max is the maximum center of the pitch line from the sprocket center. As the linkengages further, the pitch line moves nearer to the sprocket center and both pinsof the link coincide with the pitch line. The new position is given by( 180◦)R min = R max cos(14.31)N Swhere N S indicates the number of sprocket teeth. Roller chains usually have aminimum of 8 teeth and inverted-tooth chains, 17 teeth.Consider a drive sprocket rotating at constant speed n s . The pitch line velocityof the chain will change in direct proportion to its distance from the center of thesprocket, i.e., V = R × (2πn s ). The fractional velocity change V /V is given byVV= R max − R meanR max=1 − 1 + cos(180◦ /N S )21 + cos(180 ◦ /N S )2= 1 − cos(180◦ /N s )1 + cos(180 ◦ /N s )(14.32)The angular velocity (2πn s ) is canceled itself out in the ratio of Equation (14.32),but the speed ratio of a chain drive is given byn 2= N 1(14.33)n 1 N 2where n refers to the rotational speed (rpm) of the sprocket and N the numberof sprocket teeth. Subscripts 1 and 2 denote the driver sprocket and the drivensprocket, respectively. Fundamental frequencies of noise and vibration can be dueto the following:

384 14. Machinery Noise Control1. Imbalance in the driver shaft or the driven shaft or damage to one sprockettooth:f 1 = n 160 Hz and/or f 2 = n 2Hz (14.34)602. Damage to one chain link: In this case for a chain with N L links, a defectivelink will strike each sprocket n 1 N 1 /N L times per minutes. Hencef CL = n 1N 1(14.35)30N L3. Tooth engagement and chordal action:f TE/CA = n 1N 1(14.36)30When the above fundamentals are present, harmonics are likely to occur. Noiselevels and vibratory amplitudes due to tooth meshing and chordal action are functionsof the sprocket configuration and also speeds and masses of the sprocketsand chains. Also, if there are idler sprockets in the system, the values given inEquations (14.34)–(14.36) will undergo a change.Example Problem 6Given a 12-tooth drive sprocket rotating at 2400 rpm, a driven sprocket with24 teeth and a chain with 60 links, determine the velocity variation with respect tothe mean velocity and the possible vibration and noise frequencies due to this chaindrive.SolutionWe apply Equation (14.32) to the smaller sprocket, i.e., the driver, because theratio is larger:VV = 1 − cos(180◦ /N s )1 + cos(180 ◦ /N s ) = 1 − cos(180/12)1 + cos(180/12) = 0.0173Here the velocity varies by nearly ±2% from the average. Equation (14.33) is usedto find the speed of the driven sprocket’s speed:N 1n 2 = n 1 = 2400 × 12 = 1200 rpmN 2 24Equations (14.34)–(14.36) give us the following possible fundamental frequencies:f 1 = n 160 = 2400 = 40 Hz60f 2 = n 260 = 1200 = 20 Hz60

Belt Drivesf CL = n 1N 1 2400 × 12= = 24 Hz30N L 30 × 40f TE/CA = n 1N 130=2400 × 123014.12 Gas-Jet Noise 385= 960 HzBelt drives are often used instead of gears or chains in order to save on costs andprovide some degree of noise control. The elasticity of the belt prevents shock loadsat a driven machine from being transmitted back to the driver. Solid-borne noise andattendant vibration are reduced. Flat belts and V-belts mounted on smooth pulleysdepend on friction to transmit power, so there must be adequate belt tension toprevent slipping as well as to oppose centrifugal effects. When suddenly loaded,these belts can slip, causing a squealing noise. This sometimes can be fixed byincreasing the tension; but excessive tension can shorten the life of a belt and induceexcessive bending moments in connected shafts. When precise speed ratios mustbe sustained, such as the camshaft of an automotive engine, toothed belts mountedon toothed pulleys can be utilized. Toothed belts can maintain timing and phaserelationships just as well as meshing gears, but these belts isolate vibration andshock forces rather than transmitting them between the driving and driven elements.Universal JointsUniversal joints are used when the relative position of a driving element changeswith respect to a driven element, such as with an automobile transmission thatis connected through a drive shaft to a rear axle. Flexible couplings and flexibleshafts can also be used, but the former can accommodate only relatively slightmisalignments, and the latter can handle large misalignments but cannot handlelarge amounts of torque. In general, for a given rotational speed n, the frequencyassociated with noise and vibration is given byf = n 30 Hz14.12 Gas-Jet NoiseA most common and also worrisome noise source is the gas jet. This is alsoreferred to as aerodynamic noise, and examples include blowdown nozzles, gas oroil burners, steam valves, pneumatic control discharge vents, aviation jet engines,and so on. An acoustically unmitigated steam valve of a large cooker in a major foodprocessing plant can measure as much as 120 dB. Research on aerodynamicallygenerated noise began in earnest in the early 1950s as the result of the appearanceof the commercial jet engine, when it became obvious that its mechanism of soundgeneration had to be understood better in order to effect noise control (Lighthill,1952, 1978; Hubbard, 1995).

386 14. Machinery Noise ControlPrior to Lighthill’s pioneering work, there have been even earlier studies madeon aerodynamic noise in conjunction with efforts to reduce noise in axial fans. Wecan go back even further to find that the effect of jet streaming was mentioned in theearliest recorded references to sound. When the wind blew past the pillars of theancient Greek temples, eerie discrete tones were produced. The Greeks adjudgedthese tones to be the voice of Aeolus, the god of the wind, and hence these tonesare called Aeolian tones.The mechanism of the Aeolian sound can be explained by visualizing the flowof air over a cylinder. At a given velocity, the downsteam flow behind the cylinderexhibits an oscillatory pattern, as vortices are shed alternatively on one side of thecylinder and then the other. The ensuing trail of eddies form what is referred to as avon Kàrmàn vortex street, which contains strong periodic components, resulting ina sound of nearly pure tonal quality. On the basis of empirical data, the frequencyof the Aeolian tone can be predicted fromf Aeolian = αvdwherev = velocity of air, m/sd = diameter of cylinder, mα = Strouhal number, approximately 0.2Figure 14.11 displays the simplest example of a gas jet, in which the highvelocity airflow is emanating from a reservoir through a nozzle. The gas acceleratesfrom virtually zero velocity in the reservoir to a peak velocity in the core at theexit of the nozzle. The peak velocity of the jet depends greatly on the pressuredifference between the reservoir pressure p r and external (ambient) pressure p a .As the pressure ratio increases, the velocity of the gas at the discharge increasesup to a point when the pressure ratio of 1.89 (for the case of the gas being air) isreached. Once the flow velocity reaches the velocity of sound, any further increaseof the reservoir pressure will not result in an increase of the velocity at the end ofthe straight duct. When the critical pressure ratio of 1.89 is reached, the nozzle isFigure 14.11. A simple gas jet.

14.12 Gas-Jet Noise 387said to be choked. In order to increase the velocity further, the nozzle must increasein its cross-sectional area beyond the duct station where choking occurs.In the frictionless, isentropic flow of an ideal gas from one point to another, theapplicable energy equation describing this flow ise + p ρ + u2= constant (W/kg) (14.37)2where e is the internal energy of the gas, and u is the gas flow velocity. For thereservoir, u = 0 and p = p r . Also, from thermodynamics theory for an ideal gas,e + p ρ = enthalpy = c pTwhere c p is the specific heat of the gas at constant pressure and T is the absolutetemperature. Mach number M is defined byM = u c(nondimensional characteristic)But propagation speed of sound c in an ideal gas is given byEquation (14.37) becomesc 2 = γ RT = γ p ρe + p ρ + u22 = c pT + γ M2 RT2= c ppRρ= c p T(1 + γ M2 (γ − 1)2γ(1 + γ )M2 R2c p)= constant (14.38)We apply the fact that R = c p – c v in an ideal gas, and the ratio of the specificheats c p /c v = γ . Selecting two flow stations, one at a point r inside the reservoirand the other at the point in the duct where choking occurs (Mach number M =1.0), we obtain from Equation (14.37)p rρ r= p (ch1 + γ − 1ρ ch 2)orp r= ρ (ch1 + γ − 1 )p cr ρ ch 2(14.39)In an isentropic flow, according to thermodynamic theory:pρ −γ = constantThen Equation (14.37) becomes(p r= 1 + γ − 1 ) γγ −1(14.40)p ch 2The ratio γ of specific heats is equal to 1.4 for air, so Equation (14.40) yields thecritical pressure ratio for a shock to appear:p rp ch=(1 + 1.4 − 12) 1.41.4−1= (1.2) 3.5 = 1.89

388 14. Machinery Noise ControlNoise is generated from gas jets through the creation of fluctuating pressures fromthe turbulence and shearing stresses, as the high-velocity gas impacts with theambient gas. In establishing the theory of aeroacoustics, the nonlinear effects ofthe momentum flux ρu i u j (i.e., the rate of transport of any momentum ρu i acrossa unit area by any velocity component u j ) cannot be neglected as they were forlinear acoustics. The momentum flux acts as a stress, since the rate of changeof momentum constitutes a force. This momentum flux ρu i u j generates soundas a distribution of time-varying stresses. The forces between the airflow andits boundary radiate sound as distributed dipoles, and the stresses (which act onfluid elements with equal and opposite dipole-type forces) radiate as distributedquadrupoles.The nature of the noise from jets cannot be accurately predicted, owing to thecomplex nature of the jet itself and the uncertainties associated with turbulence,nozzle configuration, temperature vacillations, and so on. However, first-orderestimates can be derived from empirical data obtained for the most part fromexperimentation in the aviation industry. The earliest measurements of jet noisedemonstrated that intensity and noise power varied very closely with the eighthpower of the jet exit velocity (Lighthill’s eighth power law), and it is now generallyagreed that the overall sound power P can be expressed asP = Kρ 0U 8 D 2c 5 0(14.41)where K is a constant, with a value 3–4; D is the jet diameter (in meters); U isthe jet flow velocity (m/s); ρ 0 is the density of ambient air (kg/m 3 ); and c 0 is theambient speed of sound (m/s). The factor ρ 0 U 8 D 2 c −50is often called Lighthill’sparameter. Because the kinetic power of a jet is proportional to 1 2 ρ 0U 2 · UD 2 , thefraction of the power converted into noise is the noise-generating efficiency η,η ∝ M 5 (14.42)where M = U/c 0 , the Mach number of the flow referenced to the ambient speedof sound.Aerodynamic noise can be modeled as monopoles, dipoles, and quadrupoles.A jet pulse through a nozzle or discharge from HVAC ducts can be modeled as amonopole. In fans and compressors, the turbulent flow generally encounters rotoror stator blades, grids, and baffles; this type of flow can be modeled as a dipole.Quadrupole modeling applies to noise occurring from turbulent mixing in jetswhere there is no interaction with confining surfaces.The velocity term U in Equation (14.41) is the fluctuating velocity which variesthroughout the jet stream. Consequently, U is not easily measured nor amenableto analytical treatment. But we can consider the average velocity V and assumethat the size of the energy-bearing eddies are of the same order of magnitude as thejet diameter, and the total radiated acoustic power P is proportional to the kineticenergy of the jet flow. The total radiated power is simply a fraction of the total

14.12 Gas-Jet Noise 389power discharged from the nozzle, i.e.,whereP = ε M5 ρ 0 V 3 A2in watts (14.43)V = average flow velocity through the nozzle, m/sM = Mach number of the flow = V/c, dimensionlessρ 0 = density of ambient airA = nozzle areaε = constant of proportionality, in the order of 10 −4Equation (14.43) constitutes a first-order approximation that is applicable tomany industrial situations where the average velocity of the jetstream lies in therange of 0.15c < V ≤ c. The efficiency factor εM 5 is plotted from empirical datain Figure 14.12 over a range of Mach numbers. As a reflection of the uncertaintyassociated with turbulence, temperature, and so forth, the efficiency factor is givenas a range.Figure 14.12. Efficiency factor εM 5 versus Mach number.

390 14. Machinery Noise ControlExample Problem 7An air jet, operating through a choked 1-cm diameter nozzle, exhausts into theatmosphere. Determine the overall acoustical power and the sound power levelL W at 20 ◦ C.SolutionBecause the nozzle is choked, the average velocity of the air jet must be equalto the speed of sound at 344 m/s. The density of air at atmospheric pressure is1.204 kg/m 3 is found from the ideal gas relation,ρ = p aRT1.01325 × 105= = 1.204 kg/m3287 × (20 + 273.2)and the area of the nozzle is πD 2 /4 = π(10 –2 ) 2 /4 = 7.85 (10) –5 m 2 . FromFigure 14.12, the radiation efficiency factor εM 5 is found to be approximately3 × 10 –5 . Inserting these values into Equation (14.43) we findP = ε M5 ρ 0 V 3 A2= 3.0 × 10−5 × 1.204 × 344 3 × 7.85 × (10) −52= 0.0577 Wthus giving us the sound power level,( ) 0.0577L W = 10 log = 107.6 dB (14.44)10 −12From this last example, the acoustical power can be calculated to the first orderon the basis of the nozzle diameters and exit velocities. However, the acousticalpower of the air jet may have an accuracy of ±5 dB or thereabouts. But what if thegas jet is hot and extremely turbulent, as in the case of a gas burner? This situationcan be resolved by applying a first-order correction for the temperature:( ) TCorrection due to temperature = T = 20 logT awhere T and T a are the absolute temperatures of the gas jet and the ambient air,respectively.Example Problem 8Consider the nozzle described in Problem Example 7 above. What would be thetotal radiated acoustical power level L W if the temperature of the jet is to be raisedfrom 20 ◦ Cto260 ◦ C?

Solution14.12 Gas-Jet Noise 391Using Equation (14.44),( ) ( T (260 + 273.2) ◦ )K T = 20 log = 20 log= 5.2dBT a (20 + 273.2) ◦ KThe temperature-corrected sound power level becomesL W = 107.6 + 5.2 = 112.8dBWhen the overall power of a jet is determined, the overall sound pressure at anylocation in region surrounding the jet can be estimated through the use of therelationship between L p and L W :( ) Q(θ,φ)L p = L W + 10 log= L4π r 2 W + 10 log [Q(θ,φ)] − 20 log r − 11(14.45)where Q(θ,φ) is the directivity in three-dimensional space, and r is the distancefrom the source of L W to the location where the value of L p is desired. Thedirectivity index DI is given by( ) Q(θ,φ)DI = 10 log4π r 2In many cases of interest, the jet can be generally regarded as a point source withtypical directionality as shown in Figure 14.13. The parameter φ can be disregardedin axisymmetric flows.Figure 14.13. A fairly typical configuration for directivity DI Q in a small subsonic jet.Note that the peak levels occur approximately in the angular range of 15 ◦ to 45 ◦ from thecentral axis of the jet.

392 14. Machinery Noise ControlFigure 14.13 illustrates that peak levels occur in the range of 15 ◦ –45 ◦ from theaxis of the jet. The ordinate of Figure 14.13 giving the relative sound pressure levelconstitutes a directivity index DI θ .Example Problem 9Find the radiated overall sound pressure level at a radial distance of 15 m from thenozzle of Problem Example 7 at angular positions of 0 ◦ ,45 ◦ ,90 ◦ , and 180 ◦ .SolutionWe make use of Equation (14.45) and the results of Problem Example 7. The soundpressure level at r = 15misL p = L W + 10 log [Q(θ,φ)] − 20 log r − 11= 107.6 − 20 log(15) − 11 + DI θ = 73.1 + DI θ dBFrom Figure 14.13, the DI θ = 0atθ = 0 ◦ , and L p = 73.1 + 0 = 73.1 dB. At 45 ◦ ,DI 45 ◦ is approximately +3 dB, and L p (45 ◦ ) = 73.1 + 3 = 74.1 dB. Similarly forthe other angles, we haveDB 90 ◦ =−5dB, L p (90) ◦ = 73.1 − 5 = 68.1 dBDI 180 ◦ =−10 dB, L P (180 ◦ ) = 73.1 − 10 = 63.1 dBIt becomes apparent from the above example that the jet has a strong directionalcharacter that must be accounted for in determining the sound level pressures.Gas jets also manifest a strong frequency dependence. A first-order estimatefor the peak frequency can be obtained for a given power level from the empiricalrelationf peak = St × VD(14.46)whereSt = Stouhal number, a constant = 0.15 for a wide variety of nozzlediameters and operating conditionsV = average exit velocity of the nozzle, m/sD = nozzle diameter, mExample Problem 10Given a 5-cm diameter nozzle with an exit velocity of 344 m/s, determine the peakfrequency of the air blowdown.

14.13 Gas Jet Noise Control 393SolutionApplying Equation (14.46)0.15 × 344 m/sf peak = = 1032 Hz0.05 mThus, the peak frequency should be expected to occur in the 1 kHz octaveband.On either side of the peak frequency, the octave band distribution of the acousticalpower falls off. Experimental data indicated that the spectrum falls off at theaverage rate of –6 dB per octave above the peak frequency and −7 dB per octavebelow the peak frequency. The upshot is that the magnitude and the spectralcharacter of gas jet noise can be only estimated roughly.14.13 Gas Jet Noise ControlThe major challenge in dealing with jet engine noise is to reduce the high noiselevels with minimal impact on the thrust. The greatest progress came about byreducing the jet velocity while keeping the thrust constant because the soundpower is proportion to (thrust) × U 6 /c 0 5 , according to the Lighthill equation(14.41). Early efforts were centered on modifying the nozzle shape to variationsof the “cookie-cutter” forms or using multiple smaller nozzles. Some of thesedesigns produced up to 10 dB noise reductions with a rather small loss of thrust.Newer by-pass and fan-jet engine designs entail much lower jet velocities, largestreamlined enter bodies, and annular jets that may be subdivided in any one ofseveral ways.In the case of simple high-velocity air jets in industrial environments, such asthose used to power air tools, provide cooling or venting, parts ejection, and so on,a number of straightforward noise reduction measures can be applied. The basicsteps include the following:1. Reduction of the required air velocity by moving the nozzle closer to a partbeing ejected, while maintaining the same value of thrust.2. Adding additional nozzles, reducing the required velocity but again sustainingthe same thrust magnitude.3. Installation of newer models of quieter diffusers and air shroud nozzles.4. Interruption of airflow in sequence with ejection or blow-off timing.Methods 1 and 2 above result in noise reduction from cutting down on the jetvelocity. Reduction of the airstream velocity should be the first consideration ofany noise reduction program. Usually, the only constraint is the preservation ofthe air-jet thrust. The thrust T j of a jet is given byT j = m V (N) (14.47)wherem = mass flow rate, kg/s

394 14. Machinery Noise Control(a)(b)Figure 14.14. (a) Multi-jet diffuser and (b) restrictive diffuser nozzle.V = average jet velocityEquation (14.47) indicates that thrust can be preserved if the mass flow is increasedwhile reducing the jet velocity. Increasing the nozzle exit area and moving the nozzlecloser to the ejection target will also provide considerable velocity reduction.Adding on two or more nozzles will also permit reduction in air-jet velocity and acorresponding reduction in noise level.Multiple-Jet and Restrictive Flow Silencer NozzleFigure 14.14(a) shows an example of a multi-jet diffuser. The noise reductionaccrues mainly from the reduction in jet air core size, which also lessens theturbulence in the mixing regions. Also, the smaller inner jets flow along with theouter layer of high velocity air, thereby reducing the shearing action with the staticambient air. In the restrictive flow nozzle of Figure 14.14(b), the high-velocity coreis minimized by the sintered metal restrictor, typically mounted into the nozzleexit. The flow velocity is reduced somewhat, with a corresponding drop in theamount of noise radiation. In both of these nozzle types, there will be some loss inthe jet thrust, so additional nozzles may be required to keep up the same amountof the jet thrust.Air Shroud Silencer NozzlesIn Figure 14.15, an air shroud silencer nozzle is shown with its airflow pattern.Here the noise reduction is achieved through bypassing some of the primary airflowaround and over the nozzle. The bypassed air lowers the velocity gradients betweenthe primary jet and the ambient air, thus cutting down on the shearing action andthe resultant radiation of noise. There is usually little change in the mass flow forthis type of silencer and the jet thrust is generally preserved. A micrometer dialprovides control of amount of bypassed air. The typical noise reduction throughthe use of air-shroud silencer nozzles is 10–20 dB in the critical high-frequencyrange of 2–8 kHz for small high-velocity jets.

14.13 Gas Jet Noise Control 395Figure 14.15. Air shroud silencer. The micrometer shown here adjusts the amount ofair supply bypassing the nozzle instead of going through it. (Courtesy of ITW Vortec,Cincinnati, Ohio.).Impingement NoiseImpingement noise occurs when a gas jet is brought close and impinges upon asolid surface or object. A sharp increase in the noise level occurs, particularly inthe range of higher frequencies. In addition, the impingement of the gas jet onthe surface can bring about unsteady forces in the form of aerodynamic dipoles,which can be described as a pair of point sources of equal magnitudes separated bya small distance and oscillating with an angular phase difference of 180 ◦ . This is tosay that these two point sources are out of phase, i.e., when one of the point sourcesis positive, the other is negative. These dipoles constitute the basic mathematicalmodels used to describe the noise and radiation of many common noise sourcesincluding propellers, valves, loudspeakers, air duct diffusers, a number of musicalinstruments, and so on. There are also some directivity patterns present when thenoise evolves from dipole sources.Impingement noise, as with jet noise, is difficult to predict in the way of itsamplitude and spectral characteristics of the noise-generating mechanism. Fromboth analytical and experimental considerations, the radiated sound power forimpinging subsonic jets depends in the first order on the fifth or sixth powerof the flow velocity. Again, as with the free jet flow, even a slight reduction inthe flow velocity can bring about appreciable reductions in impingement noise.If a jet flows over a sharp edge or discontinuity, even more noise is likely tobe generated. Whistle-like edge tones will also occur. Cutting down on the flowturbulence created by jet flows over a cavity or an obstruction can lessen theperiodic components and the impingement noise. In these cases, the impingementnoise can be lessened by avoiding or eliminating the presence of cavities and byredirecting the jet away from the edge.

396 14. Machinery Noise ControlGaseous Flows in Pipes or DuctsVelocities in pipes or ducts usually occur in the subsonic range, but where there arevalves or vents present to control the flow, extremely high noise levels can occur.Noise levels have been measured as high as 140 dB downstream of reduction valvesin large steam pipes. In many cases, pressure drops across a control or regulatorvalve are sufficiently large to choke the flow at the discharge, with the result thatthe flow of the gas jet is sonic or almost sonic with corresponding generation ofhigh intensity aerodynamic noise. This noise can propagate through the pipe wallsinto the immediate surroundings, and what is even worse, it can propagate almostunabated downstream with very little attenuation.Because of the complexity of the noise source mechanism and the degree ofuncertainty in transmission loss of the pipe and ducts, it becomes quite difficultto predict the magnitude of the aerodynamic noise. But some guidance can bederived from empirical data, which can be used to establish first-order estimates.The turbulent mixing areas downstream of the valve constitute the principal regionof noise generation. But if valves are encased in thick housings, the noise levels aretypically 6 to 10 dB lower. The spectral character of the noise resembles that forhigh-velocity gas jets, i.e., there are peak levels present in the range of 2–8 kHz. Inshort, it can be expected wherever high-pressure steam and gas flows are regulatedor discharged through valves, noise levels exceeding 100 dB will most likely occur.The valves will have relatively thick walls, so the piping system itself, downstreamof the valve, is the primary source of externally radiated noise.Three basic approaches can be considered in reducing the noise from the controlvalve regions, namely, (a) revising the dynamics of the flow, (b) introducing anin-line silencer to absorb acoustic energy, and (c) increasing the transmission lossin the pipe walls.Of the three approaches just mentioned, altering the dynamics of the flow, whichmeans actually reducing noise reduction at the source, is probably the most preferredmethod. Changing the dynamics entails reducing the flow velocity via multiplestages of pressure reductions or diffusion of the primary jet. Figure 14.16shows multiple stages of pressure reduction with the use of expansion plates. Theflow velocity is lessened sequentially in each expansion chamber. These plates alsoact as a diffuser, reducing turbulent mixing. As much as 20 dB noise reduction haveFigure 14.16. Multiple stages of pressure reduction in a throttling system with use ofexpansion plates.

14.13 Gas Jet Noise Control 397Figure 14.17. A cutaway view of an in-line silencer used to reduce aerodynamic noise inpipes. (Reproduced with permission of Fisher Controls International, Inc.).been achieved. A disadvantage of this setup that a back pressure may be induced,which has the effect of impeding flows. The use of diffusers is another approach,in which the flow is diffused into smaller interacting jets.In-line silencers, one of which is shown in Figure 14.17, basically consist offlow through ducts surrounded by absorptive materials separated from the flow byperforated metal sheets. A diffusive inlet may precede the tubular portion of theabsorptive liner. The absorption materials typically consist of fiberglass or metalwool.Finally, in the third approach, that of increasing the transmission loss in pipewalls, two methods can be used. One is to increase the pipe wall thickness and theother is to swath the pipe in acoustical absorption materials. However, no simpleprocedure exists for estimating the transmission loss through pipe walls. Pipingstandards are promulgated by mechanical engineering societies, which specify wallthicknesses for high-pressure, high-velocity flow installations. These thicknessesare specified with the principal aim of preventing ruptures and other failures ofthe pipes that are subject to high pressures. For larger pipes, the standard wallthickness is approximately 3/8 inch. A wall thickness greater than that requiredto meet stress requirements can be selected so that more sound attenuation can beachieved from the presence of the thicker walls, in the range of 2–20 dB additionaltransmission loss.However, a resonance-like condition can occur at the ring frequency, at whichvalue the transmission loss virtually disappears. This is somewhat analogous tothe coincidence frequency effect associated with barriers. This dip in transmissionloss occurs in a pipe when a single wavelength of sound becomes equal to thenominal circumference of the pipe wall, a situation the ring frequency f r can bemathematically expressed asf r =π D pwhere c w is the longitudinal speed of sound in the pipe wall material in m/s; andD p the nominal diameter of the pipe (which can be considered the average of theinside and outside diameters of the pipe) in meters. For example, a 25-cm steelpipe would have a ring frequency of 6598 Hz (since c w = 5182 m/s for steel). Itwould then be expected that the peaks noise levels radiated from this pipe wouldc w

398 14. Machinery Noise ControlFigure 14.18. Lagging or jacketing of a pipe to minimize flow noise transmission.occur in the vicinity of 6.6 kHz. Higher order resonance-like conditions will alsooccur at 2 f r ,3f r , etc., but would be of little concern because those frequenciesare extremely high.Besides increasing wall thicknesses, another method for achieving noise attenuationin pipes is that of wrapping or lagging the pipes. A typical lagging setup isshown in Figure 14.18. A 2.5–8 cm layer of acoustically absorbent material (fiberglass,mineral wools, or polyurethane foam) is wrapped around the pipe wall. Thisabsorbing layer, in turn, is sheathed in sheet metal, dense vinyl, or sheet lead. Theouter dense layer is extremely important in providing a high level of noise reduction.Applying an even denser outer layer and adding additional composite layerscan result in even more noise reduction. The principal disadvantage of using laggingas the only noise reduction measure is that long lengths of piping would needtreatment, so it would be advisable to give priority to noise source reduction (i.e.,utilization of multiple pressure reduction stages or diffusers) and to use lagging asa secondary measure to bring the overall sound levels down to acceptable values.14.14 Mufflers and SilencersIn the preceding section we have discussed some of the elements of silencingair-jet flow. In this section we examine further aspects of reducing the noise fromair-jet flow. The term muffler is commonly applied to the exhaust gas silencer foran internal combustion engine, and silencer usually denotes the noise suppressorinstalled in a duct or air intake. We use these terms interchangeably in this sectionsince the same operational principle applies to both of them. Mufflers and silencerswere developed to reduce noise energy while facilitating gas flow.Both silencers and mufflers fall into one or the other of two categories: dissipativeunits and reactive units. Dissipative mufflers and silencers reduce noise energy byemploying sound-absorption materials that are flow resistive at frequencies inthe audio range. Reactive mufflers and silencers reduce noise through destructive

14.14 Mufflers and Silencers 399Figure 14.19. Expansion chamber of a reactive muffler.interference. Both reactive and dissipative principles may be combined in a singlemuffler or silencers in order to ensure effectiveness over a broad frequency range.Reactive Mufflers and SilencersConsider the basic reactive muffler in Figure 14.19. Sound waves transmit from leftto right in the inlet pipe and reflections occur in the expansion chamber, causingdestructive interference under the appropriate conditions. In this analysis (Daviset al., 1984), the following assumptions are made:1. Sound pressure is small in comparison with absolute pressure in the expansionchamber.2. No reflected waves occur in the tailpipe (i.e., the outlet pipe).3. The expansion chamber walls do not transmit nor conduct sound.4. Only plane waves exist.5. Viscosity effects are negligible.We denote the following subscripts in the description of incident and reflectedwaves: I for incident waves and R for reflected waves. The particle displacementsξ of the incident and reflected waves are described in complex format as follows:andξ I = A I e i(ωt−kx) (14.48)ξ R = A R e i(ωt+kx) (14.49)Here A I and A R are complex amplitudes. Particle velocities are obtained by differentiatingequations (14.48) and (14.49) with respect to time:u I = iω A I e i(ωt−kx) (14.50)

400 14. Machinery Noise Controlandu R = iω A R e i(ωt+kx) (14.51)We recall that for a plane wave in the direct field, sound pressure is related toparticle velocity byp = ρ cuwhich is used in Equations (14.50) and (14.51) to obtainp I = iρ cω A I e i(ωt−kx) (14.52)p R = iρ cω A R e i(ωt+kx) (14.53)Let x = 0 designate the junction of the inlet pipe and expansion chamber.Pressure must be continuous at this junction, hence(p I 1 + p R1 ) x=0 = (p I 2 + p R2 ) x=0 (14.54)Inserting Equations (14.52) and (14.53) into (14.54) givesA I 1 + A R1 = A I 2 + A R2 (14.55)Subscript 1 refers to the left of the junction and subscript 2 refers to the right.Continuity of flow requires thatA (u I 1 − u R2 ) x=0 = B (u I 2 − u R2 ) x=0Setting B/A = m, where A refers to the flow area of the inlet pipe and the outletpipe and B the flow area of the expansion chamber, we have from continuityA I 1 − A R1 = m(A I 2 − A R2 ) (14.56)In Figure 14.19, C denotes the length of the expansion chamber. At the junctionof the expansion chamber where x = C, pressure and flow continuity necessitatesthatfrom whichandresulting in(p I 2 + p R2 ) x=C = p 3A I 2 e −ikC + A R2 e ikC = A 3 (14.57)B(u I 2 − u R2 ) x=C = Au 3 (14.58)m(A I 2 e −ikC + A R2 e ikC ) = A 3 (14.59)We now have four Equations (14.55)–(14.59) and five unknowns, A I 1 , A R1 , A I 2 ,A R2 , and A 3 . These equations are solved simultaneously to correlate conditions at

(1 + m)A I 1=A 314.14 Mufflers and Silencers 401the inlet and outlet of the expansion chamber, resulting in the complex ratio(1 + 1 )( ) 1e ikC + (m − 1)mm − 1 e −ikC= cos(kC) + i (m + 1 )sin(kC) (14.60)2 mIn order to establish the transmission loss, the ratio of sound intensity at the expansionchamber inlet to transmitted sound intensity is needed:I I 1= p2 rms(I 1)I 3 prms(3)2 =A 2 I 1∣A 2 ∣ = cos2 (kC) + 1 (m + 1 ) 2sin 2 (kC)34 m= 1 + 1 (m − 1 ) 2sin 2 (kC) (14.61)4 mBecause transmission loss is a function of the ratio of power incident on the mufflerto the power transmitted, i.e.,TL = 10 log4( )WincidentW transmittedand if the inlet and outlet areas are equal, we obtain( ) ( ) [WI 1II 1TL = 10 log = 10 log = 10 log 1 + 1 W 3 I 3 4(m − 1 ) ]2sin 2 (kC)m(14.62)where k = 2π/λ is the wave number. Equation (14.62) becomes invalid if any ofthe lateral dimensions of the expansion chamber exceeds 0.8λ.Figure 14.20 displays a plot of the theoretical transmission loss versus C/λfor various area ratios. Equation (14.62) forecasts a transmission loss of zerowhen the argument of the sine function is 0, π, 2π, .2nπ, ... and a maximumtransmission loss when the argument is π/2, 3π/2, ... (2n + 1) π/2, ..., wheren = 0, 1, 2, 3, etc. It then follows that the expansion chamber works best whenlength C constitutes an odd number of quarter-wavelengths, i.e., C = λ(2n + 1)/4for maximal TL. But the expansion chamber becomes ineffective when the chamberlength C measures out to an integer number of half-wavelengths, i.e., C = nλ (i.e.,λ/2, λ, 3λ/2, etc.). For extremely low frequencies C/λ approaches zero, and thetransmission loss likewise approaches zero.Example Problem 11A reactive muffler is to be designed to produce a transmission loss of 20 dB at150 Hz. The inlet and outlet pipes are 60 mm in diameter and the average gas (air)temperature is 100 ◦ C. Determine the size of the expansion chamber.

402 14. Machinery Noise ControlFigure 14.20. Transmission loss TL versus C/λ.SolutionAt 100 ◦ C, the velocity c = 20.04 √ T + 273.16 = 387.1 m/s. The wavelength ofthe 150 Hz sound is given by λ = c/ f = 387.1/150 = 2.58 m. For the shortestexpansion chamber to produce maximum transmission loss, we examine theargument of the sine function:kC = 2πC = π λ 2 radiansfrom which we getC = λ 4 = 2.58 = 0.645 m = 645 mm4From Equation (14.62) the transmission loss is given by[TL = 10 log 1 + 1 (m − 1 ) ]2sin 2 (kC)4 m[= 10 log 1 + 1 4[= 10 log 1 + 1 4(m − 1 m(m − 1 ) ] 2m) 2 ( ) ]2π × 0.645sin 22.58(14.63)

14.14 Mufflers and Silencers 403Let(m − 1 m) 2= Then[ ( ) ]A B 2− BB A A − 1 = 0where from Equation (14.63) = 2(10 TL/10 − 1) 1/2The solution to the quadratic expression inside the bracket isBA = + √ 2 + 42Note the negative root carries no physical significance. For a transmission loss of20, = 19.90 andBA = 19.90 + √ 19.90 2 + 4= 19.952The cross-sectional area A of the inlet pipe and the outlet pipe are each π(60) 2 /4 =2827 mm 2 . The expansion chamber cross-sectional area isB = A × B A= 2827 × 19.95 = 56,399 mm2If the expansion chamber has a cylindrical cross-section, its inside diameter d isfound fromB = π d24Hence√ √4B 4 × 56,399d =π = = 268 mmπArea B represents the minimum cross-sectional area of the expansion chamberthat is required to produce the 20-dB transmission loss.Dissipative Mufflers and SilencersIn the preceding paragraphs, dissipative silencers were discussed. Lining the ductsof HVAC systems with sound absorptive materials generally provides adequatenoise attenuation. However, many industrial applications require a silencer thatcan provide a large amount of noise attenuation or insertion loss in a relativelysmall space. If the noise energy covers a narrow frequency range, reactive silencersconstitute the better solution. Reactive silencers are preferred where the gas flowscontain particles or other components that could contaminate sound absorbing

404 14. Machinery Noise Controlmaterials. Dissipative silencers are more effective over a wider range of frequenciesand in silencing fluctuating machinery noises. In a number of situations, in orderto obtain the greatest noise reduction over a wider frequency range, both reactiveand dissipative principles are combined in a single silencer.14.15 Active Noise ControlRelatively recent advances in digital electronics have made it more economicallyfeasible to apply active noise control in cutting down on noise in aircraft andautomotive interiors, pumps, compressors, electric motors, transformers, and soon (Piraux and Nayroles, 1980; Nelson and Elliott, 1992, 1997). Earlier activenoise control techniques were first used in the study of noise in fan ducts. Theadvent of inexpensive digital processors enabled the conversion of analog audiofrequency signals into digital form, then processing them through a digital filterand then converting back into an analog signal with very little time delay. Thebasis of active noise control is to duplicate the noise that is the same but 180 ◦ outof phase, so that when the offending noise and the duplicated out-of-phase noiseare combined, a cancellation will occur.Figure 14.21. An example of an active noise control system for a communication headset.

References 405Figure 14.21 illustrates a simple single-channel active noise control system.Sound is detected by a microphone and processed though a digital filter imbeddedin a special-purpose microprocessor prior to being fed into a loudspeaker thatradiates the sound that is intended to interfere destructively with the “original”unwanted sound. The characteristics of the digital filter are designed to minimizethe time-averaged signal at the error microphone on a continuous basis.The following factors, which may limit the effectiveness of active noise control,must be considered:1. Continuous and reliable measurement, signal processing, and sound generationare required.2. For maximum effectiveness the anti-noise source must be near the noise sourceor near the receiver.3. When changes occur in the relative position of the noise source and the observer,the effect could be one of sound reinforcement rather than cancellation. Thisconsideration can limit many active-noise control systems to low-frequencysounds. However, the presence of error-detection microphones combined withdigital filters that continuously adapt to changing conditions can mitigate thepossibility of unintentional sound reinforcement.4. Even under the most ideal real conditions, cancellation of sounds cannot be total,due to the statistical nature of the molecular motion in the sound propagationmedium.Active noise control systems have even been incorporated into items as small asheadphones for the purposes of shutting out external noises while allowing desiredsound to reach the ears. Such headphones are useful in aviation cockpits, passengercompartments of aircraft, and other noisy environments.ReferencesAMCA Standard 300. Reverberant Room Method for Sound Testing of Fans. ArlingtonHeights, IL: Air Movement and Control Association.ASHRAE Standard 51. Laboratory Method of Testing Fans for Rating. Atlanta, GA:American Society of Heating, Refrigeration and Air-Conditioning Engineers, Inc.ASHRAE. ASHRAE Handbook. Atlanta, GA: American Society of Heating, Refrigerationand Air-Conditioning Engineers, Inc.Davis, D. D., Stokes, G. M., Moore, D., and Stevens, G. L. 1984. Theoretical and experimentalinvestigation of mufflers with comments on engine-exhaust muffler design.(Reprint, NACA Report 1192. 1954), Noise Control, M. J. Crocker, (ed.). New York:Van Nostrand Reinhold.Fuller, C. R., Elliott, S. J., and Nelson, P. A. 1996. Active Control of Vibration. London:Academic Press.Graham, J. Barrie and Hoover, Robert M. 1991. Fan noise. In: Handbook of AcousticalMeasurements and Noise Control, 3rd ed. Harris, Cyril M. (ed.). New York: McGraw-Hill, Chapter 24.Hand, R. F. 1982. Accessory noise control. In: Noise Control in Internal CombustionEngines, Baxa, D. E. (ed.). New York: Wiley-Interscience, pp. 437–477.

406 14. Machinery Noise ControlHarris, Cyril M. (ed.). 1991. Handbook of Acoustical Measurements and Noise Control,3rd ed. New York: McGraw-Hill, Chapters 34–45.Hubbard, H. H. (ed.). 1995. Aerodynamics of Flight Vehicles: Theory and Practice, Vol. 1:Noise Sources, Vol. 2: Noise Control. Woodbury, NY: Acoustical Society of America.(A now classic compendium of theory and experimentation in the field of aeroacoustics,originally sponsored by the Aeroacoustics Technical Committee of the American Instituteof Aeronautics and Astronautics (AIAA) and reprinted by the American Instituteof Physics.)Jones, Dylan M. and Broadbent, Donald E. 1991. Human performance and noise. Handbookof Acoustical Measurements and Noise Control, 3rd ed. Harris, Cyril M. (ed.).New York: McGraw-Hill, Chapter 41.Lighthill, James M. 1952. On sound generated aerodynamically: I. General theory. Proceedingsof the Royal Society A 211:564–587. (A classic paper by the great master ofmodern aeroacoustics theory.)Lighthill, James M. 1978. Waves in Fluids. Cambridge: Cambridge University Press.(A veritable classic.)Magrab, E. B. 1975. Environmental Noise Control. New York: John Wiley and Sons.Nelson, P. A. and Elliott, S. J. 1992. Active Control of Sound. London: Academic Press.Nelson, P.A. and Elliott, S. J. 1997. Active Noise Control. In: Encyclopedia of Acoustics,Vol. 2. Crocker, Malcolm J. (ed.). New York: John Wiley & Sons, Chapter 84,pp. 1025–1037.Piraux, J. and Nayroles, B. 1980. A theoretical model for active noise attenuation in threedimensionalspace. Proceedings of Inter-Noise’80. Miami, FL: 703–708.Webb, J. D. (ed.). 1976. Noise Control in Industry. Sudbury, Suffolk, Great Britain: SoundResearch Laboratories Ltd.Problems for Chapter 141. A 6-blade fan rotates at 1150 rpm. Determine the frequencies of the noise thatwill emanate from the fan.2. Find the sound power of a fairly efficient electric motor rated at 85 hp at2400 rpm.3. Determine the frequency of the blade rate component of a diffuser-type compressorwith 24 blades in the rotor and 36 blades in the stator. The rotor rotatesat 9000 rpm.4. A 20-hp hydraulic screw-type pump operates at 1200 rpm with 4 chamberpressure cycles per revolution. Find its fundamental frequency and the soundpower output.5. A radial forward-curved fan has 36 blades and a rotor diameter of 120 cm. Itoperates at 950 rpm with an airflow rate of 24 m 3 /s under the effect of a totalpressure of 1.8 kPa. What is the total sound power at the inlet?6. Estimate the sound power of a “quiet” 100-hp electric motor operating at1200 rpm.7. A ball bearing has a stationary outer race. The inner and outer race diametersare 30 mm and 46 mm, respectively, measured at the point of ball contact. Theinner race rotates at 5200 rpm. There are 12 balls in the bearing. Determine

Problems for Chapter 14 407the noise and vibration frequencies that can occur as the result of imbalanceand defects.8. A ball-bearing is constructed as follows: it has twelve 15-mm-diameter balls,and the inner race diameter is 50 mm at the point of ball contact. The innerrace has a rotational speed of 1800 rpm and the outer race remains stationary.Find:(a) rotational speed of separator(b) speed of the balls relative to the separator(c) shaft imbalance frequency(d) outer race defect frequency(e) inner race defect frequency(f) frequency arising from damage to one ball(g) frequency attributable to imbalance in the separator.9. A roller bearing contains ten 16-mm-diameter rollers. The inner race diameteris 30 mm. The inner race rotates at 6000 rpm while the outer race remainsstationary. Find:(a) rotational speed of the separator(b) speed of the rollers relative to the separator(c) shaft imbalance frequency(d) outer race frequency(e) inner race frequency(f) frequency caused by a defect in one roller(g) frequency caused by imbalance of the separator.10. In a nonplanetary gear transmission, the input shaft gear has 40 teeth and rotatesat 1200 rpm. It meshes with another shaft through a 30-tooth gear. Predict allof the fundamental frequencies that are likely to arise in this transmission.11. The gears of a reverted gear train of Figure 14.8 carry the following specifications:gear 1 (driver), 50 teeth; gear 2, 90 teeth; gear 3, 46 teeth; and outputgear 4, 47 teeth. The input shaft rotates at 2400 rpm.(a) Determine the rotational speeds of the other shafts.(b) Compute each of the fundamental tooth-error and shaft frequencies andthe first three harmonics.(c) Find all of the fundamental tooth meshing frequencies and their first threeharmonics.(d) Determine the sideband frequencies.12. It is specified that a pair of spur gears is to be used to reduce shaft speed from2400 rpm to 1600 rpm. The driver has 18 teeth and a diametral pitch of 4. Thepressure angle is 20 ◦ in the stub teeth.(a) Predict the fundamental tooth-error frequencies.(b) Determine the fundamental tooth meshing frequency.(c) Find the contact ratio, on the basis of Equation (14.21) and discuss theresults in terms of noise. Assume that a = 0.25.13. Consider a chain drive system that contains a 15-tooth input sprocket thatrotates at 2500 rpm. The chain has 90 links, and it is specified that the outputsprocket rotates at 1500 rpm. Determine:

408 14. Machinery Noise Control(a) the number of teeth needed for the output sprocket(b) the range of output speed if input speed remains constant(c) the probable frequencies of noise and vibration due to damage to one toothor imbalance in either shaft(d) the probable frequencies due to damage to one link(e) the probable frequencies due to chordal action.14. The input speed of a chain drive is to be reduced by one-half at the output. The24-tooth input sprocket rotates at 1200 rpm. The chain contains 160 links.(a) How many teeth are needed in the output sprocket?(b) For constant input speed, determine the range of output speed.(c) Determine the likely frequencies of noise and vibration due to imbalanceof either shaft or damage to one tooth.(d) Find the possible frequencies due to damage to one link(e) Determine the possible frequencies due to chordal action.15. In a Hooke-type universal joint, in which the cross-link plane is normal to theaxis of the driving shaft, the speed ratio is given by n 2 /n 1 = 1/(cos φ), whereφ is the shaft misalignment angle, with the result that the speed range due tomisalignment is given bycos φ ≤ n 2≤ 1n 1 cos φA misalignment of 10 ◦ occurs in this type of universal joint. The input shaftrotates with a constant speed of 1200 rpm.(a) Find the speed range of the output shaft.(b) Determine the frequency resulting from vibration or noise.16. A fairly simple 3/8-inch-diameter nozzle functions under an inlet pressure of100 psi (gauge). Determine the sound level at a distance of 5 ft for an ambientair temperature of 80 ◦ F.17. A simple 25 mm nozzle operates at an inlet pressure of 600 kPa. Determinethe sound pressure level at a distance of 1.0 m at an air temperature of 20 ◦ C.18. Fifty sound level readings are taken at 10-s intervals. Find the percent exceedednoise levels L 10 , L 50 , and L 90 for the following distribution:Level [dB(A) ± 0.05]Number of Readings87 286 885 1484 1583 582 481 2

15Underwater Acoustics15.1 Sound Propagation in WaterSound waves are absorbed and scattered in water to a much lesser degree thanelectromagnetic waves. Because of this property sound waves have proven tobe particularly useful in detecting distant objects undersea by means of sonar(acronym for SOund NAvigation and Ranging). Passive sonar is strictly a “listening”device that detects sound radiation emitted (sometimes unintentionally) by atarget. In active sonar, the process entails sending out a sound pulse and listeningfor a returning echo. The loudness of the echo hinges principally on the amountof energy absorption in the water and the degree of reflection from an interceptingsurface. Some of the energy is scattered backward in a random fashion to theecho-ranging emitter, either by particles or inhomogeneities in the water, or by theocean surface or sea bottom. This scattering results in a phenomenon referred to asreverberation. The sound directly reflected back from an obstacle (target)—suchas a submarine or a whale—constitutes the echo.Recognizable echoes have been mapped for schools of fish, dolphins, whales,patches of kelp and seaweed, sunken wrecks and pronounced irregularities inshallow depths. Certain water conditions give rise to echoes at very short ranges;and ocean swells also can generate echoes. Ship wakes and other types of bubblescreens make effective targets as wells as icebergs. The most prominent use ofsonar has been in subsurface warfare: submarines, surface vessels, and underwatermines are obviously the most important targets.The reflections of the probing signals from submarines to surface vessels areevaluated in terms of target strengths, a quantitative measure of intensities of theechoes. Target strengths (TS) depend on a number of factors, viz. target size, shape,orientation with respect to the probe signal source, distance from the source to thetarget, and frequency of the probe signal. The intensity of the reflected sound isalso a function of the intensity of the probe signal striking the target, the distancefrom the target to the point of echo measurement (which is usually at virtuallythe same location as the probe signal source), and the acoustic absorptivity ofthe target. The effects of these variants can be established only if the radii ofcurvature of the sound waves striking the target and returning to a receiver are409

410 15. Underwater Acousticsof much greater magnitudes than those of the target’s dimensions, i.e., the probesignal waves should ideally be plane. In terms of ray acoustics the incident soundwave must be substantially parallel in the region of the target that they strike, andreflected sound rays should also be parallel over the area of the sonar receiver.In the military use of sonar, the applicable spectrum range covers the ultralowfrequency to the megahertz region. Acoustic mines detect the pressures below 1 Hz,which are generated by moving ships. These mines detect the acoustic radiationand explode when the acoustic level reaches a certain level in their bandpass.Such mines can be destroyed harmlessly through the use of a minesweeper that isbasically a powerful signal source towed behind a minesweeping vessel.In passive detection, the acoustic radiation of both water-surface and underwatervessels, are sensed by a hydrophone array mounted on the spy vessel or submergedat the bottom a long distance away. The receiving array must be directional in orderto be able to locate the target through sensing of somewhat higher frequencies.Modern echo-ranging (active) sonar consists of an elaborate array of equipmentto send out signals in the form of long, high-power pings in designated directionsvertically and horizontally, and newer signal processing techniques present theechoed data to the observer. Transducer arrays are often enclosed in separatehousings that are towed underwater behind the surface vessel, so that shallowthermal gradients 1 can be penetrated, and the sonar can probe in the stern (forward)direction, a procedure that cannot be achieved with a hull-mounted sonar.Peaceful uses of sonar expanded greatly immediately after World War II. Originallydeveloped for depth sounding, sonar is now being used to find fish, study fishmigration, map ocean floors, locate underwater objects, transmit communicationsand telemetric data, serve as acoustic speedometers, act as position-marking beacons,and monitor well-head flow control devices for undersea oil wells. Passivesonar provides marine biology researchers a window for tracking sounds made bycetaceans (whales, dolphins, and porpoises).15.2 Some Basic Concepts Pertaining to Underwater SoundIn an acoustic plane wave passing through a fluid medium of density ρ, the particlepressurep relates to the fluid particle velocity u as followsp = ρcu.The fluidic parameter ρc is called the specific acoustic resistance. Its value, onthe average, for seawater is 1.5 × 10 5 g/cm 2 sor1.5× 10 6 kg/m 2 s. In contrast,ρc = 42 g/cm 2 s for air. From our previous chapters we recognize that this parametercan also assume a complex value in its role as the specific acoustic impedance.1 In the 1920s and 1930s, it was observed that good echoes were obtained with the early versionsof shipboard echo-ranging equipment in the morning and poor or no results were obtained in theafternoon. This was due to the shifts in the seawater thermal gradients that caused sound to refracttoward the sea bottom and thereby place a target in the “shadow zone.”

15.3 Speed of Sound in Seawater 411In order to understand the importance of acoustic impedance, consider the factthat some sort of a piston must drive against a medium in order to generate acousticenergy. The medium presents a resistance to that drive, and in acoustic terminology,this resistance is the specific acoustic impedance. The conventional techniquesapplied in the design of the usual audio equipment, namely loudspeakers andmicrophones, are not applicable to underwater sonar. Air is a very light substance,so the driving mechanism needs to produce a large displacement with very littleforce. In the case of underwater sound, however, it is necessary to provide a verylarge force to generate even a small displacement. This means that the sonar mustpossess a very large mechanical impedance, i.e., a large ratio of the complexdriving force to the complex velocity. It is also evident that a propagating soundwave carries mechanical energy that includes the kinetic energy of the particles inmotion and the potential energy of the stresses occurring in the elastic medium. Inthe process of a wave propagating, a certain amount of energy per second crosses aunit area. This power per unit area describes the intensity of the wave. If a unit areais given an orientation with respect to reference coordinates, the intensity becomesa vector quantity represented by a Poynting vector normal to the unit area, in thesame manner as in the theory of electromagnetic propagation. In a plane wave,the instantaneous intensity relates to the instantaneous acoustic pressure in thefollowing way:I = p2ρ c .For the cases entailing transient signals, or signal distortions, or target impingementoccurring, it is more useful to use the concept of energy flux density of theacoustic wave, as defined by∫ ∞E = Idt = 1 ∫ ∞p 2 dt.0 ρ c 0The unit of intensity in underwater sound is the intensity of a plane wave havinga root-mean-square (rms) pressure of 1 micropascal (1 μPa)or10 −5 dyne/cm 2 .This amounts to 0.64 × 10 −22 W Ag/cm 2 . This pressure of 1 μPa serves as thereference level for the definition of the decibel as applied to underwater acoustics,in contrast to the decibel referred to 20 μPa, that describes the sound level ofacoustic propagation in air.15.3 Speed of Sound in SeawaterThe principal difference between the speed of sound in fresh water and seawateris that with the latter, salinity constitutes an additional factor besides pressure andtemperature. In fact, the speed of sound in seawater varies with geographic location,water depth, season, and even the time of the day. Sound velocity in a naturalbody of water was first measured in 1827 when the Swiss scientist Jean-DanielColladen (1802–1893) and the French mathematician Charles Sturm (1803–1855)

412 15. Underwater AcousticsTable 15.1. Expressions for Sound Speed (m/s) in Seawater as Functions ofTemperature, Salinity, and Depth.Expression Limits Referencec = 1492.9 + 3(T − 10) − 6 × 10 −3 (T − 10) 2 −2 ≤ T ≤ 24.5 ◦ Leroy− 4 × 10 −2 (T − 18) 2 + 1.2(S − 35) 30 ≤ S ≤ 42− 10 −2 (T − 18)(S − 35) + D/61 0 ≤ D ≤ 1000c = 1449.2 + 4.6T − 5.5 × 10 −2 T 2 0 ≤ T ≤ 35 ◦ Medwin+ 2.9 × 10 −4 T 3 + (1.34 − 10 −2 T )(S − 35) 0 ≤ S ≤ 45+ 1.6 × 10 −2 D 0 ≤ D ≤ 1000c = 1448.96 + 4.591T − 5.304 × 10 −2 T 2 0 ≤ T ≤ 30 ◦ MacKenzie+ 2.374 × 10 −4 T 3 + 1.340(S − 35) 30 ≤ S ≤ 40+ 1.630 × 10 −2 D + 1.675 × 10 −7 D 2 0 ≤ D ≤ 8000− 1.025 × 10 −2 T (S − 35) − 7.139 × 10 −13 TD 3D = depth, in meters; S = salinity, in parts per thousand; T = temperature in degrees Celsius.collaborated in striking a submerged bell in Lake Geneva and simultaneouslysetting off a charge of powder in the air (Colladen and Sturm, 1827; Wood, 1941).The intervals between the two events were timed across the lake, and they obtaineda value of 1435 m/s at 8.1 ◦ C, which is amazingly close to the modern value.Subsequent measurements over the past 40 decades entail direct measurementof speeds under carefully controlled conditions, and the speeds were mappedas functions of oceanographic parameters (Del Grosso, 1952; Weissler and DelGrosso, 1951; Wilson, 1960).Expressions have been derived in terms of three basic quantities: temperature,salinity, and pressure (hence depth). Save for the presence of contaminants such asbiological organisms and air bubbles, no other physical properties were found toaffect the speed of sound in seawater. The interdependence of these three parametersis not a simple one. Del Grosso (1974) developed an expression containing19 terms each to 12 significant figures in the powers and cross-products of the threeprincipal variables, and Lovett developed a simpler but still unwieldy expression.We list in Table 15.1 a number of simpler expressions that suffice for most practicalwork. In general, these expressions result in values with errors less than a few partsin ten thousand or approximately 0.5 m/s.The speed of sound in the sea increases with temperature, depth, and salinity.Table 15.2 gives the approximate coefficient for the rate of change in these principalTable 15.2. Approximate Coefficients of Sound Velocity.Parameter being varied Coefficient CoefficientTemperature (near 70 ◦ F)SalinityDepthc/cTc/cSc/cD=+0.001/◦ F=+0.0008/ppt=+3.4 × 10−6 /ftcT =+5ft/s◦ FcS =+4ft/s pptcD =+0.016 s−1c = speed in ft/s; T = temperature in ◦ F; S = salinity in parts per thousand (ppt);D = depth in feet.

15.4 Velocity Profiles in the Sea 413parameters. In open deep water, salinity is found to have a rather small effect onthe velocity.15.4 Velocity Profiles in the SeaThe term velocity profile refers to the variation of the sound speed with depth; itis also called the velocity–depth function. Figure 15.1 shows a typical deep-seavelocity profile, which, in turn, can be subdivided into several layers. The surfacelayer lies just below the sea surface. The speed of sound in that layer is responsiveto daily and local changes of heating, cooling, and action of the winds. The surfacelayer may consist of a mixed layer of isothermal water that is caused by action ofthe wind as it blows across the surface of the water. Sound becomes trapped in thismixed layer. On prolonged calm and sunny days, this mixed layer dissipates, to bereplaced by water in which its temperature drops with increasing depth.Figure 15.1. Deep-sea sound propagation velocity profile subdivided into principal layers.

414 15. Underwater AcousticsThe seasonal thermocline lies below the surface layer. The term thermoclinedenotes a layer in which the temperature varies with depth. The seasonal thermoclineis usually characterized by a negative thermal or velocity gradient, meaningthat the temperature and the speed of sound decreases with increasing depth, andit does vary with the seasons. During the summer and fall, when the ocean watersnear the surface are warm, the seasonal thermocline is well defined and it becomesless so during the winter and spring and in the Arctic when it tends to becomeindistinguishable from the surface layer. Below the seasonal thermocline lies themain thermocline, which hardly varies throughout the seasons. It is this layer in thedeep sea, that the temperature changes the most. Underneath the main thermocline,reaching down to the sea bottom is the deep isothermal layer having an almost constanttemperature (generally about 3 ◦ –4 ◦ C) in which the speed of sound increaseswith depth because of the effect of pressure on the sound speed. At the saddle pointbetween the negative gradient of the main thermocline and the positive gradient ofthe deep layer, there occurs a velocity minimum toward which sound traveling atgreat depth tends to bend or becomes focused by refraction. In the more northernregions, the deep isothermal layer extends almost to the water surface. The regionwhere this minimum occurs is called the deep sound-channel axis.The existence and the thicknesses of these layers vary according to latitude, season,time of day, and meteorological conditions. Figure 15.2a displays the diurnalFigure 15.2. Diurnal and seasonal variation of a surface layer near Bermuda. In (a) temperatureprofiles at various times of the day show how surface temperature increases overthe temperature at 50 ft depth. The temperature profiles for different parts of the year aregiven in (b). Temperatures and temperature differences are given in Fahrenheit degrees.

15.4 Velocity Profiles in the Sea 415Figure 15.3. Effect of latitude on sound-speed profile in deep sea.behavior of the surface layer near Bermuda. It shows how the temperature profilesvary, when the surface waters of the sea warm up during the course of asunny day and cool down during the night. These changes in temperatures affectconsiderably the transmission of sound from a surface-ship sonar, particularly inthe afternoon when echo ranging tends to be poorest. Figure 15.2b illustrates aseries of bathythermograms taken in the Bermuda area, showing how the seasonthermocline evolves during the summer and autumn. The effect of latitude onsound-speed profile in the deep sea is shown in Figure 15.3 by profiles for twodifferent locations in the North Atlantic at the same season of the year. At lowlatitudes (nearer the equator), the velocity minimum occurs at a depth of approximately3000 ft. At high latitudes the velocity minimum exists near the sea surface,and the main and seasonal thermoclines show a tendency to disappear from theprofile.In the shallow waters of the coastal regions and also on continental shelves, thevelocity profiles tend to become far less clear-cut and rather unpredictable. Thevelocities tend be greatly influenced by surface heating and cooling, changes insalinity, and the presence of water currents. Nearby sources of fresh water tend tocomplicate these effects and contribute to the spatial and temporal instability ofnumerous gradient layers.

416 15. Underwater Acoustics15.5 Underwater Transmission LossTransmission loss TL quantitatively refers to the weakening of sound between apoint 1 yard from the source and a point someplace in the sea. 2 Let I 0 indicate theintensity at the reference point 1 yard from the “center” of the acoustic source (thus10 log I 0 denotes the source level) and I 1 is the intensity at the point of interest.The transmission loss TL between the source and the point of interest isTL = 10 log I 0dBI 1Time averaging is implied in the above definition. For short pulses, a TL equivalentto that for continuous waves is given by the ratio of the energy flux density at 1 yardfrom the source E 0 to the energy flux density E 1 at the point 1 of interest, i.e.TL = 10 log E 0dBE 1If the metric units of distance are used so that the reference distance is 1 m, TLwill be 0.78 dB less than that for the TL based on the reference distance of 1 yard.Transmission loss can be subdivided into two types of losses: part of the losses isdue to spreading, a geometric effect, and the remainder is attributable to absorptionlosses that represent conversion of acoustic energy into heat.If a small source of sound is located in a homogeneous, infinite, lossless medium,the power generated by the source radiates outward uniformly in all directions.The total power radiating outward remains the same, as its wave front expands asa spherical surface with an increasing radius (the radius increases at the rate ofc × time t). Since power P is equal to intensity times the area, i.e.,P = 4πr 2 1 = 4πr 2 2 =···=constantand setting r 1 = 1 yard, the transmission loss TL to surface r 2 isTL = 10 log I 1I 2= 10 log r 2 2r 2 1= 20 log r 2which can be readily recognized as the inverse-square law of spreading, also knownas spherical spreading.When the medium is bound by upper and lower parallel planes, the spreading isno longer spherical because the sound cannot cross the boundary plates. Beyond acertain range and as shown in Figure 15.4, the power radiated by the source spreadsoutward as a wavefront that constitutes a cylindrical surface represented by anradius expanding at the rate of c × time t. The power P crossing the cylindrical2 The U.S. Navy still employs an odd melange of dimensional units in defining, say, source strengthin terms of decibels (based on the metric reference of 1 μPa for sound pressure level or 1 pW forsound power level) and the distance from the source of 1 yard rather than the 1 meter used elsewhere.Conversion to the entirely metric system requires the subtraction of 0.78 dB from the U.S. value.

15.5 Underwater Transmission Loss 417Figure 15.4. Spreading of acoustical energy: (a) in an infinite medium, (b) in a mediumbetween two parallel plates, and (c) in a tube.surfaces at radius r isP = 2π Hr1 2 = 2π Hr 2 =···=constantwhere H represents the distance between the two parallel boundary planes. Assigningr 1 = 1 yard, the transmission loss (TL) to r 2 is given byTL = 10 log I 0= 10 log r 2 .I 2In this situation the cylindrical spreading occurs in the inverse first power. Thissort of spreading occurs at moderate and long ranges when sound is entrappedwithin a sound channel in the sea.We can also consider a wave guide, essentially a conducting tube or pipe ofconstant cross section. This is a case where no spreading occurs. Beyond a certainrange, the area over which the power is distributed remains constant, and thepressure intensity and the TL are independent of range.A fourth type of spreading occurs when the signal from a pulsed source spreadsout in time as the pulse propagates through the medium. The pulse becomes elongatedby multipath propagation effects, which causes the pulse to smear out intime as it travels to a receptor. This type of effect is particularly evident in longrangepropagation in deep-ocean sound channels. If the medium is infinite andthe time stretching is proportional to the range traveled, the intensity falls off as

418 15. Underwater Acousticsthe inverse cube of the range. In the case of a sound channel of Figure 15.4b,time stretching that is proportional to the range causes the intensity to fall off asthe inverse square of the range instead of the inverse first power. In summary thespreading laws may be listed as follows:Propagation in Type of spreading Intensity varies as TL (dB)Tube None r 0 0Between parallel plates Cylindrical r −1 10 log rFree field Spherical r −2 20 log rHypersphere Free field with time stretching r −3 30 log rIt should be noted that the hyperspherical case applies in a hypothetical sense tosonar theory.Absorption varies with range in a manner different from the loss due to spreading.It occurs because acoustic energy converts into heat; and this conversionembodies a true loss of acoustic energy within the propagation medium. Considera plane wave passing through an absorbing medium. The fractional rate that theintensity of the wave decreases along distance x is proportional to the distancetraveled, i.e.dII=−kdx (15.1)where k denotes the proportionality constant and the minus sign signifies that dIdrops in the direction of increasing x. Integrating Equation (15.1) between rangesr 1 and r 2 , the intensity I 2 at r 2 is found fromRewriting (15.2) yieldsI 2 = I 1 e −k(r 2−r 1 ) . (15.2)10 log I 2 − 10 log I 1 =−10k(r 2 − r 1 ) log 10 e.Setting α = 10k log 10 e, the change of the intensity level in dB is now expressedas( )I210 log =−ω(r 2 − r 1 )I 1orα = 10 log( I 1/I 2 ).r 2 − r 1The quantity α is the logarithmic absorption coefficient and is usually expressed indecibels per kiloyard (dB/kyd) in the US or decibels per km (dB/km) in the metricsystem.

15.6 Parametric Variation of Absorption in Seawater 41915.6 Parametric Variation of Absorption in SeawaterFrom actual measurements [Wilson and Leonard (1954) among others] it becameevident that the absorption of sound in seawater was unexpectedly higher thanthat of pure water and it could not be attributable to scattering, refraction, or othereffects of propagation in the natural environment. For example, the absorption inseawater in the frequency range 5–50 kHz was found to be approximately 30 timesthat in distilled water. Liebermann (1949) suggested that this excess absorption isattributable to the sort of chemical reaction that evolves under the influence of asound wave and one of the dissolved salts in the sea.The absorption of sound in seawater depends on three effects: one is the presenceof shear viscosity, a classical effect studied by Lord Rayleigh, who derived thefollowing expression for the absorption coefficient:α = 16π 2 μ sf 2 (15.3)3ρc 3whereα = intensity absorption coefficient (cm −1 )μ s = shear viscosity, poises (approximately 0.01 for water)ρ = density, cg/cm 3 (1 for water)c = sound propagation speed (approximately 15,0000 m/s)f = frequency (Hz).According to Equation (15.3) the value of α is 6.7 × 10 –11 f 2 dB/kyd, but thisamounts to only about one-third of the absorption actually measured in distilledwater. The additional viscosity in pure water, besides that due to shear viscosity, isattributed to another type of viscosity called volume or bulk viscosity, which is theresult of a time lag for water molecules to “flow” in an expansive/compressive mannerin reacting to acoustic signals. This viscosity effect adds to the shear viscosity,so the absorption coefficient incorporating both types of viscosity becomesα = 16π 2 (μ3ρc 3 s + 3 )4 μ b f 2 (15.4)where μ b denotes the bulk viscosity. For water μ b = 2.81μ s .Below 100 kHz the predominant reason for absorption in seawater is due tothe phenomenon of ionic relaxation of the magnesium sulfate (MgSO 4 ) moleculespresent in the seawater. This association–dissociation process involves a relaxationtime, an interval during which the MgSO 4 ions in the seawater solution dissociateunder the impetus of the sound wave. Magnesium sulfate accounts for only 4.7%by weight of the total dissolved salts in seawater, but this particular salt was discoveredto be the dominant absorptive factor in seawater, rather than the principalconstituent NaCl (Leonard et al., 1949).The ionic relaxation mechanism causes a variation of the absorption with frequency,which is different from that in Equation (15.4). On the basis of more than

420 15. Underwater Acoustics30,000 measurements carried out at sea between 2 and 25 kHz out to ranges of24 kyd. Schulkin and Marsh (1962a,b) modified a frequency–dependency relationoriginally developed by Liebermann (1948). Their result waswhereα = A Sf T f 2fT 2 + f + B f 2dB/kyd (15.5)2 f TS = salinity in parts per thousand (ppt); A and B are constantA = 0.0186, a constantB = 0.0268, another constantf = frequency (kHz)f T = temperature-dependent relaxation frequency= 21.9 × 10 6 − 1.520 (T + 273)T = temperature in EC.But at frequencies

15.8 Underwater Refraction 421Figure 15.5. Attenuation and absorption processes in the sea.15.7 Spherical Spreading Combined with AbsorptionIt has been found in propagation measurements made at sea that spherical spreadingtogether with absorption provides a reasonable fit to the measured data under asurprisingly wide variety of conditions, even in situations where spherical spreadingis not supposed to occur (e.g. trapping conditions in sound channels). Whenan approximation of the transmission loss TL is sufficient, the universal sphericalspreading law plus the loss due to absorption will serve as a useful working rule:TL = 20 log r + αr × 10 −3 (15.6)In Equation (15.6) the first term denotes the spherical spreading and the secondterm the absorption effect; with 10 −3 inserted to handle the fact that r is given inyards and α is expressed in dB/kyd.15.8 Underwater RefractionRefraction is the major factor in altering simple spherical spreading of sound inthe ocean. As mentioned above, the factors affecting the sound propagation speedin seawater are temperature, depth, and salinity. Variations of salinity do occur,particularly at the mouths of large rivers where copious amounts of fresh waterintermingle with seawater, at the edges of large ocean currents such as the GulfStream, and in water near the surface, where rain, evaporation, and ice melting canimpose maximal effects. Variations in speed of sound with depth are quite small

422 15. Underwater AcousticsFigure 15.6. Diagram used for derivation of the relation between gradient G and the radiusof curvature R of a sound ray.(e.g. 0.1% over a 100 m depth). But variations in speed due to temperature changesare much greater and can fluctuate wildly, particularly near the surface.When sound varies with ocean depth, the path of a ray through the medium canbe determined by applying Snell’s law (sin φ/c = constant). Because the rays ofthe greatest interest in the study of oceans are nearly horizontal, it is more usualto restate Snell’s law as follows:cos θ= 1 (15.7)c c 0where θ is the angle of refractive deflection made with the horizontal at a depthwhere the speed of sound is c and c 0 is the speed at a depth (real or extrapolated)where the ray would become horizontal.A complex profile of the propagation velocity versus depth such as that inFigure 15.1 can be simplified for analytical purposes by separating the profile intosmall enough segments so that the velocity gradient may be considered constantover its length. Advantage is taken of the fact that the path of a sound ray througha stratum of water over which the sound speed gradient G is a constant constitutesan arc of a circle whose center lies at a depth where sound speed extrapolates tozero.In Figure 15.6 we consider a portion of the ray path with a radius of curvatureR. It follows that z = R(cos θ 1 – cos θ 2 ), and the gradient G isG = c 2 − c 1. (15.8)zWe can combine the last two Equations with Snell’s law of Equation (15.7) whichnow yieldsR =− c 0G =− cG cos θ . (15.9)

15.9 Mixed Layer 423When G is constant, and hence R is a constant, the path of the ray is therefore acircle. The center of curvature of the circle lies at the depth where θ = 90 ◦ whichcorresponds to c = 0. In the case illustrated in Figure 15.6, the speed gradient isnegative, so R is positive. Otherwise, if the speed gradient were to be positive, Rwould become negative and the path would refract upward.Once the radius of curvature of each segment of the path is established, theactual path can be traced through graphic or computerized means. Let the initialangle of deflection of the ray be designated θ 0 , and use be made of the geometryof Figure 15.6 along with Equation (15.9). The changes in range r and depth zare given byr = 1 Gz = 1 Gc 0(sin θ 1 − sin θ 2 )cos θ 0(15.10)c 1(sin θ 2 − sin θ 1 )cos θ 1(15.11)Applying the small angle approximations (cos θ ≈ 1, sin θ ≈ 0, etc.) and eliminatingθ from the last two Equations yields a convenient approximate relationshipbetween the range and depth increments along a ray for ∗θ∗ less than 20 ◦ and forr ≪|c 1 /G|:z = tan θ 0 r − G2c 0(r 2 ). (15.12)15.9 Mixed LayerWave action can cause the water to mix in the surface layer, thus creating what iscalled a mixed layer. The positive sound-speed gradient in this layer entraps soundnear the surface. After it is developed, the mixed layer tends to exist until the sunheats up the upper portion, decreasing the gradient. This heating effect engendersa negative gradient that leads to a downward refraction and the loss of sound fromthe layer. Because this occurs later during daytime, this effect became known asthe afternoon effect. During the night, surface cooling and wave mixing permitthis isothermal layer to reestablish itself.A computer-produced ray diagram (the discontinuous form of the rays are dueto the manner the velocity profile was subdivided in the computer program) fora source in a fairly typical mixed layer is shown in Figure 15.7. The conditionsfor which the diagram was plotted, the ray leaving a source at 1.76 ◦ becomeshorizontal at the base of the layer, Rays leaving the source at smaller angles stayentrapped in the layer; and rays that leave the source at greater angles are sent intothe lower depths of the sea. A shadow zone is created beneath the mixed layer ata range beyond the direct and near sound field. This zone is isonified by scattersound from the sea surface and by diffusion of sound out of the channel, causedby the nature of the lower boundary.

424 15. Underwater AcousticsFigure 15.7. A computer-produced ray diagram for sound transmission from a 50-ft(15.24-m) source in a 200-ft (61-m) mixed layer.15.10 Deep Sound ChannelIn Section 15.4 reference was made to a region constituting the deep sound channelwhere the sound propagation speed reaches a minimum in the ocean depths. All raysoriginating near the axis of this channel and making small angles with the horizontalwill return to the axis without reaching the ocean surface or bottom, thus remainingentrapped within that channel. Absorption of low frequencies in seawater tends tobe quite small, so the low-frequency components of explosive charges detonatedin this channel can travel tremendous distances, and they have been detected morethan 3000 km away. The reception of these explosive signals by two or morewell-separated hydrophone arrays can permit an accurate determination of theexplosion’s location by triangulation. Passive sonar is currently being used in deepsound channels to monitor activities in deep ocean.15.11 Sonar Transducers and Their PropertiesUnderwater sound equipment are designed to detect and analyze underwater sound.They generally consist of a hydrophone array that consists of transducers thatconvert acoustic energy into electrical energy, and vice versa, and a signal processingsystem to analyze and display the signals aurally or visually. A transducerthat accepts sound and converts it into electricity is called a receiver orhydrophone. A transducer that converts electrical energy into sound is called a projector.Some sonar systems use the same transducer to generate and receive sound.There are two principal types of transducers according to the special propertiesof their activation materials. One type of transducer depends on piezoelectricity

15.11 Sonar Transducers and Their Properties 425and its variant, electrostriction, and the other type functions on the principle ofmagnetostriction. Certain crystalline substances, such as quartz, ammonium dihydrogenphosphate (ADP), and Rochelle salt, generate a charge between certaincrystal surfaces when they are subject to pressure. Conversely, when a voltage isapplied to these substances, this causes stresses to occur in them. Such materialsare said to be piezoelectric. Electrostrictive materials are polycrystalline ceramicsthat produce the same effect and these have to be properly polarized in a strongelectrostatic field. Barium titanate and lead zirconate titanate are examples of suchmaterials.A magnetostrictive material such as nickel exhibits the same effect as piezoelectricitybut it does so under the influence of a magnetic field rather than appliedstresses. It changes its dimensions when subjected to a magnetic field and, conversely,changes the magnetic field within and around it when it becomes stressed.In other words, when a properly designed nickel element is subjected to an oscillatingmagnetic field, a mechanical oscillation is produced which generates acousticwaves in water. Magnetostrictive materials are also polarized in order to avoidfrequency doubling and to achieve a higher efficiency.Piezoelectric and magnetostrictive types of transducers are more suitable thanother kinds of transducers for use underwater due to better impedance matchingwith water. Because they are relatively inexpensive and can be readily fashionedinto the desired shapes, ceramic materials are finding increasing applicationsas underwater devices. Other types of units now include the thin-film transducers(Hennin and Lewiner, 1978) and fiberoptic hydrophones (Hucaro et al.,1977).ArraysWhile single piezoelectric or magnetostrictive elements are normally used in hydrophonesfor research or measurement purposes, much of the applications ofhydrophones entail hydrophone arrays that use a number of properly spaced elements.The following reasons exist for the use of arrays:1. The array is more sensitive, as a number of elements will generate more voltage,if connected in series, or more current, if connected in parallel.2. The array provides directivity that enables it to discriminate between soundscoming from different directions.3. An improved signal-to-noise ratio SNR over that of a single hydrophone isprovided, because the array discriminates against isotropic or quasi-isotropicnoise to favor a signal arriving from a direction that the array is pointingto.Because of the above advantages, most practical applications of underwater soundmake of arrays. Moreover, the first and second benefits listed above also apply toprojectors as well as to hydrophones.

426 15. Underwater AcousticsFigure 15.8. Cylindrical array and plane array of transducers.Examples of a cylindrical and a plane array are shown in Figure 15.8. Sphericalarrays have also been constructed for installation on submarines.Array GainThe improvement in the signal-to-noise ratio SNR, which results from the use ofarraying of hydrophones is measured by a parameter called array gain, which isdefined by()(S/N) arrayAG = 10 log(15.13)(S/N) single elementThe numerator of Equation (15.13) denotes the SNR at the array terminals and thedenominator represents the SNR of a single element of the array. It is assumed thatall the elements in the array are identical.Transducer ResponseThe effectiveness of a hydrophone in converting sound into an electric signalis called the response of the hydrophone. It relates the generated voltage to theacoustic pressure of the sound field. The receiving response of a hydrophone isdefined as the voltage produced across the terminals by a plane wave of unitacoustic pressure (the value before the introduction of the hydrophone into thesound field). The receiving response is usually expressed as the open-circuit responsethat is obtained when the hydrophone connects to an infinite impedance.The customary unit of the receiving response is the number of decibels relativeto 1 volt produced by an acoustic pressure of 1 μPa and it is written as decibelsre1V/μPa.

Example Problem 115.12 The Sonar Equations 427Find the voltage across the hydrophone terminals exposed to an acoustic rmspressure of 1 μPa if the receiving response is –80 decibels re 1 V.SolutionWe use the relationshipand since SPL =−80 dB,SPL = 20 log(p/1 μPa)p/1 μPa = antilog(−80/20),yielding p = 0.0001 μPa. For the 1 μPa sound field, the voltage is 0.0001 V acrossthe terminals.For a projector, the transmitting current response is the acoustic pressure producedat a point 1 m from the projector in the direction of the axis of its beampattern by a unit current into the projector. This means that the transmitting responseis stated in terms of the number of decibels relative to 1 μPa as measuredat the reference distance, produced by1Aofcurrent into the electric terminals ofthe projector. While the transmitting response are usually referred to a referencedistance of 1 m from the source, a correction of 0.78 dB must be added in order toconvert the transmission response expressed in terms of a reference level of 1 yardinstead of 1 m.Example Problem 2Predict the rms pressure at 1 m when the projector is driven with a current of1 rms ampere, if the response is rated at 100 dB re 1 μPa/A (referred to 1 m).Express the response in terms of a reference distance of 1 yard instead of 1 m.SolutionThe corresponding rms acoustic pressure for 100 dB is 10 5 μPa, which is producedby 1 rms ampere current. The corresponding transmitting response for 1 yard is100 + 0.78 = 100.78 dB.15.12 The Sonar EquationsThe purpose of the sonar equations, originally formulated during World War IIand derivative of similar considerations in radar, is twofold: (a) prediction of thecapabilities of existing sonar equipment with respect to detection probability orsearch rate and (b) design of new equipment to meet preestablished range ofdetection or actuation.

428 15. Underwater AcousticsIn our mathematical exposition of the sonar principle, we subscribe to the assumptionthat target strength TS is a function of the source and echo levels, respectively,as well as the transmission loss TL that occurs in the echo-ranging process.The assigned function of the sonar may be the detection of an underwater target, orit can be the homing of an acoustic torpedo at the instant when it begins to ascertainits target. Of a total signal energy received by a sensor, a portion may be desired,and is considered to be the signal. The balance of the acoustic energy is undesiredand is termed the background. The background consists of noise, the basicallysteady state portion that is not attributable to the echo-ranging, and reverberationwhich represents the slowly decaying portion of the background caused by thereturn of the original acoustic output from scatterers dispersed in the sea. In thedesign of a sonar system, it is the objective to find ways of increasing the overallresponse of the system to the signal and to decrease the response of the system tothe background, in other words, to increase the signal-to-background ratio.A sonar system serves a practical purpose such as detection, classification (establishingthe character of the target), torpedo hunting, communication, or fishfinding. In each of these tasks there will be a specific signal-to-background ratioand a level of performance in successfully detecting targets with a minimum of“false alarms” that erroneously indicate the presence of a target when no target ispresent. If the signal increases sufficiently to equal the level of the background,the desired purpose will be achieved when the signal level equals the level of thebackground, which just masks it, i.e.Signal level = Background masking level (15.14)Masking does not mean that all of the background interferes with the signal. Onlythe portion that lies in the frequency band of the signal will cause masking, just asin psychoacoustics, where a broadband noise masks out a pure tone or narrow-bandsignal presented to the human ear.The equality of (15.14) constitutes the one instant of the time when a targetapproaches or recedes from a receiver. At short ranges, the signal level from atarget should handily exceed the background masking level, while the reverse willoccur at long ranges. But it is at the instant of (15.14) when the sonar system justbegins to perform its function, which is of greatest interest to the sonar designer.The source level, SL, which is defined in terms of intensity at 1 m (formerly1 yard), was derived from physical concepts in order to express separately theeffects on the signal strength of the echo, namely, (i) the size, shape, and orientationof the target; (ii) the intensity of the source; and (iii) the range of the target. Atlong ranges, only the transmission loss TL depends on the range. At shorter ranges,target strength TS depends on the range as well as the size, shape, and orientationof the target. If the source is quite close to the target, different parts of the targetare struck by sound of different intensities, or if the receiver is so close that thespreading of the sound reflected from the target to it is not the same as the spreadingfrom a point source, the target strength term will depend on the range.

Active and Passive Equations15.12 The Sonar Equations 429The sonar parameters are determined by the equipment, the medium, and the target.We denote the following parameters, which are stated in terms of dB relative tothe standard reference intensity of a 1-μPa plane wave:Equipment parametersSL: projector source levelNL: self-noise level (also called electronic noise)DI: directivity indexDT: detection thresholdMedium parametersTL: transmission lossRL: reverberation levelNL: ambient noise levelTarget parametersTS: target strengthSL: target source level.Note that two pairs of parameters are given the same symbol because they areidentical. This set of parameters are not necessarily all inclusive, nor is this setunique, for other parameters such as sound velocity or backscattering cross sectioncould be considered. The parameters chosen above are conventional ones appliedin underwater technology.In order to understand the significance of the above listed quantities, considerFigure 15.9, which illustrates a schematic of an echo-ranging process. A transduceroperating as both sound source and receiver produces a source level of SL decibelsat a unit distance (generally 1mworldwide and 1 yard in the English system)on its axis. Let the axis of the sound source be properly aimed toward the target;the radiated sound will reach the target with a transmission loss, and the levelof the signal reaching the target will be SL − TL. On reflection or scatteringfrom the target with target strength TS, the reflected or the backscattered levelwill be SL − TL + TS (at a distance of 1 m from the acoustic center of thetarget in the direction returning to the source). This reflected signal also undergoesattenuation by the transmission loss TL as a result of its travel to the source. Thelevel of the echo reaching the source thus becomes SL − 2TL + TS. Now ifwe consider the background noise and assume it to be isotropic noise rather thanreverberation, the background level will simply be noise level. But this level willbe lessened by the directivity index (DI) of the transducer serving as a receiver,so the relative noise power at the transducer interface is NL − DI. Because theaxis of the transducer points in the direction from which the echo is traveling, therelative echo power is unaffected by the transducer directivity. At the transducerterminals, the echo-to-noise ratio becomesSL − 2TL+ TS − (NL − DL)

430 15. Underwater AcousticsFigure 15.9. Schematic of echo ranging process.Now let the sonar act as a detector, i.e., it is to give an indication on some kindof display that an echoing target is present. When the input signal-to-noise ratioexceeds a specific detection threshold DT, thus meeting preset probability criteria,a relay can be activated to indicate on a display that a target is present. Otherwise,when the input signal-to-noise ratio falls below the detection threshold DT, theindication will be that a target is absent. But when the target is just being detected,the signal-to-noise ratio equals the detection threshold DT, i.e.,SL − 2TL+ TS − (NL − DI) = DT (15.15)Equation (15.15) characterizes the active-sonar equation as an equality in termsof the detection threshold. In recognizing that only a portion of the noise powerlying above the DT masks the echo, we could rearrange (15.15) as follows:SL − 2TL + TS = NL − DI + DT (15.16)Equation (15.16) places the echo level effects on the left-hand side and thosepertaining to the noise-masking background level on the other side. Equation(15.16) constitutes the active-sonar equation for the monostatic case, one in whichthe source and the receiving hydrophone are coincident and the echo from the targettravels back to the source. In some sonar applications, a bistatic arrangement is

15.13 Noise, Echo, and Reverberation Levels 431used—i.e., the source and the receiver are separate, and the two transmissionlosses to and from the target are not generally equal. In some sonars it is virtuallyimpossible to resolve the receiving directivity index DI and detection thresholdDT, so it becomes legitimate to combine these two terms as DI − DT into a singleparameter describing the increase in signal-to-background ratio produced by theentire receiving system which includes the transducer, processing electronics, andthe display.What happens when the background consists of reverberation rather than noise?Instead of DI, which was defined in terms of an isotropic background and is nowinappropriate, the (NL − DI) term in Equation (15.16) is replaced by an equivalentplane-wave reverberation level RL observed at the hydrophone terminals.Equation (15.16) then becomesSL − 2TL + TS = RL + DT (15.17)The parameter DT will possess a value for reverberation that is different from theDT for noise.In the passive or “listening” situation, the target itself produces the signal thatis detected, and one-way transmission rather than two-way transmission is entailed.Target strength TS now becomes irrelevant and the passive sonar equationbecomesSL − TL = NL − DI + DT (15.18)Table 15.3 lists the parameters and their definitions in brief, while Table 15.4provides the terminology of commonly used terms for describing sonar parameters.A number of the parameters in Table 15.3, namely, SL, TL, TS, and the scatteringstrength (which determines RL) use 1 yard as the reference distance. To convertto 1-m reference, these quantities should be reduced by 0.78 = 20 log [1 m ×(39.37 in./m) × (1 yard/36 in.)]. The attenuation coefficient commonly expressedin dB/kiloyard should be multiplied by 1.094 to convert the coefficient to dB/km. Inthe United States, it is generally more convenient to find the range first in kiloyardsand then to divide by the factor 1.094 in order to express the range in kilometers.15.13 Noise, Echo, and Reverberation LevelsThe above sonar equations constitute a statement of equality between the signal,which is the desired portion (the echo or noise from the target) of the acoustic field,and the undesired portion, i.e., the background of noise and reverberation. Thisequality holds true at only one range; at all the other ranges, the equality will nolonger exist. This fact is demonstrated in Figure 15.10 in which the curves of theecho level, noise-masking level, and the reverberation-masking level are displayedas functions of range. The echo and reverberation levels drop off with increasingrange, but the noise remains fairly constant. The echo level curve falls off morerapidly with range than does the reverberation-masking curve, and intersects it atthe reverberation-limited range r r . This curve will also meet the noise-masking

432 15. Underwater AcousticsTable 15.3. Sonar Parameters.Parametric symbol Reference DefinitionSource level: SLTransmission loss:TLTarget strength:TSNoise level: NLReceiving: DIdirectivityindexReverberation: RLlevelDirection: DTThreshold1 yard from sourceon its acoustic axis1 yard from sourceand at target orreceiver1 yard from acousticcenter of targetAt hydrophonelocationAt hyrdrophoneterminalsAt hydrophoneterminalsAt hydrophoneterminals10 log (intensity of source/1 μPa)()signal intensity at 1 yard10 logsignal intensity at target or receiver( )echo intensity at 1 yard from target10 logincident intensity( )noise intensity10 log1 μPa⎛⎞noise power generated by an equivalent10 log ⎜isotropic hydrophone⎟⎝ noise power generated by actual ⎠hydrophone[ ](reveberation power at hydrophone terminals)10 log1 μPa⎡ ⎛⎞ ⎤signal power to just at perform⎝ ⎠a certain function10 log ⎢⎣ noise power at hydrophone terminals⎥⎦Note that some of the parameters, viz. SL, TL, TS (and the scattering strength) in Table 15.1 use 1 yardas the reference distance. To obtain their values based on the reference distance of 1 m, 0.78 dB mustbe subtracted from the values of these parameters.level curve at range r n . If the reverberation is high, the echo will be of lesservalue and the range may be considered to be reverberation-limited. But if thenoise-masking level occurs at the level represented by the dashed line in Figure15.10, the echoes will then fade away into a background of noise rather thanreverberation. The new noise-limited r n ′ will be less than the reverberation-limitedTable 15.4. Sonic Parameters: Terminology of Various Combinations.Nomenclature Parameter CommentsEcho level SL – 2TL + TS Intensity of the echo measured in thewater at the hydrophoneNoise-masking NL − DI + DTDetermines minimumlevelReverberationmaskingRL + DTDetectable echo levellevelEcho excess SL − 2TL + TS − (NL − DI + DT) Detection just occurs, when echo excessjust equals zeroPerformancefigureSL − (NL − DI)Difference between the SL and the NLmeasured at the hydrophoneFigure of merit SL − (NL − DI + DT) Equals the maximum allowable two-wayTL for TS = 0 for active sonar, orone-way TL in passive sonars

15.13 Noise, Echo, and Reverberation Levels 433Figure 15.10. Effect of range on echo level, noise masking, and reverberation masking.range r n , and the range will hence be noise-limited. Both ranges can be establishedby the use of the appropriate sonar equation.It is necessary for a sonar designer or sonar operator to know whether a sonar willbe noise-limited or reverberation-limited. In general, the curves for echo and reverberationwill not occur as straight lines because sound propagation and distributionof echo-yielding scatterers add to the complexity of the situation. In the design ofa sonar system for specific purposes, these curves should be created from the bestinformation available for the conditions that are most likely to be encountered.A convenient graphical method of solving the sonar equations for passive sonarsis the SORAP, the acronym for “sonar overlay range prediction.” Figure 15.11Figure 15.11. SORAP (Sonar Overlay RAnge Prediction) graphical method for solvingthe passive sonar equation.

434 15. Underwater Acousticsconsists of two plots that are overlaid on each other. The solid lines constitute anoverlay of a plot of SL versus frequency for a specific passive target or class oftargets. The underlay denoted by dashed lines is a plot of the sum of the parameters(TL + NL − DI + DT) for a specific passive sonar at a number of different ranges.The range and frequency at which the target can be discerned can be determinedby inspection of the plots. For example, the target will be first detected at a range of10 miles according to the line component at frequency f 1 . But suppose it is requiredthat three spectral lines must appear on the display before a detection is called; therange would be reduced to 4 miles and the lines at frequencies f 1 , f 2 , and f 3 wouldbe displayed. This procedure helps to distinguish the target parameter SL fromthe equipment and medium parameters at the location where it is used, while itaccommodates a wide range of frequencies. Targets can therefore be compared forthe same locations or locations can be compared for the same targets, and so on.15.14 Transient Form of Sonar Equations and PulsesSo far, the sonar equations have been expressed in terms of the average acousticpower per unit area or intensity of the sound radiated by the source or received fromthe target. But the time interval implied by the terminology “average” can yieldunreliable results in situations where short transient sources exist or wheneversevere distortion is incurred in sound propagation in the medium during the courseof scattering from the target.We can adopt a more general approach by writing the sonar equations in termsof energy flux density, which is defined at the acoustic energy per unit area ofthe wavefront. Consider a plane acoustic wave that has a time-dependent pressurep(t). The energy flux density of the wave is given byE = 1 ∫ ∞p 2 (t) dt. (15.19)ρ c 0Because the intensity is the rms pressure of the wave divided by the acousticimpedance ρc, averaged over a time interval T , i.e.,I = 1 ∫ Tp 2 (t)T 0 ρc dtit follows thatI = E T . (15.20)The time interval T represents the duration over which the energy flux densityof the sound wave is to be averaged to yield the intensity. In the case of long-pulsedactive sensors, this time interval equals the duration of the emitted pulse and verynearly equals the duration of the echo. But for short transient sonars, the intervalT becomes rather ambiguous, and the duration of the echo can be considerablydifferent from the duration of the transient emitted by the source. Urick (1962,

15.14 Transient Form of Sonar Equations and Pulses 435Figure 15.12. Time stretching: equivalent source level (SL) in short-pulse sonar.1983) demonstrated that under these conditions that the intensity form of the sonarequations can be applied, providing the source level is defined asSL = 10 log E − 10 log τ e (15.21)where E is the energy flux density of the source referred to 1 yard and measuredin units of the energy flux density of a 1-μPa plane wave taken over an intervalof 1 s, and τ e represents the duration of the echoes expressed in seconds for anactive sonar. As an example, E can be determined for explosives by measurementsfor a given charge weight, the depth where the explosion occurs, and the type ofexplosive.Consider a pulsed sonar emitting a rectangular pulse of constant source level SL ′over a time interval τ 0 . The energy density of a pulse equals the average intensityH duration,10 log E = SL ′ + 10 log τ 0 . (15.22)Combining Equations (15.21) and (15.22), the effective source level (SL) to beused accordingly in sonar equations isSL = SL ′ + 10 log τ 0(15.23)τ ewhere τ 0 denotes the duration of the emitted pulse of source level, SL ′ , and τ e theduration of the echo. For long-pulsed sonars, τ 0 is equal to τ e , and therefore SL =SL ′ . In the case of short-pulsed sonars, τ 0 >τ e , and thus the effective SL is lessthan the actual SL ′ by the amount 10 log (τ 0 /τ e ). This effect of time stretching isdepicted in Figure 15.12. A short pulse duration τ 0 at source level SL ′ is replacedin a sonar calculation by a longer pulse of duration τ e at a lower SL. The energyflux density source levels must be the same for these two source levels, i.e.,SL + 10 log τ e = SL ′ + 10 log τ 0which is merely a rearrangement of Equation (15.23). A pulse emitted by thesource stretches out in time and becomes reduced in level by the multipath effectof propagation and by the mechanism of target reflection. The appropriate values

436 15. Underwater AcousticsFigure 15.13. Distortion of the pressure resulting from an explosive pulse upon arrival atan extended target. The return echo in the vicinity of the source is also shown here.of the other sonar parameters in the sonar equations, such as target strength (TS)and transmission loss (TL), are those applying to long-pulse of CW conditions, inwhich the effects of multipaths in the medium and target reflection are summedup and accounted for.In active short-pulse sonars, the echo duration τ e becomes in itself an importantparameter. In Figure 15.13, a pulse is shown as a short exponential transient at thesource, as a distorted pulse at the target, and as an echo reflected back to near thesource. A shock wave pulse, for example, from an underwater explosion occurs asan exponential pulse. It may have an initial duration of 1/10 of a millisecond, andit can become distorted into an echo that is 1000 times as long.The duration of the echo can be subdivided into three components: τ 0 , which isthe duration of the emitted pulse near the source; τ m , the additional time needed fortwo-way propagation in the underwater; and τ r , the additional duration assessedby the extension in range of the target. Henceτ e = τ 0 + τ m + τ r . (15.24)Typical values of these three components of the echo duration for different circumstancesare listed in Table 15.5.Table 15.5. Examples of Echo Duration.ComponentRepresentative values (ms)Duration of the emitted pulse at short ranges Explosives: 0.1Sonar: 100Duration produced by multiple paths Deep water: 1Shallow water: 100Duration produced by a submarine target Beam aspect: 10Bow-stern aspect: 100

Example Problem 315.16 Shortcomings of the Sonar Equations 437Estimate the echo time duration of an explosive echo from a beam aspect submarinepatrolling shallow water.SolutionFrom Table 15.5 and Equation (15.24), the time duration is 0.1 + 100 + 10 =110.2 ms.15.15 Overview of the Sonar EquationsA summarization of the sonar equations may be listed as follows:Active sonars (monostatic)Noise backgroundReverberation backgroundPassive sonarsSL − 2TL + TS = NL − DI + DT. (15.25)SL − 2TL + TS = RL + DT R . (15.26)SL − TL = NL − DI + DT N . (15.27)The detection threshold (DT) differs quantitatively for reverberation and for noiseand so it carries subscripts to denote the difference.15.16 Shortcomings of the Sonar EquationsThe sonar equations expressed in terms of intensities may not always be completefor certain types of sonars. For example, the short-pulse sonars need theaddition of the echo duration to account for time stretching caused by multipathpropagation. It must also be realized that the sea is an ever-stirring medium thatcontains inhomogeneities of different sorts, with irregular boundaries with the topmostboundary on the move. Multipath propagation tends to predominate, since somany of the sonar parameters fluctuate erratically with time. Other irregularitiesmay occur because of internal changes in measurement equipment and possiblereconfiguration of the platform on which the equipment is mounted. In essence,the “solutions” proffered by the sonar equations really constitute a “best estimate”time average of what is really a stochastic problem. Thus, precise calculations to,say the nearest tenth of a decibel are exercises in futility, and the solutions mustbe considered “best guesses” or “most probable” values.

438 15. Underwater Acoustics15.17 Theoretical Target Strength of a SphereConsider a sphere of radius r as the subject target. As we shall see, the effect of thepulse length of the probe signals constitutes a very important factor. No accountis taken of the wave nature of sound, i.e., the effects of interference, diffraction,and phase differences are ignored here. Let I 0 denote the intensity of the incidentsound wave striking the target and I r the intensity of the reflected sound signalmeasured at some particular point. With all the other factors remaining constant,we can writeI0 ∝ Ir .The intensity I r of the reflection is a function of the target orientation and the locationof its measurement. Because I r is usually measured at the probe signal point,mathematical treatment becomes considerably simplified. The inverse square law,which holds for large, not small, distances is expressed asI r = KI 0r 2 (15.28)where K is a constant dependent upon the size, shape, and orientation of the target.Equation (15.28) does not apply to explosive sounds. For the incident signalI 0 = F r 2 (15.29)where F is the intensity of the projected sound1mawayfrom the source. Here itis tacitly assumed thatr ≫ Dwhere D is the order of magnitude of the size of the source. Combining Equations(15.28) and (15.29)I r = KFr . (15.30)4For an ideal medium, according to Equation (15.30), the intensity of an echois inversely proportional to the fourth power of the range, provided the echo ismeasured in the source location and the range is much larger than the dimensionsof either the target or the source.Reexpressing Equation (15.30) in logarithms, we obtain10 log I r = 10 log K + 10 log F − 40 log r. (15.31)In the course of a signal traveling to a target and the echo reflecting back to thesignal source, some attenuation in the signal intensity must occur. This drop inintensity in each direction has been defined as the transmission loss TL that shouldequal 20 log r in the idealized case of Equation (15.31). The total TL is therefore2TL, i.e. = 40 log r, which represents the attenuation of the signal traveling to andfrom the target. It can be recalled that losses do not occur solely owing to the effectof the inverse square law; they are also attributable to absorption and scattering

15.17 Theoretical Target Strength of a Sphere 439in sea water, bending by temperature gradients (with consequent focusing andspreading out). Therefore, the actual transmission loss does not equal 20 log ralone, but includes other effects of attenuation. The function 2TL represents amore general situation than 40 log r in Equation (15.31), which is then recast as10 log I r = 10 log K + 10 log F − 2TL. (15.32)As mentioned earlier, because of the great variance of oceanographic conditions,the quantity 2TL can be established only through careful measurements. Settingtarget strength TSand the echoand source levelTS = 10 log K,E = 10 log I r ,SL = 10 log F,we obtain the first sonar equation of Table 15.4, which constitutes the fundamentaldefinition of signal strength:TS = E − SL + 2TL. (15.33)As it entails only directly measurable quantities Equation (15.33) is particularlyuseful in the computation of TS from data measured at sea.A sphere presents a perfectly symmetric target. The echoes it returns to a soundsource are completely independent of its own orientation, and it is for this reasonthat spheres make convenient experimental targets in echo-ranging measurements.In a simple derivation, we consider a plane wave of intensity I o striking a sphereof radius a and cross sectional area πa 2 . The total sound energy intercepted bythe sphere is thus πa 2 I 0 in the ideal case of perfect reflection. Now let us assumeuniform reflectivity in all directions. At a distance r from the sphere’s center theacoustic energy will be spread uniformly over the surface of a sphere of radius r orover the surface area 4πr 2 . Because the intensity I r of the reflected sound equalsthe total energy πa 2 I 0 reflected by the target sphere per unit time divided by 4πr 2over which it is distributed, then at a distance r from the sphere’s centerI r = πa2 I 04πr 2= a24r 2 I 0.But from Equation (15.28), I r = KI 0 /r 2 , where r represents the distance from thetarget to the measurement point,and soK = a24TS = 10 log K = 20 log(a/2).

440 15. Underwater AcousticsReferencesBezdek, H. H. 1973. Pressure dependence of sound attenuation in the Pacific Ocean.Journal of the Acoustical Society of America 53: 782.Browning, D., Fechner, M., and Mellon, R. 1981. Regional dependence of very lowfrequencysound attenuation in the deep sound channel. Naval Underwater SystemsCenter Technical Document 6561.Colladen, J.-D. and Sturm, J. C. F. 1827. Memoir on the compression of liquids. Ann.Chim. Phys. 36: 113–159, 225–257.Del Grosso, V. A. 1952. The velocity of sound in sea water at zero depth. U.S. NavalResearch Laboratory Report 4002.Del Grosso, V. A. 1974. New equations for the speed of sound in natural waters withcomparisons to other equations. Journal of the Acoustical Society of America 56: 1084.Hennin, C., and J. Lewiner. 1978. Condenser electret hydrophones. Journal of the AcousticalSociety of America 63: 279.Hucaro, J.A. et al. 1977. Fiber-optic hydrophone. Journal of the Acoustical Society ofAmerica 62: 1302.Leonard, R. W., Combs, P. C., and Skidmore, I. R. 1949. Attenuation of sound in syntheticsea water. Journal of the Acoustical Society of America 21: 63.Leroy, C.C. 1969. Development of simple equations for accurate and more realistic calculationof the speed of sound in sea water. Journal of the Acoustical Society of America46: 216.Liebermann, L. N. 1948. Origin of sound absorption in water and in sea water. Journal ofthe Acoustical Society of America 20: 868.Liebermann, L. N. 1949. Sound propagation in chemically active media. Physics Review76: 1520.Mackenzie, K. V. 1981. Nine-term equation for sound speed in the oceans. Journal of theAcoustical Society of America 70: 807.Medwin, Herman. 1975. Speed of sound in water for realistic parameters. Journal of theAcoustical Society of America 58: 1318National Defense Research Committee. 1946. Application of oceanography to subsurfacewarfare. Div. 6 Summary Tech. Rep., vol. 6A: Figures 17, 32.Schulkin, M., and Marsh, H. W. 1962a. Absorption of sound in sea water. Journal of theBritish IRE 25: 293.Schulkin, M., and Marsh, H. W. 1962b. Sound Absorption in sea water. Journal of theAcoustical Society of America 34: 864.U.S. Navy. 1975. An Interim Report on the Sound Velocity Distribution in the NorthAtlantic Ocean. U.S. Navy Oceangoing Office Technical Report 171.Urick, Robert J. 1962. Generalized form of the sonar equations. Journal of the AcousticalSociety of America 34: 547.Urick, Robert J. 1983. Principles of Underwater Sound, 3rd ed. New York: McGraw-Hill.(A classic text in the field by one of the leading researchers.)Weissler, A., and Del Grosso, V. A. 1951. The velocity of sound in sea water. Journal ofthe Acoustical Society of America 23: 219.Wilson, W. A. 1960. Speed of sound in sea water as a function of temperature, pressure andsalinity. Journal of the Acoustical Society of America 32: 219. Also see extensions andrevisions in Journal of the Acoustical Society of America (1960), 32: 1357 and (1962),34: 866.

Problems for Chapter 15 441Wilson, O. B., and Leonard, O.B. 1954. Measurements of sound absorption in aqueoussalt solutions by a resonator method. Journal of the Acoustical Society of America 26:223.Wood, A. B. 1941. A Textbook of Sound. New York: Macmillan Company; p. 261.Yeager, E., Fisher, F. H., Miceli, J., and Bressel, R. 1973. Origin of the low-frequencysound absorption in sea water. Journal of the Acoustical Society of America 53: 1705.Problems for Chapter 151. Determine the speed of sound in seawater at 20 m depth and at 20 ◦ C and 34 pptusing the Leroy formula.2. Using the same conditions as in Problem 1, find the speed of sound on thebasis of the Medwin formulation.3. Do Problem 1 using the MacKenzie empirical expression.4. Compare the velocities of sound in seawater at a depth of 5000 ft for thelatitude of 61 ◦ N and the latitude of 19 ◦ N. Convert the depth to meters andspeed of sound to km/s.5. If the sound intensity 1 m from a source is 10 W/m 2 in the sea, find thetransmission loss (TL) and the intensity 50 m from the source due strictlyto spherical spreading. What other factors usually affect the transmissionloss?6. If sound is transmitted in a tube of water, what would be the transmission loss?In real life, would there be transmission loss and why?7. Predict the value of the attenuation coefficient for seawater in which salinityS = 40 ppt, frequency of the signal is 50 kHz, and the temperatureis 18 ◦ C.8. Compare the spherical spreading over r = 100 yards with and withoutabsorption by computing the transmission loss under the conditions ofProblem 7.9. Given a change of seawater sound velocity of 20 m/s for a depth of 150 m,what is the gradient and what would be the radius of curvature in terms of c 0 ?10. If an element of an array has a signal to noise ratio of 40 dB, what would bethe array gain of 25 similar elements in such an array?11. A transducer is rated at a receiving response of –95 dB re 1 V. If an acousticpressure of 3 μPa is applied, what will be the resulting voltage at the terminals?12. A projector is driven with a 1.8 rms ampere current. Its response is rated at120 dB re 1 μPa/A. Find the response in dB.

16Ultrasonics16.1 IntroductionAs a subcategory of acoustics, ultrasonics deals with acoustics beyond the audiofrequency limit of 20 kHz. Although ultrasonics has been employed for most ofthe twentieth century, the tempo of new and improved applications has reachedvirtually explosive proportions only in the past few years, particularly in medicaldiagnostics and therapeutics.Applications of ultrasonics fall into two categories—low intensity and high intensity.Low intensity applications carry the purpose of simply transmitting energythrough a medium in order to obtain information about the medium or to conveyinformation through the medium. Nondestructive testing, medical diagnostics,acoustical holography, and measurements of elastic properties of materials fallinto this category. Even marine applications are included in this category, despitethe large energy input into operating sonar submarine detectors, depth sounders,echo ranging processors, and communication devices. 1High-intensity applications deliberately affect the propagation medium or itscontents. Uses of high intensities include medical therapy and surgery, atomizationof liquids, machining of materials, cleaning, welding of plastics and metals,disruption of biological cells, and homogenization of materials.Human beings are not alone in the use of ultrasonics, and even in this respect theyhave been preceded by thousands of years by other species in the animal kingdom.Certain animals are capable of generating and detecting ultrasonic signals in orderto locate and identify food, navigate their way through their environment, anddetect danger. In fact, the study of these animals have helped and is still helpingscientists to develop and improve techniques in the application of ultrasonic energy.Bats are known to emit pulses in the 30–120 kHz range, and it has been hypothesizedthat the bats judge range by sensing the time delay between an emittedpulse and the echo. Small bats can fly at full speed through barriers constructedof 0.4 mm vertical wiring spaced only one wingspan apart. They are capable of1 Emsinger (1988) suggested that a third category not based on intensity be designated to specificallycover underwater applications.443

444 16. Ultrasonicscatching small insects at the rate of one every 10 s for as long as half an hour. Theirability to discriminate between objects such as food and raindrops or foliage canbe described as being nothing less than phenomenal; and yet when a large numberof bats fly in close proximity to each other in a potentially confusing backgroundof ultrasonic noise, they continue to locate prey and avoid collisions with eachother.The Noctillio bat of Trinidad catches small fish by dipping its feet below thesurface of the water, after emitting a series of repetitive pulses. It has been conjecturedthat the characteristic ripples created by the fish are being detected by the batrather than echoes from an object beneath the water surface. This is particularlyremarkable in view of the fact that the sound must penetrate a barrier with a veryhigh-reflection coefficient. As with bats, the echolocation of porpoises appears tobe unaffected by the presence of interfering noises or jamming.Moths, a prime target of bats, use ultrasonics for self-defense. When a moth detectsa sonic pulse from a bat, it immediately takes evasive action through zigzaggingin its flight and executing power dives. Moths can detect bats as far away as13 m, and their ears can detect the cries of approaching bats at this distance, butwhen the bats are moving away their ears stop registering at about this distance.It was also observed by Roeder and Treat that when a bat makes a straight-onapproach, it was observed to emit an uneven, sporadic signal; this was interpretedas an indication that the bat was counter-maneuvering by varying the intensity ofits sonar pings.Cetaceans constitute a group of sea mammals, which includes whales, dolphins,and porpoises. They are extremely intelligent as well as beautiful creatures, andare of the greatest interest to acousticians. Porpoise sounds have been describedas whistling, barking, rasping, repetitive clicks or pulses, mewing. The cetaceansemit signals for the purpose of echolocation and communication with each other.Signals as high as 170 kHz have been observed in the clicks of porpoises, whichvary in repetition rate from five clicks to several hundred per second. Extensiveobservations are being conducted to observe the emission patterns of differentcetacean species, which vary from location to location at different times of theyear even with the same group tracked on an almost continuous basis.At least two species of birds, Steatornia and Collocalla, are known to be echolocators.Ornithologists have reported that calls by various birds may be comprehendedonly by members of their own local flocks. Gulls or crows will alwaysrespond to calls from members of their own flocks, but they may or may notrespond to distress or assembly calls from gulls or crows from another region.The hearing of dogs extends well beyond the frequency range of human hearing.The “silent” dog whistles that generate ultrasonic output can thus be used to summona dog. Although dogs are generally endowed with especially keen senses ofsmell, hearing, and sight, there is no evidence that they make use of echolocation.Certain effects become more evident in wave propagation through a mediumas the acoustic signal extends into the ultrasonic range. The attenuation of thesignal’s amplitude occurs not only because of the spreading of the wave front, butalso because of the conversion of the acoustical energy into heat and scattering

16.2 Relaxation Processes 445from irregular surfaces. The process of relaxation, which represents the lag betweenthe introduction of a perturbation and the adjustment of the molecular energydistribution to the perturbation, requires a finite time; and the energy interchangeapproaches equilibrium in an exponential fashion. Considerable information regardingthe nature of matter can be derived from the study of relaxation phenomena.High-intensity ultrasound can result in energy absorption that yields considerableamount of heat, to the extent that glass or steel can be melted quickly.Ultrasonic waves can also generate stresses, resulting in cavitation in fluids.Cavitation is also capable of producing free chemical radicals, thus fosteringspecific chemical reactions. The stresses produced in the cavitation process aresufficiently concentrated to erode even extremely sturdy materials. Cavitation alsoprovides the mechanism of ultrasonic cleaning.16.2 Relaxation ProcessesRelaxation entails molecular interactions in gases and liquids. These interactionsaffect absorption and velocity dispersion, both of which depend on frequency(and on pressure, in the case of gases). Chemical reactions also entail relaxationprocesses on their own, but they will not be considered in this chapter, except forthe effect of ultrasound on reaction rates.To better understand the phenomenon of relaxation, let us consider an ideal gasmade up of diatomic molecules. The individual molecules move translationwisein three principal directions in a nonquantitized fashion, i.e., any translational energyis allowable. In addition, the molecules rotate about three perpendicular axes(actually two, since one axis has zero moment of inertia, hence zero rotationalenergy), and the molecules also vibrate along the direction of the bond joining theatoms. These molecules collide with one another in translation motion, exchangingenergy among themselves. A single collision is usually sufficient to transfertranslational energy from one molecule to another, but a certain period of time isneeded to randomize the energy associated with excessive velocity in a particulardirection. This amount of time is referred to as the translational relaxation time(Herzfeld and Litovitz, 1959), and it is given byτ tr = η p = 1.25 τ cwhere τ tr is the translational relaxation time, p is the gas pressure, η is the viscosityof the gas, and τ c is the interval between collisions. As the gas pressure is lowered,the rate of collisions decreases in the same proportion.Unlike the case of translational motion, rotation and vibration are quantitized.When a collision occurs, a change in rotational or vibrational state will occuronly when the change of energy of another state is sufficient to permit at least onequantum jump. For rotational energy transfer the spacing between energy levelsis given by 2(J + 1)B, where J denotes the rotational quantum number (whichmust be an integer), B = ¯h 2 /2I , I is the effective moment of inertia and ¯h isthe

446 16. UltrasonicsPlanck constant 1.055 ×10 −34 J s. Let us assume that J is the most probable valueaccording to the Boltzmann distribution. The value of 2(J+ 1)B for a typicalmolecule (for example, O 2 ) in temperature units is about 1 K. This indicates thatin a gas above 1 K all collisions will have enough translational energy to engendermultiple changes, with the result that rotation rapidly equilibrates with translation.Hydrogen, however, constitutes an exception, because it has much larger rotationalenergy level spacing due to its small moment of inertia. According to Winter andHill (1967), as many as 350 collisions may be necessary to transfer a quantum of rotationalenergy in the hydrogen molecule. As a pressure of 1 atm, this correspondsto a relaxation time of 2 × 10 −8 s. It should be understood that a specific collisioneither does or does not transfer a quantum of rotational energy in the hydrogenmolecule. The 350-collision average indicates that only one of the 350 collisionspossesses the proper geometry and energy to execute a transfer of one quantum ofrotational energy. The number of collisions necessary, on the average, to engenderthe transfer one quantum of rotational energy is termed the collision number Z.Where rotational energy is entailed, the collision number is written as Z rot . Theinverse of this dimensionless parameter represents the probability of transferringa quantum in a collision, symbolized by P rot . Rotational energy levels are spacedunevenly, i.e., a 1 → 2 transition should be more probable than a 2 → 3 transition.orProt 2→3 .As a rule, the probability for transferring a quantum of energy through a collisiondrops off rapidly with the size of the quantum transferred. Because vibrationalenergy levels are much more widely spaced than rotational energy levels,the vibrational relaxation times are considerably longer than rotational relaxationtimes. Vibrational levels in single vibrational mode are virtually evenly spaced,so energy can therefore be exchanged between levels (e.g., the vibrational quantumnumber goes up in one molecule and goes down in another) with hardlyany energy exchanged between vibration and translational. Thus, the vibration-tovibrationexchanges occur very rapidly. The amount of time it takes for energy totransfer between translation and the lowest lying vibrational level determines thevibrational relaxation time.Because this energy level varies greatly for different molecules, the probabilityof vibrational energy transfer during a collision also varies greatly. In the case ofN 1 molecules colliding with N 2 molecules, Z 10 (the number of collisions needed totransfer energy from the lowest vibrational energy to translation) is approximately1.5 × 10 11 , so the relaxation time is close to 15 s (Zuckerwar and Griffin, 1980).Larger molecules possess very closely spaced vibrational energy levels, so fewercollisions are required to transfer a quantum of first-level vibrational energy intotranslational energy. For example, a relatively large molecule such as C 2 H 6 needsto undergo about 100 collisions to execute this type of transfer.Let us consider the case of a vibrational state that is excited to an energy E v (thesubscript v denotes vibration) that is greater than energy E v (T tr ) that would existin a Boltzmann equilibrium with translation. This excess vibrational energy willequilibrate with translational energy, in accordance with the following standardThe probability of these events occurring is distinguished by the symbolsP 1→2rot

16.2 Relaxation Processes 447expression for relaxation:dE v= 1 dt τ [E v − E v (T tr )] (16.1)This reversion to equilibrium occurs through individual molecular collisions inwhich energy transfers result.Let k 10 be defined as the rate at which the molecules descend from the firstexcited state to the ground state, owing to collisions at 1 atm pressure. This issimply the collision frequency M multiplied by the probability of energy transferP 1→0 multiplied by the mole fraction x 1 of the molecules in the first excited state.The reverse process will also occur, i.e., some molecules in the ground state willbecome excited at a rate k 01 . In equilibrium both rates are equal, and we can writek 10 x 1 = k 01 x 0The energy is quickly shared from the first excited level of the vibrational mode tohigh-level modes through vibrational exchanges. On the basis of quantum mechanicsgoverning the probabilities of energy exchanges between vibrational levels ofa harmonic oscillator, Landau and Teller (1936)derived the following expression:− dE v (= k 10 1 − e−hυ/kT ) [E v − E v (T tr )] (16.2)dtwhere h is the Planck constant (6.626 × 10 –34 J s), v is the vibrational frequencyof the relaxing mode, k is the Boltzmann constant (1.38 × 10 16 ergs K −1 ), and Tis the absolute temperature in K. Through comparison with Equation (16.1), therelaxation time τ for Equation (16.2) is found to be1τ =(16.3)k 10 (1 − e −hν/kT )Relaxation time and ultrasonic absorption and dispersion are interlinked. Therelaxation process causes the specific heat of the gas to be frequency-dependent.The specific heat of a simple gas can be traced to translational, vibratory, androtational contributions. Let us now consider the situation where any acousticallyimposed temperature variation is followed by both translational and rotationalenergy equilibrating extremely quickly. The specific heat C ′ is deemed independentof temperature or the temperature never deviates appreciably from the equilibriumvalue T 0 ; so we can write[E v − E v (T tr )] = C ′ (T ′ − T 0 ) (16.4)and insert this last expression into Equation (16.1) to yielddT ′= 1 (T ′ )− T tr (16.5)dt τNow consider the case when the translational temperature is suddenly raisedat time t = 0 from T 0 to a value T 1 . Two cases can be ascertained: The externaltemperature is kept constant at T 1 after t = 0, the energy seeping from the outsideinto the internal (rotational and vibrational) degrees of freedom. The solution of

448 16. UltrasonicsEquation (16.5) isT ′ = T 1 + (T 0 − T 1 )e −t/τBut if the external energy is not constant and is lessened by the amount flowinginto the internal degrees of freedom, we have0 = ¯C(T tr − T 1 ) + C ′ (T ′ − T 0 ) (16.6)where ¯C refers to the specific heat arising from external degrees of freedom andC ′ denotes the specific heat belonging to the internal degrees. Equation (16.6) isrewritten as follows:0 = ¯C(T tr − T ′ ) + ¯C(T ′ − T 1 ) + C ′ (T ′ − T 0 )= ¯C(T tr − T ′ ) + C(T ′ − T 0 ) (16.7)whereT 2 = ¯CC T 1 + C′C T 0where C is the total specific heat, and T 2 the final equilibrium temperature. Weeliminate (T tr − T ′ ) from Equation (16.5) with the use of Equation (16.7) to obtain− dT′ = C¯C1 (T ′ )− T 2dt τwhich yields the solutionT ′ = T 2 + (T 0 − T 2 )e −t/τ ′The apparent relaxation time isτ ′ = ¯CC τ = C − C′CτEquation (16.5) can be written asor−τ dT′dt= T ′ − T 0 − (T tr − T 0 )T ′ − T 0 + τ d dt (T ′ − T 0 ) = T tr − T 0 (16.8)If T tr − T 0 is periodic in time, i.e., it is proportional to e iωt , T ′ —T 0 will likewisebe proportional to e iωt . After the transient dies out, then we haveT tr − T 0 = (1 + iωt)(T ′ − T 0)orT ′ − T 0 = T tr − T 01 + iωτwhich can also be rewritten asdT ′ 1=dT tr 1 + iωt

The effective specific heat can be established from( (dE =(C v ) eff dT tr = ¯C v dT tr + C ′ dT ′ = ¯C v)vand16.2 Relaxation Processes 449+ C′dT′dT tr)dT tr(C v ) eff = ¯C v +1 + iωτ = C v − C′ iωτ1 + iωtThe acoustic propagation constant k can be written in the formk 2 ( 1ω = 2 c − iα ) 2= ρ 0κ TωC ′γ eff(16.9)where c represents the acoustic velocity, α is attenuation coefficient, ρ 0 is theequilibrium density, κ T is the compressibility of the gas, and γ eff is given byγ eff ≡ (C v) eff + R(C v ) effwhere R is the gas constant. In the case of this simple single relaxation, for α/ω≪ 1andwhereλa = π( c0c( cc 0) 2εωτ1 + (ωτ s ) 2 (16.10)) 2= 1 −ε(ωτ s ) 21 + (ωτ s ) 2 (16.11)ε = c2 ∞ − c2 0c∞2c 0 = speed of sound for ωτs ≪ 1c ∞ = speed of sound at frequencies ≫ relaxation frequencyλ = wavelengthThe adiabatic relaxation time τ s is related to the isothermal relaxation time τ asfollows:τ s = C v + RCv∞ + R τThe frequency at which the maximum absorption per wavelength occurs is calledthe relaxation frequency, symbolized by f r . It is related to the adiabatic absorptiontime τ s as follows:f r = 1 c ∞2πτ s c 0We take as an example the gas Fl 2 at 102 ◦ C in Figure 16.1 that shows curves for absorptionper wavelength and velocity dispersion due to a single-relaxation process(Shields, 1962). Measured values are also plotted for the purpose of comparingwith theory.

450 16. UltrasonicsFigure 16.1. Sound absorption per wavelength and velocity dispersion in fluorine at102 ◦ C.In the case of polyatomic gases or mixtures of relaxing diatomic gases, the relaxingmodes can be coupled by vibration-to-vibration exchanges. These multiplerelaxation processes follow the general behavior given by Equations (16.10) and(16.11), but the magnitude of the absorption and dispersion and the associated relaxationfrequencies assumes a different connotation. For this case of the multiplerelaxing internal energy modes, Equations (16.10) and (16.11) are changed intothe following formats:( cαλ) 2 ∑=−πc ∞ j( c) 2 ∑= 1 +c ∞jδ j k s /k ∞ s1 + (ωτ s, j ) 2 (16.12)δ j k s /k ∞ s1 + (ωτ s, j ) 2 (16.13)

16.3 Cavitation 451where δ j k s /ks∞ is a relaxation adiabatic compressibility (which has a negativevalue) and j denotes that there may be more than a single-relaxation processentailed. In such complex cases, τ s, j and δ j k s /ks∞ can no longer be respectivelyassociated with a single-transfer reaction and relaxation energy of a specific mode,because the various modes and reaction pathways are coupled. So the sums inEquations (16.12) and (16.13) should cover all eigenvalues of the energy transfermatrix, which also accounts for all reactions. These two equations constitute thestandard equations for calculating sound absorption in moist air as function offrequency and temperature. Equations (16.12) and (16.13) can also be used inthe reverse manner, where the measured values of absorption and velocities canbe used to derive the transition rates. But when the number of relaxation modesincreases, the number of possible relaxation paths multiplies very rapidly, thuslimiting this procedure to only a few special cases.While relaxation processes similar to the energy exchanges in gases occur in liquids,there are important differences that arise from the greater density of moleculesinherent in liquids and the consequential multibody interactions. The concept of arate equation is less applicable, but the existence of relaxation time as a measure ofthe time for a system to revert to an equilibrium state in sustaining a perturbationremains valid.In a few cases such as CS 2 and a number of organic liquids, the relaxationmechanism appears to be the same as that for gases, i.e., the internal energy ofthe individual molecules is excited by “collisions.” These types of liquids arecalled Kneser liquids, and they generally have a positive temperature coefficientof absorption. With other liquids, the molecules bond temporarily to form largegroups that reconfigure themselves when an ultrasound wave passes through. Suchfigurative relaxations tend to be very rapid, and there is the possibility of thefrequency dependence of absorption and dispersion, which results in a distributionof relaxation times. Such liquids are termed associated liquids. Water is a primeexample of such a liquid.Chemical reactions complicate matters even more: in a reversible chemicalreaction with heat of reaction H. H plays a role in the relaxation equationsin the same manner as E for vibrational relaxation. Because chemical reactionsincrease the possibility of the number density of the molecules changing, additionalrelaxation absorption and dispersion tend to occur.16.3 CavitationThe phenomenon of cavitation, the rupture of liquids, is readily observed in boilingwater, turbines, hydrofoils, and in seawater in the vicinity of a ship’s rotatingpropeller. It occurs in those regions of liquids that are subject to high-amplitude,rapidly vacillating pressures. Cavitation also occurs in a liquid irradiated withhigh-energy ultrasound.Consider a small volume of liquid through which sound travels. During thenegative half of the pressure cycle the liquid undergoes a tensile stress, and during

452 16. Ultrasonicsthe positive half the liquid undergoes compression. Bubbles entrapped in the liquidwill expand and contract alternatively. When the pressure amplitude is sufficientlygreat and the initial radius of the bobble is less than a critical value R 0 given by(R 0 = 1 p√ 0 + 2T )stR 0 3 γ (16.14)ωρthe bubble collapses suddenly during the compression phase. In Equation (16.14),the symbols used are defined as follows:ω = angular frequency of the signalp 0 = hydrostatic pressure in the liquidγ = ratio of the principal specific heats of the gas in the bubbleT ts = surface tension at the surface of the bubbleThis sudden collapse of bubbles constitutes the phenomenon of cavitation and itcan result in the very sudden release of a comparatively large amount of energy.The severity of this cavitation, as measured by the amount of the energy released,depends on the value of the ratio R m /R 0 , where R m denotes the radius of the bubblewhen it has expanded to its maximum size. Obviously this ratio depends on themagnitude of the acoustic pressure amplitude, i.e., the acoustic intensity.The presence of bubbles facilitates the onset of cavitation, but cavitation canalso occur in gas-free liquids when the acoustic pressure exceeds the hydrostaticpressure in the liquid. During a part of the negative phase of the pressure cycle,the liquid is in a state of tension. This causes the forces of cohesion betweenneighboring molecules to become opposed, and voids are formed at weak pointsin the structure of the liquid. These voids expand and then collapse in the samemanner as gas-filled bubbles. The cavities produced in this fashion contain onlythe vapor of the liquid. Cavitation in a gasless liquid can be induced by introducingdefects in the structure of the liquid by adding impurities or by bombarding theliquid with neutrons.A hissing noise often accompanies the onset of cavitation. This noise is referredto as cavitation noise. Th minimum intensity or pressure amplitude required toestablish cavitation is termed the threshold of cavitation. Figure 16.2 displays howthe threshold intensity varies with both aerated and gas-free water. The thresholdintensity is obviously considerably greater for gas-free water than for the aeratedwater. This parameter remains fairly constant up to about 10 kHz, then it undergoesa steady increase up to about 50 kHz and a more pronounced exponential increasebeyond 50 kHz.Generally speaking, the threshold intensity usually increases with increasingpressure and decreases with increasing temperature. But a number of exceptionsto this rule exist (Hunter and Bolt, 1955). The threshold intensity decreases as thetime of exposure to sound is increased. This is the result of a time delay betweenthe acoustic excitation and the onset of cavitation. For pulsed waves, the threshold

16.3 Cavitation 453Figure 16.2. Variation of threshold intensity with frequency in aerated and gas-free waterat room temperature (20 ◦ C).intensity reduces in value as the pulse length is increased to an upper limit, beyondwhich it becomes independent of pulse length. This frequency-dependent upperlimit would be of the order to 20 ms for a frequency of 20 kHz.The amount of energy released by cavitation depends on the kinetics of thebubble growth and collapse of the bubbles. This energy should increase withsurface tension at the bubble interface and lessen with the vapor pressure of theliquid. Water has a comparatively high surface tension, so it can be a very effectivemedium for cavitation. It can be made even more effective by the addition of 10%alcohol—this results in an appreciable increase in vapor pressure but at the costof a decrease in the surface tension, but the former effect outweighs the lattereffect.Weak emission of light has been observed in cavitation. This phenomenon isknown as sonoluminescence. Frenzel and Schultes first observed its effects in waterin 1934 (Frenzel and Schultes, 1934). Two separate forms of sonoluminescenceare thought to exist: multiple-bubble sonoluminescence (MBSL) and single-bubblesonoluminescence (SBSL). When a sufficiently strong acoustic field propagatesthrough a liquid, placing it under dynamic stress, preexisting microscopic inhomogeneitiesserve as nucleation sites for liquid rupture. Most liquids such as waterhave thousands of potential nucleation sites per milliliter, so a cavitation field canharbor many bubbles over extended space. This cavitation, if sufficiently intense,will produce sonoluminescence of the MBSL type. It was more recently discoveredthat under certain conditions, a single, stable oscillating gas bubble can beforced into large amplitude pulsations that it produces sonoluminescence emission

454 16. Ultrasonicson each and every cycle (Gaitan and Crum, 1992; Gaitan et al., 1992). This is theSBSL-type of sonoluminescence.Sonochemistry deals with high-energy chemical reactions that occur duringultrasonic irradiation of liquids. The chemical effects of ultrasound do not resultfrom direct molecular interactions but occur principally from the effects ofacoustic cavitation. Cavitation provides the means of concentrating the diffuseenergy of sound, with bubble collapse producing intense, local heating and highpressures that are extremely transient. Among the clouds of cavitating bubbles,the highly localized hot spots have temperatures of roughly 5000 ◦ K, pressuresexceeding 2000 atm, and heating and cooling rates greater than 10 7 K/s. Ultrasonicscan serve as useful chemical tool, as its chemical effects are diverse andit can provide dramatic improvements in both stoichiometric and catalytic reactions.In a number of cases, ultrasonic irradiation can increase reactivity by amillion-fold.The chemical effects can be categorized into three areas: (a) homogeneoussonochemistry of liquids, (b) heterogeneous sonochemistry of liquid–liquid orliquid–solid systems, and (c) sonocatalysis (which constitutes an overlap of thefirst two categories). Chemical reactions have generally not been observed in theultrasonic irradiation of solids and solid–gas systems.16.4 PhononsIn quantum mechanics, energy states are considered to occur only at discrete levelsor eigenstates, not at any arbitrary values. In the analytical treatment of crystallinesolids, the concept of phonons, often referred to as a “quantitized sound waves,”is used to represent the effects of a transition between the eigenstates of a systemof coupled quantum mechanical oscillators. Phonons generally apply to discretestrictly linear systems, while “classical” sound waves derive from continuous,intrinsically nonlinear systems within the limits of small amplitudes. Althoughphonons can occur in all states of matter, they are most easily discerned in crystallinesolids.In 1819, Dulong and Petit discovered the first evidence for phonons in solidswhen they observed that the specific heat of a solid is twice that of the correspondinggas. This finding suggested the fact that solids have a way of storingpotential energy, in addition to the kinetic energy that is so apparent in gases.Einstein first postulated an acoustic theory of the specific heat of solids by assumingthat the kinetic energy and potential energy arose from atoms oscillatingabout the equilibrium positions in the crystalline lattices. Applying the Planckquantum theory, Einstein related the energy to the frequency but he had made thesimplifying assumption that each atom oscillated independently of the other, so hisformula for the specific heat was therefore incorrect. Peter Debye in 1912 correctlyinferred that these atomic oscillators are coupled, and later Max Born, Theodorevon Kármán, and Moses Blackman refined the theory to the extent of matching theexperimental results of the temperature dependence of the specific heat of solids.

16.4 Phonons 455Debye described the role of phonons in his explanation of thermal conductivity,and in that same year (1912) Frederick Landemond correlated lattice vibrationsto thermal expansion and melting of solids, both of which are attributable to thenonlinearities of forces between atoms in solids.Consider a one-dimensional array of masses m j ( j = 0,...,N + 1) interconnectedwith ideal springs of stiffness s j ( j = 0,...,N). A spring s j interconnectsmass m j and mass m j+1 . Let ξ j denote the displacement of mass m j . The displacementsξ 0 and ξ N+1 provide the boundary conditions at each end of the system.Within the boundaries, the motion of each mass is assumed to follow Hooke’s lawof linear elasticity and Newton’s law:d 2 ξ jm jdt =−s j−1(ξ 2 j − ξ j−1 ) + s j (ξ j+1 − ξ j ) (16.15)The motion can be assumed amenable to Fourier analysis, so that it assumes atime dependence e iωt . The left term of Equation (16.15) then becomes –m j ω 2 ξ j .The solution to this equation, through the application of the Bloch theorem forthe normal modes of a coupled system, is a linear combination of the normalmodes:ξ j = ∑ (eik( ja) X k + e −ik( ja) )X −k eiω k tkwherek = eigenvalues of the system determined by the boundary conditionsa = periodic spacing, or the “stretch” of the “springs” connecting the massesX k = ξ 1 − ξ 0 e −ika2i sin(ka)The values of ξ 0 , ξ 1 , X k and the restrictions on k are established by boundary andinitial conditions. For example, consider a system clamped at the ends, i.e., x 0 =0, x N+1 = 0. Then( )ξ1ξ j = sin[k( ja)]sin(ka)The eigenvalue k is quantitized withk =n πwith n = 1, 2,...,NN + 1 aIn an infinite system or in a system with periodic boundary conditions, it isreadily established that X k = 0orX −k = 0. If we apply a coordinate system withthe mass m 0 located at its origin, then the location of the jth mass is x = ja, andwe haveξ k (x) = e ikx X k (16.16)Equation (16.16) represents the customary Bloch wave result. The subscript k wasadded in Equation (16.16) to serve as a label for the normal mode. The solution is

456 16. Ultrasonicscomplete as a linear combination of the normal modes, i.e.,ξ j = ∑ k(e ik( ja) X k + e −ik( ja) X −k )e iω kt(16.17)The normal modes are orthogonal, so the normalization constants can be obtainedfor any boundary conditions, whether they be clamped, periodic, and so on. Thetotal energy E of the system can be found fromE = m ∑ k|X k | 2 ω 2 kCoupled Quantum ParticlesIn dealing with harmonically coupled particles it is more expeditious to use aLagrangian formulation rather than Newtonian formation of Equation (16.15).The Lagragian for such a set of connected particles isL = 1 ∑( dξ jm j2 dtj) 2− 1 ∑s j (ξ j+1 − ξ j ) 22With the canonical momenta p j = m j (dξ j /dt), the Hamiltonian becomesH = ∑ ( ) dξ jp j − LdtjThe system is quantitized with the commutation relations[ ]ξ j , p j ′ = i¯hδ j, j; (16.18)jwhere δ j, j; represents the Dirac delta function that equals unity when j =j ′ andzero when j ≠ j ′ . We then assume a periodic system and set m j = m and s = s j ,and also make use of Equation (16.17) without the coefficient e −iωkt . We obtainthe HamiltonianH = 1 ∑P k P −k + 1 2m2 m ∑ ωk 2 X k X −kkwhereP k = m dX ( )−kka, ω k = ω 0 sindt2The above Hamiltonian could be used to construct a Schrödinger wave equationfor a field ψ(X), where X represents a point in the 2N-dimensional X k space and( ) ∂ψP k =−i¯h∂ X kAnother approach is to construct the properties of the eigenfuctions and eigenvaluesthrough the use of the commutation relations of Equation (16.18) for x j and p j .

16.4 Phonons 457Definingand√ ( ma k = ω k X k + i )2¯hω k m P −ka ∗ k = a∗ −k ,the Hamiltonian assumes the formwhereH = ∑ kω k =−ω −k(¯hω k N k + 1 )2N k = a ∗ k a kA general state of the system is constructed from a superposition of the eigenstates:ψ = ∑ ∑C kn ψ knk nwhere |C kn | 2 represents the probability that the system is in the state ψ kn . Fromthe customary quantum mechanical relation Eψ = Hψ, the expectation value ofthe total energy of the system is〈ψ |E| ψ〉 = ∑ ∑|C nk | 2¯hω (k n k + 1 )k n2The expectation value of the square of the momentum operator is found from〈ψ ∣ ∣P 2∣ ∣ ψ〉 =m ∑ k〈ψ |X k |ψω 2 k = ∑∑ |C nk | 2¯hω k n k =〈ψ |E| ψ〉−E 0where E 0 is the zero-point energy.From the expectation state of the position operator, a state ψ nk can be coupledonly to the state ψ (n+1)k or ψ (n−1)k . Consequently the time dependence will behaveas 2 cos (ω k t). The term phonon can be defined in the following manner: a phonon isemitted or absorbed when a system of harmonically coupled quantum mechanicalparticles executes a transition from a state ψ nk to a state ψ (n–1)k (which results inemission) or ψ (n+1)k (which results in absorption).The absorption of ultrasonic waves in solids are attributable to a number ofdifferent causes, each one of which is characteristic of the physical propertiesof the material concerned. They can be classified as (a) losses characteristic ofpolycrystalline solids, (b) absorption due to lattice imperfections, (c) absorption inferromagnetic and ferroelectric materials, (d) absorption due to electron–phononinteractions, (e) absorption due to phonon–phonon interactions, and (f) absorptiondue to other possible causes. It is also interesting to note that a rapid decrease in attenuationoccurs at the critical temperature for superconductivity. This variation ofattenuation with temperature has been explained in a Noble-prize winning paperby Bardeen, Cooper, and Schrieffer through the B.C.S. theory (so-called after the

458 16. Ultrasonicsinitial of their surnames) that predicts a temperature-dependent energy gap 2ε widearound the Fermi level at the critical temperatures of T c and less (Bardeen et al.,1957). As the temperature is reduced, the gap increases toward a maximum at zeroabsolute temperature, where the predicted value of ε is equal to 1.75 × κ T c , whereκ is the Boltzmann constant. In the realm of l e >λ/2π, the B.C.S. theory predictsthatα s 2=α n e ε/κT + 1where l e denotes the mean-free path of an electron, α s and α n represent the valuesof absorption in the super-conducting and normal state, respectively, at absolutetemperature T . This variation was confirmed experimentally by Morse (1959) andBohm for indium at 28.5 MHz. Gibbons and Benton measured the velocities oflongitudinal waves in both normal and superconducting tin; they found a verysmall reduction in velocity (about 1/500,000th) for the superconducting state.Application of a magnetic field to a metal at low temperatures affects the meanfreepath and thus affects acoustic attenuation. For l e

16.5 Transducers 459probe is to gauge the characteristics of an acoustic field, so its dimensions must besufficiently small so as not to affect the field. A probe diameter is typically onlyabout one-tenth of the wavelength.Piezoelectric CrystalsIn 1880, the Curie brothers discovered that when a crystal having one or morepolar axes or lacking axisymmetry is subjected to mechanical stress, an electricalpotential difference occurs. Consider a segment of such a crystal, in the form of aslab or a disk, that is cut with its parallel surface running normal to a polar axis.When this segment undergoes a mechanical stress, equal and opposite electriccharges arise on the parallel surfaces. The magnitude of the charge density (i.e.,dielectric polarization) is directly proportional to the applied stress, provided theapplied stresses do not strain the crystal beyond its elastic limit.The opposite effect, predicted by Lippmann in 1881 and verified experimentallyby the Curie brothers the same year, occurs when an electric field is applied in thedirection of a polar axis, causing a mechanical strain in the crystal segment. Theamount of strain is directly proportional to the intensity of the applied electric field.From the viewpoint of the principle of conservation of energy, the piezoelectriceffect and its converse may be deemed to be equal and opposite. Such effectsoccur in crystals such as quartz (a member of the trigonal system, as shown inFigure 16.3), Rochelle salt, and lithium sulfate.Quartz is very commonly applied for ultrasonic generation. A quartz crystalis shown in Figure 16.3, with a hexagonal cross section normal to the nonpolarFigure 16.3. A hexagonal quartz crystal with x-cut rectangular and circular plates.

460 16. Ultrasonicsoptic axis, denoted by the z-axis. The axes joining opposite edges are designatedas x-axes, and the associated axes, which are perpendicular to these and joiningopposite faces are termed y-axes. The x- and y-axes are polar axes, and slabs cutwith their faces perpendicular to them manifest the piezoelectric effect. Crystalswhich are cut with their faces perpendicular to an x-axis or y-axis are termedx-cut and y-cut crystals, respectively. The x-cut crystals are generally utilized topropagate compression waves, and the y-cut crystals are applied to generate shearwaves.Now consider an x-cut crystal in the form of a rectangular prism shown inFigure 16.3. Applying an electric field along the x-axis produces compressionin that direction, while expansion occurs simultaneously along the y-direction.If the direction of the field is reversed, expansion occurs along the x-axis withan associated compression along the y-axis. No strain, however, occurs along thez-axis. If a pair of surfaces normal to either of the polar axes (x- and y-axes)is coated with a conductive material to form electrodes, small-amplitude oscillationswill result when an alternating voltage of frequency f is applied acrossthem. When the frequency f equals one of the natural frequencies of mechanicalvibration for a particular axis, the response amplitude jumps to a considerablyhigher value. Crystals are generally operated at resonant frequencies for either“length” or “thickness” vibrations, as denoted by the resonance occurring in thedirection parallel with or normal to the radiating surfaces, respectively. The naturalfrequency for mechanical vibrations is proportional to the inverse of the dimensionalong which they occur, so it becomes obvious the lower frequencies are generatedby “length” vibrations along the direction of the longer dimension whereasthe higher frequencies are produced by “thickness vibrations” along the directionof the smaller dimension.Maximum acoustic intensities are obviously obtained by operating at the fundamentalnatural frequencies. But material constraints in crystals may necessitate theuse of higher harmonics to obtain higher frequencies. For example, an x-cut quartzplate can be only 0.15 mm thick in order to generate a fundamental “thickness”mode for 20 MHz. Such a quartz plate is extremely brittle and it can shatter underthe impetus of a exceedingly high-applied voltage, or its dielectric properties maybreak down. To avoid this situation, it is customary to use thicker slabs of crystalswith lower resonance frequencies and operate at one of the upper harmonics. Anexample is the vibration of a 1-cm thick quartz crystal at its 191st harmonic togenerate 55 MHz ultrasound.The piezoelectric effect occurs only when opposite charges appear on the electrodes,and for that reason, only odd harmonics can be generated. At the nth harmonic,the thickness of the crystal is divided into n equal segments with compressionsand expansions alternating in adjacent sections, as illustrated in Figure 16.4.For even harmonics in the nth mode, compressions occur in n/2 segments andexpansions occur in the other n/2 segments, with the result no net strain existsin the crystal. When n is odd, the (n–1)/2 compressions offset the same numberof expansions, leaving either a compression or an expansion in the remainingsegment.

16.5 Transducers 461Figure 16.4. Crystal divided into segments.The Electrostrictive EffectThe electrostrictive effect, which is the electrical analog of the magnetostrictiveeffect discussed in a later section, occurs in all dielectrics, but it is not a very pronouncedphenomenon in most materials except for a certain class of dielectrics. Theeffect is much more apparent in this class namely, the ferroelectrics. An electricfield applied along a given direction produces a mechanical strain. The magnitudeof the strain is proportional to the square of the strength of the applied electricfield and is therefore independent of the sense of the field. A positive strain maythus result for both positive and negative values of the excitation field. For a sinusoidallyvarying electric field, the waveform of the strain assumes that of a rectifiedbut unsmoothed alternating current, and its frequency is twice that of the appliedfield.It is possible to obtain a sinusoidal variation in the strain, and this is done bypermanently polarizing the transducer, namely, one that has magnetostrictive properties.The transducer is heated to a temperature above the Curie point, causing themagnetostrictive effect to vanish, and then it is cooled slowly in a strong direct fieldoriented in the direction along which it is intended to apply the exciting field. Ifthe exciting field is kept small compared with the initial polarizing field, the strainshould vary sinusoidally at the frequency of the exciting field. Because a polarizedferroelectric transducer appears to manifest the same effect as a piezoelectric

462 16. Ultrasonicstransducer, it has mislabeled as being “piezoelectric.” Among the principal ferroelectrics,barium titanate, lead meta-niobate, and lead zirconate titanate are greatlyused for electrostrictive applications. To construct this type of transducer, manysmall crystallites of ferroelectric substances are bonded together to form a ceramicof the appropriate shape. Because such materials are polycrystalline, they may beconsidered as being isotropic and thus do not have to be cut along specific axes.This renders possible the construction of a concave transducer so that the ultrasonicradiation can be focused without the need for an auxiliary lens system.Fundamental Piezoelectric RelationshipsFor a given temperature, consider a piezoelectric element having cross-sectionalarea A, thickness t, with electrodes attached to the opposite faces. A voltage V isapplied across the electrodes to generate an electric field E = V/t, and a constanttensile stress σ is applied to the surfaces. Within the elastic limits, the resultantmechanical strain s relates to the stress as follows:s = aσ + bE (16.19)and we also haveD = cE + dσ (16.20)where D represents the electric displacement and a, b, c, d are coefficients definedbelow.Let us now short circuit the electrodes so that E = 0. Equation (16.20) nowreadsD = dσWe note here that the electric displacement D equals the dielectric polarization P,or the charge per unit area. HenceP = dσ (16.21)under short-circuit conditions. The coefficient d constitutes the piezoelectric strainconstant, which is defined as the charge-density output per unit applied stress underthe conditions of short-circuited electrodes. Now if the stress σ is reduced to zero,Equation (16.19) modifies toσ = bEThe principle of conservation of energy dictates that b = d, resulting inσ = dE (16.22)for the no-load condition. The coefficient d may also be described as the mechanicalstrain produced by a unit applied field under the conditions of no loading and it isexpressed either in units of coulomb/newton (C/N) or meter/volt (m/V). Equation(16.19) becomes altered tos = aσ + dE (16.23)

16.5 Transducers 463If no piezoelectric effect is present, the term d vanishes from Equations (16.20)and (16.23), which yields the familiar relationshipsands = aσ = σ/Y (16.24)D = εE (16.25)where Y is the elastic constant (or Young’s modulus) for the material and ε is thecorresponding electrical permittivityUnder short-circuit conditions for the crystal, Equations (16.21) and (16.24)lead toP = es (for short-circuit conditions) (16.26)where e = d/Y.When a compressive stress is applied to the crystal, Equation (16.23) becomess =−aσ + dE (16.27)When the crystal is clamped to keep strain zero and when stress is applied, wesee from Equations (16.26) and (16.27) thatσ = eE (for the constraint s = 0) (16.28)where e is the piezoelectric stress constant which is expressed on C/m 2 or N/V m.It must be realized at this stage that the piezoelectric phenomenon is a threedimensionalone. Not only we have to consider the changes in voltage, stress,strain, and dielectric polarization in the thickness direction of the crystal, we mustalso take into account the effects in any direction. A stress applied to a solid in agiven direction may be resolved into six components: three tensile stresses σ x , σ y ,σ z along the principal axes x, y, and z, respectively, and three shear stresses τ yz ,τ xz , and τ yz about axes x, y, and z. In the notation for shear, the subscripts indicatethe action plane of the shear—thus yz denotes a shear in the yz plane acting aboutthe x-axis. Also, we note that τ yz = τ zy , and so on. The corresponding componentsof strain are ε x , ε y , ε z , ε yz , ε xz , and ε yx . In general, the following stress–strainrelationship of Equation (5.2) can be generalized as followsσ ji = c jk ε ij (16.29)where c jk denotes the elastic modulus or stiffness coefficient. This yields 36 valuesof c jk :⎤σ xx = σ x = c 11 ε x + c 12 ε y + c 13 ε z + c 14 ε yz + c 15 ε xz + c 16 ε xyσ yy = σ y = c 21 ε x + c 22 ε y + c 23 ε z + c 24 ε yz + c 25 ε xz + c 26 ε xyσ xx = σ z = c 31 ε x + c 32 ε y + c 33 ε z + c 34 ε yz + c 35 ε xz + c 36 ε xyτ yz = c 41 ε x + c 42 ε y + c 43 ε z + c 44 ε yz + c 45 ε xz + c 46 ε xy(16.30)⎥τ xz = c 51 ε x + c 52 ε y + c 53 ε z + c 54 ε yz + c 55 ε xz + c 56 ε xy⎦τ xy = c 61 ε x + c 62 ε y + c 63 ε z + c 64 ε yz + c 65 ε xz + c 66 ε xy

464 16. UltrasonicsTable 16.1. Value of theAdiabatic Elastic Constantsfor Quartz ×10 10 dyne/cm 2or ×10 9 N/m 2 .c 11 = 87.5c 33 = 107.7c 44 = 57.3c 12 = 7.62c 13 = 15.1c 14 = 17.2Because c mn= c nm, the number of these constants is reduced from 36 to 21. Symmetryof the axes will lessen this number even further. In the case of quartz, onlysix elastic constants are independent of one another, and the values of c mn can beexpressed as a matrix as follow:c 11 c 12 c 13 c 14 0 0c 13 c 11 c 13 −c 14 0 0c 13 c 13 c 23 0 0 0c 14 −c 14 0 c 14 0 0∣ 0 0 0 0 c 44 c 14 ∣∣0 0 0 0 c 14(c 11 − c 12)2The values of these constants for adiabatic conditions are given in Table 16.1The Dynamics of Piezoelectric TransducersA body undergoing forced vibrations can be considered analogous to an electriccircuit that is activated by an electromotive force, with the current i correspondingto body velocity u and the voltage V to the applied force F. In terms of the strains, velocity u = l(ds/dt), and the current is expressed as i =(dQ/dt) = A(dP/dt).Here l is the body length, Q is the electric charge, A is the cross-sectional area.We invoke Equation (16.26) to obtainwhich givesdPdt= e dsdti = Ae u = α T u (16.31)lwhere the transformation factor α T = Ae/l, which constitutes a characteristicconstant for a specific transducer. Because P has three components and S has sixcomponents, we can write Equation (16.31) in the general matrix formi = a T u∣

From Equation (16.28) we haveorF = Ae V = α T VlF = a T V16.5 Transducers 465The mechanical compliance C m is analogous to the electrical capacitance C, i.e.,C m = slF = slσ A = YlAWhen an applied force F causes a strain s, mechanical energy W m is stored in thetransducer, according towhereW m = 1 2 Fsl = 1 2 F 2 C m = 1 2 α2 T V 2 C m = 1 2 CV2 (16.32)C = α 2 T C mThe electrical capacitance C e between the electrodes of the transducer followsthe relationship:C e = ε A(16.33)lThe corresponding electrical energy W e is equal to 1 / 2 C e V 2 . From Equations(16.32) and (16.33) the ratio of mechanical energy stored in a piezoelectric transducerto the electrical energy provided to it is given byW mW e= C C e= α 2 TC mC e= k 2 eHere the electromechanical coupling factor ke 2 constitutes a measure of the efficiencyof the transducer.The Q FactorThe Q factor of either a mechanical or an electrical system determines the contourof the frequency response curve for that system. A low value of Q results ina resonance spreading over a wide frequency band. At higher values of Q, aresonance will be confined to a considerably narrower frequency band. Two Qfactors exist in a transducer, one mechanical and the other electrical, denoted byQ m and Q e , respectively. The mechanical Q factor is defined byQ m = mω rR mwhere ω r represents the resonance frequency of the transducer, m its mass and R mthe mechanical resistance. In the simplest case for the radiating transducer surface,

466 16. Ultrasonicsthe mechanical resistance is given byR m = σ Au= ρcAin terms of the stress, radiating surface velocity u, material density ρ, and soundpropagation speed c. For the electrical Q factor, electrical capacitance C e betweentransducer electrodes must be taken into consideration. At resonant frequenciesthe only effective mechanical impedance is R m . ThusQ e∼ = Ce ω r R = C e ω rR mα 2 T= π 2 /2k 2 eQ mMagnetostrictive TransducersMagnetostriction occurs in ferromagnetic materials and certain nonmetals thatare termed ferrites. When a magnetic field is applied, a bar of ferromagnetic orferrimagnetic material undergoes a change in length. Conversely, a mechanicalstress applied to the bar will cause a change in intensity of magnetization. Theformer effect was discovered by Joule in 1847 and the converse effect by Villari in1868.Magnetostriction occurs prominently in materials such as iron, nickel, andcobalt. Whether there occurs an increase or decrease in length fully depends on thenature of the material as well as on the strength of the applied magnetic field. Thechange in length does not depend on the direction of the magnetic field. The magnitudeof the strain varying as a function of the applied magnetic field is shownin Figure 16.5 for four different materials, viz. cast cobalt, permendur, nickel,and iron. Figure 16.6 shows strain varying as a function of magnetic polarization.The magnetostrictive effect generally decreases with a rise in temperature anddisappears altogether at the Curie temperature.Figure 16.5. Mechanical strain as a function of magnetic field.

16.5 Transducers 467Figure 16.6. Mechanical strain as a function of magnetic polarization.When a sinusoidally varying magnetic field is applied in the direction of the axisof a bar of ferromagnetic material, the bar will oscillate at double the frequency ofthe applied field. In accordance with the relevant curve in Figure 16.6, a decrease inlength occurs when a field is applied to nickel, regardless of the sense of the field.A negative strain occurs every half cycle. The waveform of the strain occurs as arectified sine curve, with the result that unwanted harmonics may be generated. Apurely sinusoidal wave corresponding to the frequency of the applied field, alongwith a markedly increased energy output, will be obtained if the bar is polarized.This is achieved by simultaneously applying the alternating field and a directmagnetic field of sufficiently high intensity for the value of the resultant field toremain above zero.The maximum output for magnetostriction occurs by operating at the fundamentalfrequency f r of the bar, given by√f r = 1 Y(16.34)2L ρwhere Y is the Young’s modulus for the bar material, ρ is the density of the material,and L is the length of the bar. The term √ Y/ρ in Equation (16.34) also happens tobe the propagation velocity of sound in the material. At resonant frequencies themechanical strains reaches the order of 10 –4 rather than magnitudes in the orderof 10 −6 that prevails in operating at nonresonant frequencies.

468 16. UltrasonicsThe Physics of Magnetostrictive TransducersMagnetrostriction theory is highly analogous to that of piezoelectricity, but, in thiscase, account is taken of the polarizing field H 0 . We now consider a ferromagnticrod undergoing polarization throughout its length with a magnetic field H 0 , with B 0denoting the associated flux density. The resultant strain ε 0 is directly proportionalto the square of the flux density, i.e.,ε 0 = CB0 2 (16.35)where C is a constant. It is seen from Equation (16.35) that the sign of the resultantstrain is independent of the direction of the field. Now if we apply an excitingmagnetic field of strength H, which is appreciably less than H 0 with an associatedflux density B, we can writeB = μ i H = B 0 ≪ B 0 (16.36)where we have denoted μ i as the incremental magnetic permeability. For a stateof constant stress, we have for the resulting strain by differentiation of Equation(16.35):ε = ε 0ε = 2CB 0 B 0orε = 2CB 0 B = βμ i H (16.37)Here β = 2CB 0 constitutes the magnetostrictive strain coefficient (given in units ofm 2 /weber) that applies to small strains. Equation (16.36) is analogous to Equation(16.22) for no-load conditions. When no alternating field is applied, the value ofthe strain ε is given by Hooke’s law,ε = σ/Y (16.38)in terms of stress σ and Young’s modulus Y . We can now obtain the analog toEquation (16.23) by using H instead of the electric field E, Y instead of 1/a, andβμ i instead of d. This gives usε = σ Y + βμ i H (16.39)for a rod undergoing simultaneously a tensile stress σ and a magnetic field H. Theanalogy extended to Equation (16.20) yieldsB = σβμ i + μ i H (16.40)Clamping the rod causes strain ε to be zero in Equation (16.38), yieldingσ = Yβμ i H = Bwhere σ denotes the compressive stress and = Yβ represents the magnetostrictivestress constant (given in units of newton/weber). The reciprocal of , preferredby some authors, is called the piezomagnetic constant (units of weber/newton).

16.6 Transducer Arrays 46916.6 Transducer ArraysA single-element ultrasound transducer tends to radiate a rather narrow beam orreceive signals over a narrow spatial range. In order to cover a wider area througha process called scanning, and, in many instances, to emit more powerful signalsthan is possible with a single element, an especially arranged group of transducersor arrays are used to extend the versatility of transducers.Array transducers are also used to focus an acoustic beam. Variable delays areapplied across the transducer aperture. The delays are electronically controlled ina sequential or phased array and can be changed instantaneously to focus the beamon different areas.With linear-array transducers, which are far more versatile than piston transducers,the electronic scanning entails no moving parts, and the focal point canbe changed readily to any position in the scanning plane. A broad variety of scanformats can be generated, and received echoes can be processed for other applicationssuch as dynamic receive focusing, correction for phase aberrations, andsynthetic aperture imaging. The principal disadvantages of linear arrays obviouslylie in the greater complexity and increased costs of the transducers and scanners.In order to ensure high quality imaging, many (as high as 128 and on theincrease) identical array elements are required. Each array element tends to beless than 1 mm on one side and is connected to its own transmitter and receivingelectronics.Phased Arrays: Focusing and SteeringWe examine how a phase-array transducer can focus and steer an acoustic beamalong a specified direction. An ultrasound image is created by repeating the scanningprocess more than a hundred times to probe a two-dimensional (2D) or athree-dimensional (3D) locale in the medium. In Figure 16.7(a), a simple sixelementarray is shown focusing the transmitted beam. Each array element maybe considered a point source that radiates a spherical wavefront into the medium.Because the topmost element is the farthest away from the focus in this example, itis activated first. The other elements are triggered progressively at the appropriatetime so that the acoustic signals from all the elements reach the focal point simultaneously.According to Huygens’ principle, the resultant acoustic signal constitutesthe sum of the signals that came from the source. The contributions from eachelement add in-phase at the focal point to yield a peak in the acoustic signal. Elsewhere,some of the contributions add out-of-phase, lessening the signal relative tothe peak.On receiving an ultrasound echo, the phased array works in reverse. An echois shown in Figure 16.7(b) originating from receive focus 1. The echo is incidenton each array element at a different time interval. The received signals undergoelectronic delay so that they add in phase for an echo originating at a focal point.Echoes originating elsewhere have some of their signals adding out of phase,thereby reducing the receive signal relative to the peak at focus.

470 16. UltrasonicsFigure 16.7. The phased array illustrated above provides for steering and focusing of anultrasonic beam. In (a) the six-element linear array is shown in the transmit mode. In thereceive mode of (b), dynamic focusing allows the scanner focus to track the returningechoes.In the receive mode, a dynamic adjustment can be made with a focal point so thatit coincides with the range of returning echoes. After the transmission of a pulse,the initial echoes return from targets nearest the transducer. The scanner thereforefocuses the phased array on these targets, located at focus point 1 as shown inFigure 16.7(b). As echoes return from the more distant targets, the scanner focusesat a greater depth (e.g., focus point 2). Focal zones are achieved with adequatedepth of field so that the targets always remain in focus in receive mode. This isthe dynamic receive focusing process.Array-Element ConfigurationsThe dynamic receive focusing process is repeated many times to form an ultrasonicimage in the scan of a 2D or 3D region of tissue. In defining the 2D image, thescanning plane is the azimuth dimension; the elevation dimension is normal tothe azimuth scanning plane. In linear sequential arrays, as many as 512 elementsconstitute a sequential linear array in currently available scanners. A subaperturecontaining as many as 128 elements is selected to function at a given time.In Figure 16.8(a), the scanning lines are directed perpendicularly to the transducerface; the acoustic beams is focused but not steered. The advantage is that

16.6 Transducer Arrays 471Figure 16.8. Various configurations of array elements and the corresponding regionsscanned by the acoustic beam: (a) sequential linear array scanning a rectangular region;(b) curvilinear array scanning a sectored region; (c) linear-phased array sweeping a sectoredregion; (d) 1.5D array scanning a sectored region; and (e) 2D array sweeping a pyramidalregion.elements have high sensitivity when beam is directed straight out. The disadvantageis that the field of view is limited to the rectangular region directly facingthe transducer. Linear-array transducers also require a large footprint to obtain asufficiently wide field of view. Another type of array configuration is that of the

472 16. Ultrasonicscurvilinear array. Because of its convex shape [Figure 16.8(b)], the curvilinear (orconvex) array scans a wider field of view than does a linear-array configuration.The curvilinear array operates in the same manner as the linear array in that thescan lines are directed normal to the transducer face. In the linear-phased array ofFigure 16.8(c), which may contain as many as 128 elements, each element is usedto emit and receive each line of data. In Figure 16.8(c), the scanner steers the beamthrough a sector-shaped region in the azimuth plane. These phased arrays can scana region considerably wider than the footprint of the transducer, thus renderingthem suitable for scanning through acoustically restricted windows. This is idealfor use in cardiac imaging, where the transducer must scan through a small windowin order to avoid obstruction by ribs and lungs.The 1.5D array is structurally similar to a 2D array but operates as 1D. The1.5D array consists of elements along both the azimuth and elevation directions.Dynamic focusing and phase correction can be implemented in both dimensionsto enhance image quality. Steering is not possible in the elevation dimension sincea 1.5D array contains a fairly limited number of elements in elevation (usually3 to 9 elements). Figure 16.8(d) shows a B-scan conducted with a 1.5D phasedarray. It is also possible to use linear sequential scanning with 1.5D arrays. In the2D phased array, a large number of elements are employed in both the azimuthand elevation. This permits focusing and steering of the acoustic beam along bothdimensions. With the application of parallel receive processing, a 2D array canscan a pyramidal volume in real time to yield a volumetric image, as illustrated inFigure 16.8(e).Linear-Array Transducer SpecificationsIn the design or selection of an ultrasound transducer, a number of compromisesare entailed. The ideal transducers have high sensitivity and SNR (signal-to-noiseratio), excellent spatial resolution and freedom from spurious signals. Additionallyan individual array element should possess wide angular response in steeringdimensions, low cross-coupling with another element, and an electrical impedancematching the transmitter.16.7 Basic Instrumentation: Scanning MethodsFigure 16.9 illustrates a “generic ultrasonic instrument” in block diagram format.The synchronization generator establishes a repetition rate (typically 1,000 repetitionsper second) for a pulse input and a display module. The pulser provideselectrical energy to the transducer through a pulse limiter, which clips the voltageexperienced by the amplifiers down to a tolerable threshold but allows the fullvoltage to impact the transducer. The transducer generates the ultrasonic wavesand receives the echoes. The resulting voltage of the echoes, not clipped by thepulse limiter because their voltages are usually below the clipping threshold, goeson directly to the amplifier and then onward to a display. The computer (whichmay or may not be present) acts to process algorithms.

16.7 Basic Instrumentation: Scanning Methods 473SYNCGENERATORPULSERPULSELIMITERAMPLIFIERDISPLAYTRANSDUCERCOMPUTERWORKPIECEBEAMBACK FACEA-ScanFigure 16.9. Block diagram of a “generic ultrasonic instrument.”In a flaw-detection instrument the display may be an oscilloscope with limitedadaptability. The display, shown in Figure 16.10(a) resulted from a well-dampedtransducer. If the transducers incorporate less damping or if they are electronicallytuned, the echoes will contain more RF (radio frequency) cycles. In generalterms, the result is rectified and detected prior to the display stage, as shown inFigure 16.10(b). This results in the A-scan, which is simply amplitude versus timefor the echoes falling in the range of the transducer.Analog processes are incorporated in commercial flaw-detection instruments toanalyze data. In addition to the rectification of detection of the echoes describedabove, another process is that of a baseline suppressor that eliminates a few decibelsof the amplitude just above zero amplitude. This process serves to emphasizethe more significant flaw echoes, which are now easier to ascertain. A third processconstitutes the range gate that selects a time range from t 1 to t 2 along the A-scanbaseline. The signals within the range gate may be either (a) suppressed or (b) analyzedto the exclusion of all other signals. A computer, an internal alarm trigger, oran external analog instrument may execute the analysis. An alarm may be activatedif echoes fall within the range gate exceeds a certain amplitude or threshold.The A-scan procedure is also used to measure gauge thickness on the basis ofthe known velocity of the pulse inside the workpiece and the time it takes for anecho to be received from the backside of the workpiece.

474 16. Ultrasonics(a)amplitudeinput pulseflowbackface(b)timeFigure 16.10. A-scan display for the generic ultrasonic instrument of Figure 16.9. Theamplitude is shown presented as RF in (a) pr as rectified and detected in (b), versus time,which is proportional to the distance into the workpiece.B-ScanBy introducing a mechanism to physically move a transducer over a workpiece inan appropriate manner, a 2D electronic or electromechanical display can provide apicture of a slice of the interior of the workpiece. Figure 16.11 shows a schematicof a scanning mechanism in an immersion tank. The transducer moves in the

16.7 Basic Instrumentation: Scanning Methods 475TRANSDUCERxyzR 2R 3WORKPIECER 1Figure 16.11. Schematic of a scanning mechanism in an immersion tank. The transducercan be translated and angulated to aim at a target.x- and y-directions while propagating signals in the z-direction (travel time isalong the z-direction).A B-scan portrays an x–z slice of the workpiece at a specific position y i and thenmakes another x–z survey at y z+1 , and so on. The transducer travels along x at arate allowing the ultrasonic pulse to interrogate the workpiece many times over thex-direction of the workpiece. The pulse repetition rate must be kept sufficiently

476 16. Ultrasonicsslow to allow the previous pulse to drop below the noise level before the next pulseis sent. One of the coordinates on the display is the position x B representing thelocation the transducer was at the time of arrival t B of a refection from a reflectorR (R 1 or R 2 or R 3 ) as shown in Figure 16.11, depending on whether the beamin the slice y passes over them. It follows that t B represents the other coordinateof the point (x B , t B ). If the display is assigned a gray scale, the brightness ofthe display corresponds to the amplitude of the reflection. In order to establish an“on-off” scale, a threshold may be assigned to that only echoes exceeding a certainamplitude are displayed. The B-scan procedure is extensively used in medicalultrasonics, especially in obstetrics.C-ScanThe C-scan procedure differs from B-scan in that, while the transducer scansalong the x-axis in consecutive steps y i , the coordinate on the display is (x, y).The display is nonzero only if a reflecting surface R lies within the beam rangealong z centered at (x, y). This configuration would ordinarily result in the echoesfrom the front and back surfaces of the workpiece swamping the display, but theyare disallowed by setting up a range gate to accept and transmit echoes only ifthey arise from regions below the top surface and above the bottom surface of theworkpiece. It is also apparent that the range gate can be narrowed to accept signalsfrom a slice thinner in the z-direction. A threshold may also be prescribed withinthe gate so that only signals above a set amplitude will be displayed. A gray scalecan also be assigned.In both the B-scan and A-scan procedures, a computer can be utilized for datastorage and image enhancement. Color monitors can be useful for pseduocolorgray scales. 2ReferencesAtchley, A. A. and Crum, L.A. 1988. Acoustic cavitation and bubble dynamics. In: Ultrasound:Its Chemical, Physical and Biological Effects. Suslink, K. S. (ed.). New York:VCH Publishers, pp. 1–64.Bardeen, J. B., Cooper, L. N., and Schrieffer, J. R. 1957. Theory of superconductivity.Physics Review 108:1175.Blake, J. R., Boulton-Stone, J. M., and Thomas, N. H. (eds.). 1994. Bubble Dynamics andInterface Phenomena. Dordrecht, the Netherlands: Kluwer.Blitz, Jack. 1967. Fundamentals of Ultrasonics, 2nd ed. New York: Plenum Press. (A bitdated, but this remains a very clear exposition of the ultrasonic field.)Brown, B. and Goodman, J. E. 1965. High Intensity Ultrasonics. London: Iliffe.Crum, L. A. 1994. Sonoluminescence. Physics Today. September: pp. 22–29.2 “Color” displays for ultrasound analyses do not represent optical colors of the ultrasound targets, perse. Rather, they indicate values assigned to specific parameters of ultrasound reflections.

Problems for Chapter 16 477Ensminger, Dale. 1988. Ultrasonics: Fundamental, Technology, Applications, 2nd ed. NewYork: Marcel Dekker, Inc. (A comprehensive, well-done modern text giving an overviewof the field of ultrasonics.)Frenzel, H. and Schültes, Z. 1934. Physical Chemistry 27B:421.Gaitan, D. F. and Crum, L. A. 1990. In: Frontiers of Nonlinear Acoustics. Hamilton, M.and Blackstock, D. R. (eds.). New York: Elsevier Applied Science, pp. 49–463.Gaitan, D. F., Crum, L. A., Roy, A., and Church, C. C. 1992. Journal of the AcousticalSociety of America 91:3166.Herzfeld, Karl F. and Litovitz, Theodore A. 1959. Absorption and Dispersion of UltrasonicWaves. New York: Academic Press. (A classic reference on ultrasonic phenomena.)Hunter, T. F. and Bolt, R. G. 1955. Sonics. New York: John Wiley & Sons, p. 234.Landau, Lev and Teller, Edward. 1936. Zur Theorie der Schaledispersion. Physik ZeitschriftSowjet Union 10:34.Leighton, T. G. 1994. The Acoustic Bubble. London: Academic Press.Maeda, K. and Ide, M. 1986. IEEE Transactions: Ultrasonics, Ferroelectrics, and FrequencyControl, UFFX-33(2):179–185.Mason, Warren P. (ed.). 1964. Physical Acoustics—Principles and Methods, Vol. 1, PartA. New York: Academic Press, Chapters 3–5.Morse, R. W. 1959. In: Progress on Cryogenics. Mendelssohn, K., (ed.), Vol. I. London:Heywood.Neppiras, E. A. 1973. The prestressed piezoelectric sandwich transducer. Proceedings ofthe International Ultrasonics Conference: pp. 295–300.Shields, F. D. 1962. thermal relaxation in fluorine. Journal of the Acoustical Society ofAmerica 34(3): 271.Sternberg, M. S. 1958. Physics Review 110:772.Suslick, Kenneth S. and Crum, Lawrence A. 1997. Sonochemistry and sonoluminescence.In: Encyclopedia of Acoustics. Crocker, Malcolm J. (ed.), Vol. 1. New York: John Wiley& Sons, Chapter 26, pp. 71–281.Winter, T. G. and Hill, G. L. 1967. High temperature ultrasonic measurements of rotationalrelaxation in hydrogen, deuterium, nitrogen, and oxygen. Journal of the AcousticalSociety of America 42(4): 848.Zuckerwar, A. J. and Griffin, W. A. 1980. Resonant tube for measurement of sound absorptionof gases at low frequency/pressure ratios. Journal of the Acoustical Society ofAmerica 68(1):218.Problems for Chapter 161. For a frequency of 0.1 MHz, find the threshold frequency in gas-free water forthe onset of cavitation. Compare with the value of the threshold for aeratedwater subjected to the same frequency.2. At a hydrostatic pressure of 135 μPa and a 35 kHz sinusoidal signal, find thecritical radius for cavitation to occur. Assume a value of 1.4 for the ratio ofspecific heats and a surface tension of 75 dynes/cm.3. At a magnetic field of 500 oersteds, find the mechanical strain of nickel.4. At the magnetic field of 500 oersteds, find the mechanical strain of permendur.What is the major difference between the mechanical strain of this problem andthat of Problem 3?

478 16. Ultrasonics5. In the cases of Problems 3 and 4 for a crystal of 20 mm length, what will be therespective changes in the length of the crystals fabricated from the materials ofthe last two problems?6. Compare the mechanical strains of ferrous ferrite with that of 12 Alter for amagnetic polarization of 20 weber m 2 .7. Given a magnetostrictive material with a known Young’s modulus of 110 GPaand a density of 7650 kg/m 3 :(a) what is the velocity of sound in that material?(b) find the fundamental frequency for magentostriction for a length of 50 mm.8. Why is A-scanning inadequate for a sonogram of a fetus?

17Commercial and MedicalUltrasound Applications17.1 The Growth of Ultrasonic ApplicationsIt is recognized among technologists that the field of ultrasonics is still in its infancy,and many new uses for ultrasound will continue to accrue in commercialand medical fields. Ultrasound is proving its continued worth in diagnostics, as it isgenerally nondestructive at lower intensities and does not engender the deleteriouseffects on materials and living tissues that X-rays do. At higher levels of operation,ultrasound is used in manufacturing operations in cleaning, cutting, and weldingmaterials and to provide therapeutic treatment and alleviate the trauma of surgeryby making possible increased precision in incisions, targeting of tumors and infectedtissues for selective ablation, and stanching bleeding. Improvements havebeen made and are still continuing; as the results of better imaging, more versatiletransducer arrays, and more powerful analytical algorithms are rendered possibleby more powerful computers and larger data storage systems. In this chapter asurvey is conducted of the uses of ultrasound in manufacturing and agriculturalprocesses, quality control, and in medical (including dentistry) diagnostics andtherapeutic procedures. In addition to manufacturing, the petroleum, chemical,and pharmaceutical sectors also make use of ultrasound to monitor and facilitateproduction. The safety of ultrasound also constitutes an important topic in thissection.17.2 Industrial Applications of UltrasoundUltrasonic CleaningUltrasonic cleaning is the oldest industrial application of power ultrasound. Applicationsspan a wide variety of industries ranging from castings to semiconductors.Ultrasonic cleaning works best on relatively hard materials such as metals, glass,ceramics, and plastics, which reflect rather than absorb sound.Both cavitation and the agitation of the fluid by the waves are entailed inthe process of ultrasonic cleaning. At lower frequencies, cavitation acts as the479

480 17. Commercial and Medical Ultrasound ApplicationsFigure 17.1. Conveyor belt for ultrasonic cleaning.principal agent but at higher frequencies the cleaning effect occurs mainly fromagitation. Most cleaning applications are executed in the frequency range of 20–50 kHz, where cavitation effects occur more strongly. Either piezoelectric or magnetostrictivesources are used. The workpiece being cleaned is immersed in atank containing a liquid selected on the basis of its susceptibility to cavitation,its detergent properties, ability to degrease, and so on. Trichlroethylene and cyclohexaneare among the more satisfactory fluids used for ultrasonic cleaning.Standard ultrasonic cleaners range in power from 100 to several thousand wattswith corresponding tanks capacities of 4–160 liters. Multi-kilowatt special systemswith tank capacities of several hundred liters are not uncommon. In recentyears, low-power, inexpensive (below $100) ultrasonic cleaners have been madeavailable, thereby rendering ultrasonic cleaning accessible to small shops andlaboratories.Ultrasound cleaning lends itself to continuous processing in which a series ofworkpieces can be transported on a conveyor belt through a series of processesin separate tanks as shown in Figure 17.1. Ultrasonic cleaning has supersededother older usual methods of cleaning, particularly when these methods are ineffectiveand liable to cause damage. Applications include the removal of lappingpaste from lenses without scratching after grinding, the flushout of greaseand machining particulates from otherwise inaccessible small crevices in enginecomponents, removal of blood and other organic material from surgical instrumentsafter use, and so on. Very delicate parts that can be damaged by cavitationare cleaned by wave agitation at much higher frequencies, from 100 kHz to1 MHz.In general, modern ultrasonic cleaners employ solid-state electronic power suppliesincorporating automatic tuning; thus they do not require operator’s attention.A problem prevalent with all ultrasonic cleaners is the gradual deterioration of thetank due to cavitation erosion. This depends principally on the application, andwell-designed systems can provide years of satisfactory service.

17.2 Industrial Applications of Ultrasound 481Flaw Detection and Thickness MeasurementsA method of nondestructive testing, the pulse technique, is used extensively todetermine the propagation constants of solids, particularly in the MHz frequencyrange. This method consists of sending a short train of sound waves through amedium to a receiver. In the transmission mode of the pulse technique, the receiveris placed at a measured distance from the source. In the echo mode, a reversibletransducer acts as both source and receiver, with a reflector used to reflect thepulses. The speed of sound in a medium can be determined from the time of travelof the pulse over a given length of acoustic path. Longitudinal waves are generallyused. In gauging the thickness of a specimen, advantage is taken of the fact that abeam of pulses will reflect from the specimen surface opposite to the side of thereversible transducer.The use of the single-pulse method for flaw detection is fairly straightforwardwhen the specimen has two parallel surfaces and the defect is linear and roughlyparallel to these surfaces but not too close to a surface or another defect. If thepulse is followed on an oscilloscope and there is no defect present, two peaks,say, A and B, will appear on the screen. Peak A represents the instant of thetransmission of the pulse, and peak B that of its return after a simple echo. PeakB is referred to as the bottom echo. When a defect is present, a discontinuityof the characteristic impedance and some, or possibly all, of the sound energy isprematurely reflected back to the transducer. Another peak will then occur betweenA and B. The distance AC indicates the depth where the flaw exists and the heightof the peak C determines the extent of the defect.Figure 17.2 illustrates a schematic of a longitudinal wave probe used to detectflaws. A crystal transducer is normally used, and it is encased in a suitable housing.The crystal is mounted for heavy damping, which results in the propagation ofFigure 17.2. Longitudinal wave probe for detecting flaws.

482 17. Commercial and Medical Ultrasound Applicationssmall pulses to allow for greater accuracy in locating defects and better resolutionof neighboring defects from one another. The transducer is protected by a plasticcover that is coupled to the crystal with oil so as to prevent wear by friction betweenthe surfaces of the crystal and the material being tested. It would be ideal for thecharacteristic impedances of the transducer material, the material of the protectivecover, and the oil should be similar. Sometimes it is more desirable to apply theimmersion method, whereby both the specimen and the probe are immersed inwater, with the probe at a fixed distance from the upper surface of the sample. Anadditional peak that represents the echo from the interface between the immersionliquid and the sample will appear on the oscilloscope screen.When a defect is not parallel with a surface, it is preferable to use an angledprobe, which consists of a transducer mounted on a wedge. This enables the soundwaves to be incident normally to the defect, and a greater degree of sensitivity isthus achieved. Using variable-angled probes makes it easier to better gauge thedirection of defects that might otherwise remained undetected. For samples havingirregular shapes, two probes are used, one acting as a sender and the other servingas a receiver in order to locate the defects. Care should be taken that couplingbetween the transducers should involve the medium and nothing else.Flaws, such as those that occur in butt welds oriented at right angles to thesurface, are commonly detected through the procedure of forward scanning. Herea beam of transverse waves is propagated, after refraction at the boundary, in themedium at a shallow angle to the surface. In Figure 17.3(a) the transducer is shownFigure 17.3. Transducer mounted on plexiglass wedge and setup of transverse waveprobes for locating a defect in a specimen.

17.2 Industrial Applications of Ultrasound 483mounted on a Plexiglas wedge and the longitudinal waves are directed to the surfacewith an angle of incidence greater than the first critical angle (this is the angle ofincidence sufficiently large that the refracted ray is directed along the boundaryrather than into the medium). The figure shows that the wedge is shaped in such amanner that the longitudinal waves that are reflected at the surface become totallyabsorbed by subsequent reflections. In Figure 17.3(b) a similar probe is positionedat a suitable location to receive the waves from the defect after a reflection atthe base of the specimen. In a given substance, the transverse wave velocity isgenerally about half that of the longitudinal wave velocity, so the sensitivity of thismethodology is twice that for longitudinal procedures.Surface defects can be discerned through the means of surface waves. Theseare produced by a probe similar to that shown in Figure 17.3(a), but the incidentlongitudinal waves are directed to the surface at the second critical anglewhere the transverse waves are refracted at an angle of 90 ◦ (i.e., along the boundarysurface). Laminar defects that exist just below the surface, which are hardto detect by normal longitudinal wave methods, can be located by Lamb waves(Worlton, 1957). According to Lamb, a solid plate can resonate at an infinitenumber of frequencies. The portion of specimen between the surface and a laminationclose to it forms such a plate. If surface waves are directed toward thisplate, it will resonate and generate a signal that can show up on an oscilloscopescreen.Determination of Propagation Velocity and Attenuationthrough an InterferometerThe interferometer is a continuous wave device that can accurately measure velocityand attenuation in liquids and gases that can sustained standing waves. Itconsists of a fluid column that contains a fixed, air-backed piezoelectric transducerat one end and a moveable rigid reflector at the other end. A fixed frequencyis selected. The reflector is moved with respect to the transducer by a micrometeradjustment mechanism. As the reflector moves, the reflected waves becomeperiodically in and out of phase with the transmitted waves, as a result of the correspondingconstructive and destructive interference. The effect of the interferenceon the crystals influences the load impedance detected by electronic system. Theload current in the electronic amplifier fluctuates accordingly. The wavelength ofthe sound is established by the distance the micrometer moves the reflector overone cycle of load current fluctuation, with the distance between two successivemaxima being equal to half-wavelength λ/2.Optical interference methods also have been used in this fashion to accuratelymeasure the wavelengths of standing waves at high frequencies (near 1.0 MHzor above). An accuracy of 0.05% is typical for the interferometer, which dependson the quality of micrometer readings, the parallelism between transducer andreflector surfaces, and the accuracy of the frequency determination. The velocityof sound is found by simply multiplying the frequency by the wavelength. The

484 17. Commercial and Medical Ultrasound ApplicationsFigure 17.4. Magnetostrictive delay line.attenuation is found from the decay of the maxima of the periodic amplitude plottedas the function of the distance x between the transmitter and the reflector increases.Ultrasonic Delay LinesDelay lines are used to store electrical signals for finite time periods. These are usedin computers to store information to be extracted for a later stage of calculation. Amethod for generating the delay is to convert those signals into ultrasonic wavesthat then travel through a material to be reconverted into their original forms. Thesimplest ultrasonic delay line is a crystal transducer radiating into a column ofliquid, such as mercury, that terminates at a reflector. An adjustment of the delaytime can be effected by changing the position of the reflector relative to the crystal.As liquid delay devices are not always convenient to use, solid delay lines aremore common. For delay times of a few microseconds, only a few centimeters ofa solid rod or block is sufficient. The delay time may be doubled by using shearwaves instead of longitudinal waves. If even longer delay times are required, say,in the order of a few milliseconds, very long acoustic paths are necessary. Thesecan be achieved by using materials in the form of polygons in which large numbersof multiple reflections can occur. The solid delay line has the disadvantage thatthe delay time usually cannot be varied. However, the magnetostrictive delay line,illustrated in Figure 17.4, can be varied in length and unwanted reflections areavoided by coating the ends of the rods with grease, which completely absorbs thesound waves. The line may be a wire, a rod, or some sort of ribbon of ferromagneticmaterial such as nickel. An electrical signal applied to coil A induces through amagnetostricive effect a sound pulse in the line. The pulse travels along the lineand induces an electrical signal in coil B by the reverse magnetostrictive effect.The permanent magnets C and D provide the requisite polarization.Measuring Thicknesses through the Pulse TechniqueWhen the pulse technique is used for gauging thicknesses, better results areachieved through the use of a variable delay line. Two pulses are generated simultaneously.One pulse is sent through the sample and the other through a delay

17.2 Industrial Applications of Ultrasound 485line, which can be a length of nickel wire or a column of liquid terminating in areflector. The latter pulse is indicated on the oscilloscope by a trace following thatrepresenting the pulse passing through the sample. The delay line is adjusted inlength by means of a micrometer device until the two traces on the screen coincidepositionwise. The thickness of the specimen is derived from the predeterminedcalibration of the delay line.A major advantage of using ultrasound for thickness measurement is that accessto only one surface is needed. This is especially useful in measuring the extentof corrosion in infrastructures such as viaducts, sewers, gas pipes, and chemicalconduits. The thicknesses of ship hulls can be monitored at sea without resorting totaking sample borings, an expensive and tedious process that is not 100% effective.In the livestock industry, ultrasonic thickness measurement is used to measure theamount of fat on the bodies of live animals.Mechanical Stress MeasurementsWhen a solid material undergoes a change in mechanical stress, changes alsooccur in its elastic moduli and hence in its acoustic velocities (Shahbender, 1971).This method can also determine the variations of stress in real time. In orderto render possible the use of this procedure, a calibration curve is necessary toprovide the reference data, namely, plots of velocity (in terms of percentage) ina direction as a function of stress applied normal to that direction. Figure 17.5displays typical calibration curves for longitudinal waves and also for shear wavespolarized perpendicular to the direction of the stress and also polarized parallelto the direction of the stress. It can be inferred that the variations of the twocomponents of shear velocity with time are different. Thus, at any given timethe corresponding wave vectors will be out of phase with each other. This phaseFigure 17.5. Calibration curves for longitudinal waves.

486 17. Commercial and Medical Ultrasound Applicationsdifference φ is given byφ = Lωc 0[ (cc 0)n( ]c−c 0)pwhere ω denotes the angular frequency, L is the acoustic path length, c 0 is the shearwave velocity in the unstressed medium, and (c/c 0 ) n and (c/c 0 ) p representthe fractional shear velocity for polarizations in directions normal to and parallelwith the direction of stress, respectively. The value of L is found simply fromlongitudinal wave measurements while φ is obtained by the means of a suitablephase-shift network. This methodology is also applied to determine third-orderelastic constants.The Ultrasonic FlowmeterThe Doppler principle constitutes the operating basis of the ultrasonic flowmeter.Two reversible transducers are submerged in the liquid along the line of flow. Onetransducer acts as a signal source of ultrasonic pulses and the other acts as a receiver.At short regular intervals the roles of the transducers are reversed, so that the sourcebecomes the receiver and the receiver becomes the source. The wave velocities arec + u along the direction of the flow and c − u in the opposite direction, where crepresents the propagation velocity of sound in the fluid and u the velocity of thestreamline flow of the liquid. A number of techniques have been used to comparethe upstream and downstream propagation rates. A “sing-around” method uses apulse generator to produce a short train of ultrasonic waves. The received signal isamplified and used to retrigger the pulse generator. If we neglect delay times dueto the electronic system and the distance the pulse travels beyond the fluid stream,the difference between the downstream and upstream pulse repetition rates isf 1 − f 2 = 2udwhere d represents the distance between the two transducers.Another type of ultrasonic flowmeter is based upon the deflection of an ultrasonicbeam by the fluid flow (Dalke and Welkowitz, 1960). In Figure 17.6, a transmittingtransducer located on one side of the fluid conduit emits a continuous signal into thefluid stream. A split transducer on the other side of the pipe determines the amountof beam deflection. A differential amplifier is used to determine the difference inthe outputs of the two receiving transducers. If there is no flow, the beam fallsmidway between the two receiving sections, and the two sections generate equalvoltages, and the output from the differential amplifier will be zero. When fluid flowoccurs, the beam shifts in the direction to the flow by an amount corresponding tothe flow speed, and the outputs from the two sections differ. The difference voltagethen corresponds to the rate of flow. Assuming a constant flow rate across the pipe,the deflection φ of the beam is computed from(φ = tan −1 u)c

17.2 Industrial Applications of Ultrasound 487Figure 17.6. Schematic of a beam-deflection ultrasonic flowmeter.Ultrasonic meters are being used to measure flow rates of rivers, nuclear reactorheat-exchanger fluids, gas-containing emulsion, slurries, corrosive liquids, andwind velocities. These measurement devices carry the advantage that they introducenegligible pressure loss in a system and they are economical to operate andcan handle a wide range of flow rates, pipe diameters, and pressures.Resonance Method of Measuring Sound Propagation SpeedThe resonance method is similar to the interferometer method for measuring velocity,but it can be applied to solids as well as to fluids. This involves the use ofa fixed transducer and a fixed reflector or two transducers spaced a fixed distanceapart. The transducer is driven to sweep through a range of frequencies to determinesuccessive resonances. In a nondispersive medium, the difference betweentwo successive resonant frequencies equals the fundamental resonant frequencyof the medium, i.e.,c = 2 lfwhere l is the distance between the transducer and the reflecting surface and fdenotes the difference between successive resonant frequencies.Motion and Fire SensingOne of the few ultrasonic applications in open air is that of the motion and firesensor, which is restricted to the lower kilohertz range, where attenuation is notvery much. A magnetostrictive transducer placed at some point in a room emitspulses in all directions. The reflected signals from the walls and furniture areeventually picked up by a receiver, from which a constant indication is generated.Any variation in the sound field, caused by an intruder or a change in temperature,

488 17. Commercial and Medical Ultrasound Applicationsgives rise to a change in this indication, which triggers an alarm. One version ofthis device is a light switch that goes on automatically when someone enters anempty room and goes off automatically when nobody is present in the room aftera predetermined period of time.Working of Metals and PlasticsUltrasound have been successfully used in the treatment and working of metalsand plastics. As a molten metal is cooled, bubbles should be removed beforesolidification sets in. Otherwise defects will occur. Irradiation of the melt withultrasound sets the fluid particles in motion so that bubbles tend to coalesce and,when sufficiently large, tend to rise to the surface. As cooling progresses, crystalsbegin to form at the solidification temperature. The crystal growth depends onthe rate of cooling and the presence of specific impurities in the metal. The highenergy of the ultrasound-induced cavitation causes the crystals to be break up asthey form. The solid that is finally formed has a much finer grain structure than itwould have if cooling had taken place with the melt undisturbed.Ultrasonic machining is one of the first industrial applications of high-intensityultrasound. In one technique, a slurry consisting of abrasive particles suspended ina low-viscosity liquid flows over the end of a tool shaped in the desired geometryof the impression to be made. The axial motion of the tool tip and the ensuingcavitation impart high accelerations to the abrasive particles and thus erode awaythe material from the workpiece. This method is suitable for machining of brittleceramic materials as well as of powder-metal components. But in the latter application,there has been evidence of highly accelerated electrolytic erosion thatadversely affects the tool.Ultrasonic drills do not have transducers in immediate contact with the workpiece.An intermediate material is necessary, and this has the task of matching theimpedance. In order to achieve maximum velocity amplitude in the tool, a taperedrod that acts as an acoustic transformer in the same manner as a loudspeaker hornis used. The rod is made of a material that has a characteristic impedance matchingthat of the transducer to which it is rigidly fixed. It is exactly one wavelengthlong in order to achieve maximum transfer efficiency and clamped at a distance ofone-fourth wavelength from the transducer.The soldering of metals without any flux can be achieved through an ultrasonicsoldering iron illustrated in Figure 17.7. The soldering iron is similar in design toan electronic drill, but the tool at the end of the drill is replaced by the electricallyheated bit. The vibration of the bit produces cavitation in the solder, thus effectivelycleaning the surface of the workpiece and removing any oxide coating. Until fluxeswere developed for soldering aluminum, ultrasound had provided the only effectivemethod of soldering this metal. Ultrasound soldering is now being extensively usedfor soldering joints in miniaturized printed circuit boards.Ultrasonic welding is effected when two materials are pressed together in intimatecontact and the ultrasound vibrations produce shearing stresses at the interface,thereby generating a great deal of heat. This phenomenon lends itself well to

17.2 Industrial Applications of Ultrasound 489Figure 17.7. Ultrasound soldering iron.produce spot and seam welds in metals such as 304 and 321 stainless steel, aluminum,brass, copper, zirconium, titanium, gold, molybdenum-0.5% titanium, andplatinum. This type of welding can be applied to very small wires to seam weldingof metal plates up to 0.5 cm in thickness. Both similar and dissimilar metals canbe welded. While the formation of a thin molten film in the interface seems to bethe primary mechanism of bonding, there is some evidence of solid-state bondinginstigated by diffusion in the welding zone at low amplitudes and high clampingpressure.Bonding of thermoplastic materials 1 through ultrasound is even easier than formetals, since the necessary equipment is smaller and requires appreciably lesspower. Ultrasonic welding of thermoplastic materials has gained wide acceptance,particularly in situations where thick materials are joined together and the toxicityof adhesives must be avoided. In addition to bonding plastic parts, the same equipmentcan be used to insert metal parts in plastic pieces. A hole slightly smallerthan the metal is drilled or incorporated in a molding part and the metal is driveninto the hole ultrasonically. During the insertion the melted plastic encapsulatesthe metal piece and fills flutes, threads, undercuts, and so on.Ultrasound welding combines the ideal ingredients sought in modern manufacturing.The process is fast and clean, requires no consumables, does not callfor skilled operators, and lends itself to automation. It is used extensively in theautomotive industry of assembly of taillights dashboards, heater ducts, and othercomponents where plastics have replaced the traditional use of metal and glass.1 Plastics fall into two categories: thermoplastic and thermosetting. The former plastic type will soften atsufficiently elevated temperatures and reset upon cooling. Thermosetting plastics retain their rigidityat elevated temperatures.

490 17. Commercial and Medical Ultrasound ApplicationsIn contrast with ultrasonic cleaning, ultrasonic plastic welding requires muchgreater power densities, typically at hundreds of watts per square centimeter at theweld and at the contact of the tool with the workpiece. Modern plastic weldersoperate predominantly around 20 kHz at power outputs below 1 kW, lock automaticallyon the horn resonance, and keep vibrational amplitude constant for varyingmechanical loads.Commercial gear for ultrasonic metal welding was introduced in the late 1950sand originally used in the semiconductor industry. Later advances in equipmentdesign and the need for joining high-conductivity metals spurred greater interestin the process. Standard equipment now available can weld parts up to0.32 cm (1/8 in.) thick and larger, depending on the material and part configuration.Ultrasonic metal welds are characterized by low heat (the material doesnot melt as in the process of arc welding) and relatively low distortion. Becausewelding temperatures are well below the melting temperatures, embrittlementand formulation of high-resistance intermetallic compounds are avoided.The equipment for ultrasonic metal welding range from low power microbonders(used in semiconductor industry) operating between 40 and 60 kHz to machinesof several kilowatt output capacity operating between 10 and 20 kHz forwelding of larger parts. It is interesting to note that on high-conductivity materialsultrasonic welding can be over 20 times more energy efficient than resistancewelding. A5kWwelder may be equivalent to a 100-kVA resistancewelder.AgricultureOver the last decade, ultrasound technology has considerably impacted the meatindustry, with a commanding role in the rating of product value. Ultrasound is beingused to measure the thickness of fat layers in pigs and cattle as part of livestockmanagement. It is utilized to predict carcass traits such as fat cover, ribeye area,and intra-muscular fat (marbling) in live animals. Ultrasound also has been usedto improve the quality of homogenized milk. A related agricultural application ispest control that includes killing of insects.Extraction ProcessesHigh-intensity cavitation is widely used for biological cell disruption in researchand low volume processing. Near field cavitation breaks down cell walls, causingrelease of the cell contents into the surrounding fluid. This method is appliedto extract active antigens for making vaccines and as a general tool forstudying cell structure. Other extraction uses include the extraction of perfumefrom flowers, essential oils from hops, juices from fruits, and chemicals fromplants. The equipment is normally in the range of 100 –500 W, operating at around20 kHz. High amplitude horns are used to produce power densities of around80 W/cm 2 .

Atomization17.2 Industrial Applications of Ultrasound 491Ultrasonic atomizers can produce small droplets of predictable size. For a givenliquid, droplet size depends on atomizer frequency and it gets smaller with increasingfrequency. Ultrasonic nebulizers are widely used for medical inhalation andoperate between 1 and 3 MHz to produce droplets between 1 and 5 μm. Increasedcombustion engine efficiency and reduction of pollution have been made possibleby ultrasound atomization of fuel, operating between 20 and 200 kHz.Emulsification and Flow EnhancementThe principal advantage of ultrasonic emulsification lies in the ability to mix someimmiscible liquids without additives (surfactants). Liquid flow through porousmedia can be increased by ultrasound and can find use in filtering and impregnation.Ultrasonic ViscometerThe ideal liquid should not support a shear stress, but the fact is that liquids dohave viscosity that gives rise to shear waves. A viscoelastic liquid combiningthe attributes of both fluid and solid behaviors (which produces shear stresses) isdescribed by− ∂ε∂y = 1 η p y + 1 G ṗy (17.1)where p y represents a variable shear stress; ṗ y is the time rate of the shear stress, εis the fluid particle displacement, η is the viscosity coefficient; and G is the shearcoefficient. The solution to Equation (17.1) is( ) ∂εp y =−∂y0(1 − e tη/G)(17.2)Equation (17.2) indicates that a periodically varying shear produces a relaxationprocess characterized by a time constant τ = η/G. The associated relaxation frequencyis given byf 0 = 12πτ = G(17.3)2πηFrom Equation (17.2) the attenuation of shear waves in a viscous liquid at a givenfrequency decreases with increasing viscosity. The damping of a vibrating shearwave transducer submerged in the liquid is a function of the coefficient of viscosityof the liquid. One way of measuring this quantity is to apply a pulse toa Y-cut crystal or to a torsionally vibrating rod immersed in the liquid so thatit vibrates freely with damped harmonic motion. When the amplitude of the vibrationdrops to a predetermined level, another pulse is generated. The rate ofpulse repetition increases with damping and hence decreases with increase in viscosity.The device is calibrated by using liquids having known coefficients ofviscosity.

492 17. Commercial and Medical Ultrasound Applications17.3 Ultrasound ImagingThe application of ultrasound to imaging processes is extremely important in industryand in medicine. Because imaging entails low-intensity ultrasound energy,it provides a valuable nondestructive testing technique. Ultrasonic imaging, whichmay be defined as any technique of providing a visible display of the intensity andphase distributions in an acoustic field, falls into a number of categories: (1) theelectronic-acoustic imaging; (2) B-scanning, most commonly used in medical diagnosis;(3) C-scanning, widely used in nondestructive testing and inspection offlat and cylindrical surfaces; (4) liquid-surface-levitation presentations; (5) liquidcrystal display (LCD), photographic or similar display; (6) light-refractionmethods; and (7) acoustical holography.Electron-acoustic Image ConvertersThe concept of an electron-image converter originated by the Russian scientist S.Ya. Sokolov (1937) who envisioned a device, similar to a video camera tube, inwhich the photosensitive element is replaced by a pressure-sensitive piezoelectricplate. Secondary electrons are given off when the plate is struck by a scanningbeam of electrons. An ultrasound field aimed at the plate influences the electricalpotentials on the faces of the plate. These potentials are proportional to the impressedacoustic pressure, with the result that the electrical potentials modulatethe secondary emission of electrons. The secondary emission that results from theimpingement of an electron beam is a function of the velocity of the primary electronsand also of the plate material. The ratio of the number of secondary electronsleaving the plate to the number of electrons impinging on the plate is called thesecondary emission ratio. When the primary electron voltage is increased fromzero, the secondary emission rate also increases from zero, passes through a maximum,and then decreases. With some piezoelectric materials, the maximum ratioexceeds unity and there are two velocities, one above and one below the point ofmaximum secondary emission, at which the ratio becomes unity. In other materials,the maximum secondary emission ratio never exceeds unity.Smythe et al. (1953) improved on the Sokolov concept by developing anultrasonic-imaging camera that could be operated in either an amplitude-sensitivemode or a phase-sensitive mode. In the amplitude-sensitive mode, a relatively lowvoltage(160–200 V) scanning beam is used. The secondary emission ratio is lessthan zero, which allows the potential of the surface of the (quartz) piezoelectricplate to nearly equal that of the electron gun cathode, in absence of ultrasound.After a scan without ultrasound, the ultrasound is activated and the plate scannedagain. The electrons are distributed over the surface of the plate in direct ratio to thepiezoelectric voltage present at each point of the surface. A corresponding imagecharge forms on an anode located externally to the tube, and the video image isobtained directly from the anode charges. The ultrasound is then turned off and thenegative charge on the piezoelectric surface is removed by inundating the surface

17.3 Ultrasound Imaging 493with gas ions, a process which occurs in about 0.2 s. High-voltage scanning canalso be applied to remove the negative charge, followed by a low-voltage scanningto restore the surface back to the cathode potential, with the ultrasound turnedoff. This procedure removes the need for ion recharging and allows the use of ahigh-vacuum tube.For the phase-sensitive mode of operation, higher-voltage (600 V) beams areused in conjunction with an auxiliary grid on which the secondary emission currentis collected. The inner quartz surface is stabilized at approximately the same voltageas the auxiliary grid. The piezoelectric charge in the quartz plate tends towardneutralization and, consequently, the current to the quartz plate and to the collectorgrid fluctuates at the frequency of the ultrasonic waves. The output pressure at eachpoint on the plate is therefore proportional to the incident ultrasonic pressure.Other methods have been developed to reproduce the image on the piezoelectricplate in order to circumvent problems posed by high-velocity scanning-beam stabilizationof an insulating surface. One method entails the use of a photoemissivesurface deposited on the piezoelectric surface, with the scanning being executedby a moving beam of light. Mechanical methods also were developed, and theyproved to be appreciably more sensitive than scanning with an electron beam. Thesurface of a piezoelectric plate subjected to an ultrasonic field is scanned mechanicallyusing a capacitive, noncontacting electrode or by sliding a small electrodeover the surface itself.Jacobs (1962) introduced the use of electron multipliers in the camera tube ofultrasound image converters based on Sokolov’s method. A schematic diagramof Jacob’s device is shown in Figure 17.8. The construction provides shieldingfor the low-voltage circuits and virtually eliminates ground loops. The secondaryemission beams produced by the scanning beam are attracted toward a positivechargedelectrode in the electron-multiplier unit. The electron multiplier amplifiesFigure 17.8. Jacobs device for ultrasonic scanning.

494 17. Commercial and Medical Ultrasound Applicationsthe signal currents by about 100,000 times greater than the threshold values beforethey are fed into the associated amplifiers. Thus, the electron multiplier servesas a wide-band amplifier with good noise characteristics. Its output constitutesthe video signal that is processed through a conventional closed-circuit televisionsystem.Two sealed ultrasonic camera tubes were developed by Jacobs: one version incorporatesquartz crystals permanently sealed to the end of the tube and the featuresinterchangeable crystals which permit operation over the frequency range from 1to 15 MHz. Color also has been introduced to increase the sensitivity of the ultrasonicimage converter, particularly in view of the fact that the human eye is moresensitive to changes in color than to differential changes in display brightness. Thecolor display indicates both relative amplitude and phase of the ultrasonic signalundergoing analysis. In North America, the video method used is the NationalTelevision System Committee (NTSC) color broadcasting standard, and the operatingfrequency is 3.58 MHz. 2 A more dominant video broadcasting method is thePAL system that is used in the United Kingdom, Germany, Spain, Portugal, Italy,China, India, most of Africa, and the Middle East. Several distinguishing featuresof the PAL system are: (a) a better overall picture than NTSC because of the increasedamount of scan lines and (b) because color was part of the standard fromthe beginning, color consistency between stations and TVs is considerably better.There is a down side to PAL, however, as there are fewer frames (25) displayedper second, compared with nearly 30 frames per second of the NTSC system.Acoustic LensAcoustic lens are necessary in the use of electron-acoustic image converters.Acoustic lens can (a) increase the sensitivity of an imaging system through energyconcentration and (b) provide coverage of a larger area by concentrating the imageon the receiving piezoelectric element of the image converter in almost the samefashion as an optical lens reduces a larger picture into a smaller area. For optimalperformance, the velocity of sound in the lens material must differ considerablyfrom that of the surrounding media, and the reflection of energy at the boundarybetween the lens and the surrounding should be minimal. The latter condition isfulfilled automatically when the acoustic impedances of the lens materials andthe surrounding medium match each other. Liquid lens of carbon tetrachloride orchloroform have the same acoustic impedance as water, but their toxicity generallyprecludes their use in industry. Plastic lenses also have been developed, butthe sound propagation velocity in these materials exceeds that of water and theypresent an impedance mismatch between the lenses and water, but not to the degreeof rendering them useless. The relatively high absorption of plastics limitstheir use to frequencies less than 15 MHz. Metallic lenses, which can be made ofaluminum and other metals, possess low-absorption characteristics, so they can2 In the year 2006 analog NTSC television broadcasts are scheduled to cease in the United States infavor of the ATSC Digital Television Standard. This should result in higher definition video imaging.

17.3 Ultrasound Imaging 495be used at frequencies exceeding 15 MHz, but the impedance mismatch betweenmetals and water essentially prevents their effective use.The velocity of sound in solids is higher than in liquids, so solid concave lensesare convergent and solid convex lenses are divergent—unlike the case of lighttraveling in a vacuum or a gas through optical lenses.Schlieren ImagingSchlieren imaging has been used for many years as a tool to visualize sound fields.Advantage is taken of the physical fact that pressure gradients in an ultrasonic wavecause density gradients in the medium. When a light beam passes through thesegradients, it becomes refracted. The refracted light is used in schlieren apparatusto produce an image of the sound field. Either of the two methods can be used toproduce the sound field image: (a) interruption of the refracted part of the beamin order to remove it from the beam and focusing the remainder of the field ontoan image detector (e.g., a ground glass screen or camera film), or (b) focusingthe refracted rays on the image plane and eliminating the remainder of the beamof light from the image. The result of the first method is a dark image on a lightbackground, and that of the second method is a light image on a dark background.A schlieren device for direct viewing during nondestructive testing is schematicallyillustrated in Figure 17.9. The image is viewed directly on a ground plateglass screen. At the right-hand side of Figure 17.9, the lamp (e.g., zirconium arc),condenser lenses, a filter, first knife edge, and first collimator are mounted on asingle 1-m optical bench. The second collimator, second knife edge, and camerasystem are mounted on another 1-m optical bench. The two benches themselves arepositioned on a 0.25 m wide, 5.75 m long steel girder. The water tank is mountedapart from the optical system so that any movement or changes in the tank will notcause the optical system to go out of alignment.The optical system must be mechanically isolated from all the other parts of theapparatus. When the device is being used, the image of the light source is centeredon the first knife edge, and then the collimating lenses, tank windows, second knifeedge, camera lenses, and viewing screen are centered in the light beam. CenteringFigure 17.9. Schlieren device for direct viewing in nondestructive testing.

496 17. Commercial and Medical Ultrasound Applicationsof the optics must be precise to ensure maximum sensitivity and uniform field.Through careful alignment and focusing, the field can be made to go from lightto gray to dark uniformly by adjusting the second knife edge further into the lightbeam until it intercepts it completely. In the range of gray settings, convectioncurrents both in air and water become clearly visible. The ultrasonic field is bestrendered visible when the second knife edge is set to intercept all of the main beam,thus allowing only light refracted in the ultrasonic field to pass and illuminate theviewing plane. The optical system may also be adjusted for bright-field operation,in which situation the ultrasonic perturbations appear as dark shadows.Color schlieren photography is useful in the study of ultrasonic waves and shockwaves. Color can indicate the various pressure levels in an ultrasonic field. Onemethod of color schlieren photography uses a spectroscopic prism between a slitlocated at the position of the first knife in Figure 17.9 and the first collimating lens.This method produces colors ranging from red or from blue to green. Anothermethod developed by Waddell and Waddell (1970) produces a complete colorspectrum. They eliminated the spectroscopic prism and used a color-filter matrixin place of the second knife edge and a vertical slit in place of the first knife edge.Illumination was provided by a high-pressure mercury arc lamp. The color matrixconsisted of three filters, which represent the primary colors.Liquid Crystal ImagingLiquid crystals exhibit properties of solid crystals that are not apparent in ordinaryliquids. When a stress is placed upon a liquid crystal, its optical properties change.A certain class of liquid crystals known as nematic crystals is used to indicate thepresence of an ultrasonic field with sensitivity equal to that of schlieren systems,and with high resolution, large-area capability, and handling ease.Ultrasonic HolographyHolography is a form of three-dimensional imaging that was conceived and developedby Dennis Gabor who received the Nobel Prize in physics for his efforts.Gabor applied his discovery to electron microscopy to overcome the problem ofcorrecting spherical aberration of electronic lenses. The principle of holography isas follows: a diffraction diagram of an object is taken with coherent illuminationand a coherent background is added to the diffracted wave. A photograph so takenwill contain the full information on the changes sustained by the illuminating wavein traversing the object. The object can be reconstructed from this diagram by removingthe object and illuminating the photograph by the coherent backgroundalone. The wave emerging from the photograph will contain a reconstruction ofthe original wave, which seems to issue from the object. A hologram, therefore, isa recording or a photograph of two or more coherent waves. If one recorded waveis from an illuminated object and another is a reference wave, simply illuminatingthe hologram with the reference wave reconstructs twin images of the originalobject, thus giving the illusion of three dimensions.

17.4 Medical Uses of Ultrasound 497Figure 17.10. Ultrasonic holography making use of liquid levitation.The wave used in the reconstruction does not have to be the original, and thisallows the use of ultrasonic waves and the subsequent reconstruction of the imageusing light from a laser. But the size of the image changes in proportion to theratio of the wavelength of the reconstructing wave to the wavelength of the originalilluminating wave. There are a number of methods of making acoustical holograms.One method as shown in Figure 17.10 uses liquid levitation, where the acousticwaves from below the surface of a liquid forms an ultrasonic image at the surfaceof the liquid that can be rendered visible on a photographic plate. The referencebeam may be obtained either by reflecting a portion of the irradiating beam ontothe surface or by generating a separate wave through a second transducer. Theheight of the bulge of the liquid surface due to the impetus of an acoustic wave iscritical; it should be small compared to the wavelength of the light.17.4 Medical Uses of UltrasoundUltrasound use in the medical arts can be classified as being diagnostic or therapeutic.While the diagnostic procedures involving ultrasound have been in usefor a number of years, ultrasonic therapeutics constitutes a newer, rapidly growingdomain. The ultrasound frequencies used in medical applications range from approximately25 kHz used in dental plaque removal to the megahertz range that isrequired for medical imaging. The merits of ultrasound diagnostics include safety,convenience, and capability of detecting medical conditions to which X-rays andother means of diagnosis are insensitive. Because ultrasound is much safer than

498 17. Commercial and Medical Ultrasound ApplicationsX-rays, it is used in fetal monitoring, detection of aneurysms, and echocardiography.The production of heat in the body through ultrasound is applied for its therapeuticvalue. The selectively greater absorption of ultrasound in cancerous tissueshas proven its usefulness in hypothermic treatment of cancer. Emerging developmentsinclude the use of ultrasonic waves to perform noninvasive or “bloodless”surgery, stop internal bleeding in trauma patients, and control delivery of drugs orother compounds. Enormous advances in electronic miniaturization are resultingin fairly compact handheld diagnostic units.Diagnostic Uses of UltrasoundDiagnostic medical applications are based on the imaging procedures described inSection 16.7. One diagnostic technique is based on the pulse method and seconddiagnostic technique is based on the Doppler effect where the reflected wave isshifted in frequency from that of the incident wave impinging on a moving target. Inthe reflection or pulse-type equipment, A-scan, B-scan, or a combination of thesetwo methods are utilized to present data on an oscilloscope display or processthe data for permanent record. The A-scan presents echo amplitude and distance,and it is used principally in echoencephalography for the detection of midlineshifts traceable to tumors or concussions. It has also been applied in obstetrics,gynecology, and ophthalmology in conjunction with B-scanning techniques. InB-scanning, the radar/sonar techniques of data processing are applied to synthesizethe reflected signals into a pattern on the oscilloscope display that corresponds toa cross section of the region scanned lying in a plane parallel to the direction ofbeam propagation.The position of the probe is synchronized with the sweep of one of the axes of theoscilloscope, and the echo amplitude appears as a spot with a specific intensity at aposition on the screen corresponding to the position of the plane causing the echo.The TM-mode (or M-mode) is a diagnostic ultrasound representation of temporalchanges in echoes in which the depth of echo-producing interfaces is displayedalong one axis, and time (T) is displayed along the second axis, thus recordingmotion of the interfaces toward and away from the transducer.In order to resolve small structural details, the transmitted pulse should be asshort as possible, which means that the transducer must be highly damped. In orderto promote good coupling between the transducer and the body, a film of oil orgrease is applied at the selected spot, and care must be taken that the contactingfilm is free of air bubbles. In immersion procedures (such as that for kidney stonepulverization), coupling is effected through a bath of liquid, usually water. Theimpedance match between water and soft tissue is good, and little energy is lost inirradiating soft tissue. But the match between water and bone is poor and also theattenuation in bone is high. The result in echoencephalography is that considerableenergy is lost, even where almost immediate contact is made with the skull bone.Ultrasound densitometry is used to measure bone density in the heel, shinbone,or kneecap. It is used as a screening tool and while currently not as precise asabsorptiometry techniques, it is still effective, inexpensive, portable and uses no

17.4 Medical Uses of Ultrasound 499radiation. This technique is primarily used as a screening tool to predict bonefracture risk or to determine the need for single-energy X-ray absorptiometry(SXA) or peripheral dual-energy X-ray absorptiometry (pDEXA) to ascertain thepresence of osteoporosis (degeneration or decalcification of the bone).Ultrasound imaging is being extensively used to study cardiac functions. Oneuse of the A-scan technique is to monitor for early signs of rejection following aheart transplant. As the heart fills at the onset of rejection, the muscle walls swelland stiffen. Measurements are made using a 2-cm diameter, 2.25 MHz transducerat a pulse repetition rate of 1000/s. Echo indications from the anterior wall and theposterior wall supply the measurement information, as the distance between thetwo walls indicates overall heart size. Echocardiography is useful in diagnosingpericardial effusion (escape of fluid from a rupture in the pericardium) becausethe echo received from the posterior wall of the heart is split when the transduceris located on the anterior chest surface.Echocardiography is also useful in assessing the degree of stenosis (narrowingof opening) in mitral valves. The transducer is aimed at the anterior mitral leafletand the echo signal can record on a strip chart so that an upward movement of therecorder pen corresponds to a movement toward the transducer and a downwardmovement corresponds to a movement away from the transducer. The slope of thetracing indicates the velocity of motion. The degree of stenosis affects the bloodflow rate through the opening. A number of symptoms can be discerned with theuse of this method; for example, rigidity or calcification of the mitral valve isindicated by decrease in the total amplitude of the anterior mitral leaflet betweenthe closed position during ventricle systole and the position of maximum openingin early diastole.Intercardiac scanning is used to obtain plan-position (C-scan) displays for theinterior of the heart. A tiny probe is inserted into the right atrium through the externaljugular vein or the femoral vein. An advanced version of the probe consists ofmany elements so that scanning can be achieved by sector techniques in which thepositions of points on the recorded image are correlated with beam direction. Thedata from the transducer can be processed by computer and displayed. A-scanningmay be combined with C-scanning to provide 3D information. The motion of selectedregions of the heart can even be viewed on a video display, particular withthe use of focusing transducers, which can provide high-resolution images of theheart.A tomographic method of observing interior structures of the heart threedimensionallyis based on a stereoscopic display of two-dimensional images. Theultrasonic device functions in synchronism with the cardiac cycle to obtain phasespecifictomograms, which then can be displayed on a storage CRT. Tomographicsystems are also used to investigate other internal organs such as right ventricle,atrium, and kidney cysts that have irregular shapes.Accurate diagnosis is rendered possible in the field of ophthalmology throughthe use of ultrasound to diagnose conditions existing in the soft tissues of the orbit ofthe light-opaque portions of the eye. Focused transducers are used with frequencytypically being at 15 MHz. This method can outline tumors, and detached retinas,

500 17. Commercial and Medical Ultrasound Applicationsmeasuring the length of the axis of the eye, and detect foreign bodies close to theposterior eye wall. One instrument combines a diagnostic transducer for locatingforeign bodies with a surgical instrument for removing an object, which enablesrapid removal of foreign bodies from the eye by directing the surgical tool to theobject with least damage to the eye.In the field of neurology, ultrasonic echoencephalography provides an immediatemeans of detecting lateral shifts in the midline septum caused by tumorsor concussion. It is notable that every emergency ambulance in Japan is outfittedwith echoencephalographic equipment in order to identify victims with possiblesubdural hemorrhage so that they may be transported directly to special neurologicalunits for treatment. Posttraumatic intercrannial hemorrhage and skull andbrain trauma can be quickly diagnosed and lesions can be located rapidly withoutdiscomfort to the patient. Ultrasonic pulses are transmitted through the temples. Inthe A-scan mode, echo indications from the midline and the opposite temple areasof the skull are presented on an oscilloscope screen. A shift in the midline is readilydiscernible. B-scans, which could provide more information, are more effectivewith small children and infants because their skulls are soft and have attenuationcoefficients lower than those in adults.The ultrasonic B-scan technique for examining the abdomen is useful for detectingpelvic tumors, hydatiform moles, cysts, and fibroids. It is also used to diagnosepregnancy at 6 weeks (counted from the first day of the last menstrual period) andafterwards. The obstetrician can follow the development of the fetus throughoutthe pregnancy, including size and maturity and positioning of the placenta. Thepresence of twins or multiple pregnancy is also revealed. This avoids the need forX-rays and the attendant danger of irradiation to mother and child. Fetal death canbe confirmed much earlier by ultrasound than it can by radiography.With improvements in ultrasonic and computer technology, 3D visualizationbegan to appear in the early 1980s. Some work came from the domain of cardiologistswhere initial efforts were directed to determining the volume of cardiacchambers. Real-time scanner probes mounted on articulated arms were often employedwhere positions of the probe can be accurately established. The principleof 3D imaging has always been to stack parallel image sections together with theirpositional information into a computer.Ultrasonic diagnosis by echo methods extends to all parts of the body. Air andother gases have much lower impedance than either liquid or solids, so air cavitiesproduce distinct echoes. Stationary air embolisms can be so identified becausethey are located in areas usually filled with fluids. Moving embolisms can beidentified by Doppler methods. Gallstones have an acoustic impedance equivalentto that of bone and so can provide good ultrasonic echoes. The ultrasonic B-scantechnique has been proven effective in diagnosing thyroid disorders. Disordersdetected ultrasonically include neoplastic lesions including cystic nodules, solidadenoma and carcinoma, nonneoplastic lesions, subacute thyroidiis, and chronicthyroiditis.The Doppler method takes advantage of the fact that a shift in frequency occurswhen an ultrasonic wave is reflected from a moving target. Also any variation in

17.4 Medical Uses of Ultrasound 501fluid motion that causes a beam of ultrasound to be deflected causes a Doppler shiftin frequency. One application of the Doppler shift principle is for the detection offetal blood flow and heartbeat, which can provide a mother the exciting assurancethat her unborn baby is alive by listening to its heartbeat through a set of headphonesconnected to the detecting equipment. Blood flow is measured through intact bloodvessels through the use of the Doppler principle with the carrier frequency rangingfrom 2–20 MHz. The Doppler frequency signal is calibrated in terms of velocity,according to the pitch of the Doppler signal. Doppler-type diagnostic units areused to determine the severity of atherosclerosis, locating congenital heart defects,and continuously monitoring fetal heartbeat during birth.In 1991 researchers at Duke University developed a matrix array scanner forimaging the heart. Three years later, the world’s first electronically steered matrixarray3D ultrasound imager was developed and it became commercially availablein 1997. The matrix array transducer, which steered the ultrasound beam in threedimensions, contained 2000 elements of which 512 were used for image formation.Newer units became available commercially for providing 3D scans offetuses in “color.” This resulted in a new market for “re-assurance” scans (sometimescalled “entertainment scans”), as the attraction of being able to see a babybefore birth apparently is proved to be quite irresistible to parents-to-be (as wellas grandparents-to-be). Three-dimensional ultrasound has been helpful in determiningthe topographical geography of fetal heart valves while blood is flowing.Another application is the imaging of microcirculation, which is important in thestudy of artherosclerosis, diabetes, and cancer—this means that 3D ultrasound isa powerful tool for ascertaining the progression of diseases in the body. As recentlyas 2003, researchers at Norway’s SINTEF Unimed Ultrasound (Kaspersen,2003) have demonstrated that real-time 3D ultrasound can be used as a diagnosticimaging tool during surgery. These researchers have applied 3D ultrasound tolaparoscopic, neurological, and vascular surgeries.Use of Contrast AgentsIn the last decade, researchers in academia, pharmaceutical companies, and scannermanufacturers invested manpower and funding in developing efficacious contrastagents and new contrast specific modalities. Contrast agents can improve the imagequality of sonography either by decreasing the reflectivity of the undesired interfacesor by increasing the backscatter echoes from desired regions. In the formerapproach, the contrast agents are taken orally, and for the latter approach, the agentsare introduced vascularly. In the upper GI tract, sonographic assessment is limitedby the gas-filled bowel, which produces shadowing artifacts. One agent, recentlyapproved by FDA, is SonoRx r○ , a simethicone-coated cellulose. Vascular enhancingultrasound entails the injection of contrast agent, such as perfluorochemicalsthat have low solubility in blood and high vapor pressure.One of the most important clinical uses of ultrasound contrast is in cardiology,where it will potentially compete with thallium nuclear scans. Newer agents suchas Optison, Definity, and Sonazoid can produce myocardial perfusion images in

502 17. Commercial and Medical Ultrasound Applicationshumans. This is clinically significant, because visualization of the myocardial flowpermits direct assessment of underperfused or unperfused regions (i.e., areas ofischemia or infarction) in patients with a history of chest pain. Myocardial imagingusing ultrasound contrast agents provides an assessment of the coronary arteriesand of the coronary blood flow reserve, as well as collateral blood flow that mayexist.Any body cavity that can be accessed can, in principle, be injected with vascularcontrast. According to Shi et al. (2005), the most successful application inthis category is hysterosalpingo-contrast sonography (abbreviated HyCoSy) forthe evaluation of fallopian tube patency. Degenhardt et al. (1996) reported on103 patients with fertility problems who underwent transvaginal sonography andHyCoSy with Echovist 14. In 58 cases, HyCoSy was compared with conventional,more invasive techniques such as chromolaparoscopy and 91% agreement wasfound. The upshot is that HyCoSy is rapidly becoming the screening test of choiceto determine tubal patency.Safety of Ultrasonic DiagnosisExtensive studies on the safety of ultrasound in Japan and elsewhere have led toissuance of Japanese industrial standards (JIS) for diagnostic ultrasound devices.These standards place limits on various diagnostic procedures:1. Ultrasonic Doppler fetal diagnostic equipment, 10 mW/cm 2 or less.2. Manual scanning B-mode ultrasonic diagnostic equipment, 10 mW/cm 2 or lessfor each probe.3. Electronic linear scanning B-mode ultrasonic diagnostic equipment,10 mW/cm 2 or less in a single aperture.4. A-mode ultrasonic diagnostic equipment, 100 mW/cm 2 or less: This standard islimited to diagnosis of the adult head and is not for pregnancy where B-mode isthe principal technique used. This higher permissible level is due to attenuationthrough the adult skull bone.5. M-mode ultrasonic diagnostic equipment, 40 mW/cm 2 or less: The M-mode isused in clinical diagnosis of the heart. For a combination of the M-mode withthe B-mode, the intensity is limited by the B-mode standard, i.e., 10 mW/cm 2 .In 2000, the Safety Group of the British Medical Ultrasound Society issued thefollowing statement on the safe use and potential hazards of diagnostic ultrasound(Ter and Duck, 2000): Ultrasound is now accepted as being of considerable diagnostic value. There isno evidence that diagnostic ultrasound has produced any harm to patients in thefour decades that it has been in use. However, the acoustic output of modernequipment is generally much greater than that of the early equipment and, inview of the continuing progress in equipment design and applications, outputsmay be expected to continue to be subject to change. Also, investigations into the

17.4 Medical Uses of Ultrasound 503possibility of subtle or transient effects are still at an early stage. Consequentlydiagnostic ultrasound can only be considered safe if used prudently. Thermal hazard exists with some diagnostic ultrasound equipment, if used imprudently.A temperature elevation of less than 1.5 ◦ C is considered to presentno hazard to human or animal tissue, including a human embryo or fetus, evenif maintained indefinitely. Temperature elevations in excess of this may causeharm, depending on the time for which they are maintained. A temperature elevationof 4 ◦ C, maintained for 5 minutes or more, is considered to be potentiallyhazardous to a fetus or embryo. Some diagnostic ultrasound equipment, operatingin spectral pulsed Doppler mode, can produce temperature rises in excess of4 ◦ C in bone, with an associated risk of high temperatures being produced in adjacentsoft tissues by conduction. With some machines colour Doppler imagingmodes may also produce high temperature rises, particularly if a deep focus or anarrow colour box is selected. In other modes, temperature elevations in excessof 1 ◦ C are possible, but are unlikely to reach 1.5 ◦ C with equipment currently inclinical use, except where significant self-heating of the transducer occurs. Non-thermal damage has been demonstrated in animal tissues containing gaspockets, such as lung and intestine, using diagnostic levels of ultrasound (mechanicalindex values of 0.3 or more). In view of this, it is recommended thatcare should be taken to avoid unnecessary exposure of neonatal lung , andto maintain MI as low as possible when this is not possible. In other tissuesthere is no evidence that diagnostic ultrasound produces nonthermal damage,in the absence of gas-filled contrast agents. However, in view of the difficultyof demonstrating small, localised, regions of damage in vivo, the possibility ofthis cannot be excluded. The Mechanical Index, if displayed, acts as a guide tothe operator. The use of contrast agents in the form of stabilised gas bubblesincreases the probability of cavitation. Single beam modes (A-mode, M-mode,and spectral pulsed Doppler) have a greater potential for nonthermal hazardthan scanned modes (B-mode, Colour Doppler), although the use of a narrowwrite-zoom box increases this potential for scanning modes.Therapeutic Uses of UltrasoundWe may very well be witnessing at this time only the beginning of the use ofultrasound for therapeutic purposes. Some techniques, such as the use of 25 kHzultrasound combined with a water jet to remove plaque from teeth and the cleaningof dental and medical tools with ultrasound, have been well established for a numberof years. Ultrasonic nebulizers of pharmaceuticals operate without producingdestructive temperature levels. Athletic centers and sports medicine specialistsmake use of ultrasound devices to heat sore muscles. Newer techniques are arrivingon the market or are still in the testing stages. One example is the use ofultrasound in catheters to ream out arteriosclerositic deposits in arteries that is stillvery much in the experimental stage.One therapeutic use of ultrasound already in widespread clinical use, extracorporealshock wave lithotripsy, has completely changed the treatment of kidney

504 17. Commercial and Medical Ultrasound Applicationsstones. Kidney stones are calcified particles that tend to block the urinary tract.In this type of treatment, the patient is immersed in water to equalize as much aspossible the acoustic impedances between the transducer and the patient’s body.A focused, high-pressure ultrasonic pulse is directed through the water and intothe patient’s torso to break the stone into small pieces. The pulverized materialcan now pass out of the body unhindered. Lithotripsy causes very little damage tokidney tissue.A promising procedure for therapeutic ultrasound is the laser-guided ablativeacoustic surgery, in which sound supersedes the scalpel in destroying benign ormalignant tissues. The ultrasound focused by a specially shaped set of transducersconverges inside the body to create a region of intense heat that can destroy tumorcells. The spot of destruction is so small that a boundary of only six cells liesbetween the destroyed tissue and completely unharmed tissue, which connotes aprecision far beyond any current method of surgical incision.Acoustical surgery offers a potentially better means of treating cancerous tumorsbecause it does not require an anesthetic, can be administered in a single treatment,and causes no observable side effects. In a Phase I clinical trial at Marsden Hospitalin London, focused sound waves destructed parts of liver, kidney, and prostate tumorsin 23 patients. In the next phase the researchers will attempt to fully destroytumors in the liver and prostate. Also under testing is the Sonablade TM systemby Focus Surgery of Indianapolis, IN, which incorporates proprietary transducertechnology in a transrectal probe that provides imaging for tissue targeting andhigh-intensity focused ultrasound (HIFU) for tissue ablation. After the operatordefines the area of periutheral tissue to be ablated, the treatment process beginsunder computer control, where the focus of the dual-function transducer is electromechanicallystepped through the designated volume of the tissue. HIFU resultsin thermally induced coagulative necrosis only in the intraprostatic tissue encompassedby the focal volume, with effect on intervening tissues. Confirmation oftargeting accuracy is provided through continuously updated images. The necrotictissue is either sloughed during urination or reabsorbed, along with the cessationof patient symptoms.Ultrasound can also be used to stop internal bleeding through an effect calledacoustic hemostatis. With sufficient power, ultrasonic pulses can elevate the bodytemperature at selected sites from 37 ◦ C to between 70 ◦ C and 90 ◦ C in an extremelyshort time, less than 1 s. This causes the tissue to undergo a series of phasetransitions, and the protein-based bodily fluids and blood coagulate as the resultof the proteins undergoing cross-linking (a process similar to cooking an egg).A research team at University of Washington’s Applied Physics Laboratory isinvestigating the use of ultrasound to stop internal bleeding during surgery and fortreating trauma cases. The present method of stemming bleeding in delicate organs,such as the liver, pancreas, or kidney, is through cauterization on the surface withion or microwave systems. The focused ultrasound waves, however, can penetratedeeply into the organ and “cook” the tissue in a layer as thin as 1 mm. It follows thattrauma patients could be treated without the need for a sterile environment of anoperating room and without the danger of infection that accompanies conventional

17.4 Medical Uses of Ultrasound 505surgery. To date, success has been achieved by the University of Washington groupin identifying patients with internal bleeding and in the use of HIFU in the operatingroom to stop bleeding in the organs and vessels of animals.Another promising technology still under development testing for use in biopsiesis that of augmented reality system, in which 3D ultrasound technology is combinedwith virtual reality (State et al., 1996). This concept uses ultrasound echographyimaging, laparoscopic (referring to the use of a small video camera to performsurgery) range imaging, a video see-through head-mounted display (HMD), anda high-performance graphics computers to generate live images through the useof computer-generated imagery with the live video. An augmented reality systempresents live ultrasound data or laparoscopic range data in real time while recordingdata from the part of the patient that is being scanned.Advances in magnetic resonance imaging (MRI) and diagnostic ultrasoundimaging will allow these two areas to be combined. Manufacturers of MRI equipmentare developing models that combine therapeutic functions such as ultrasoundwith imaging processes. One company (Therus Corporation, Seattle, WA)is combining ultrasound diagnostic imaging with a separate therapeutic ultrasoundcapability to produce a relatively small, portable system that can be carried byparamedics and rescue workers for use at the site of disaster.A newer method of targeted drug delivery is that of sonophoresis, which usessound waves instead of needles to inject drugs such as insulin and interferonthrough the skin. The high-frequency waves open tiny holes in cell membranes, thusrendering the cells temporarily permeable in localized regions and allowing betterpenetration of the drug into the blood vessels below the skin. This results in greatereffectiveness of the drug, lessens the dosage requirements and toxicity, and allowsfor more precise localization of drug delivery. Although the mechanisms by whichultrasound augments these effects are only partially understood, it is known thatultrasound produces biophysical reactions yielding hydroxyl radicals that in turnaffect cell membranes. A system developed by Ekos Corporation of Bothell, WA, todissolve life-threatening blood clots constitutes an early application of ultrasounddrug delivery. The Ekos device injects thrombolytic drugs through a catheter usinglow-energy, localized ultrasound to the target site in the body. Thrombolytics,which are generally administrated to break up clots, are dramatically more effectivewith the use of ultrasound because the sound waves help to concentrate the drugsat the site of the clot. Ekos’s device and similar products are meant to providetreatment of cardiovascular obstructive diseases (e.g., stroke and arterial and deepveinthrombosis). In these types of ailments, rapid response is critical, and therapynecessitates applications of massive doses of clot-dissolving drugs over 72 h.Ultrasound may also be used with antiresrenosis agents, which are intended toprevent coronary arteries from reclosing after angioplasty—a procedure in whicha balloon-tipped catheter is inserted into a clogged vessel and the balloon is inflatedto open up the vessel.Another long-term application is the use of ultrasound to deliver insulin throughthe skin for treating diabetes. It may be even possible through ultrasound to penetratethe blood–brain barrier, which insulates the brain from foreign substances

506 17. Commercial and Medical Ultrasound Applicationsand also prevents many drugs from reaching diseased tissues there, so that theeffects of chemotherapy can be enhanced.Higher frequencies of ultrasound can cause the transfer of large molecules suchas DNA to migrate among cells, as the result of sonoporation, which enhances theporous effect on cell membranes by induction of ultrasonic shock waves through alithotripter. One pharmaceutical company (ImaRx, Tucson, AZ) developed a genedeliveringsystem for improving gene expression, i.e., revising a gene’s geneticcode to make a specific protein. The ImaRx system employs acoustically drivenmicrobubbles that carry the gene and fluorinated compounds that serve as markersto detect the DNA. The ultrasound assists in delivering the DNA to targeted areas inthe body and in tracking its progress through the use of contrasting agents injectedwith the microbubbles of genetic material.ReferencesAIUM. 2000. Mechanical bioeffects from diagnostic ultrasound: AIUM consensus statements.Journal of Ultrasound in Medicine 19/2. Bethesda, MD: American Institute ofUltrasound in Medicine.Bachman, Donald M., Crewson, Philip E., and Lewis, Rebecca S. 2002. Comparisonof heel ultrasound and finger DXA to central DXA in the detection of osteoporosis.Implications for patient management. Journal of Clinical Densitometry 5(2): 131–142.Barnett, S. B. and Kossoff, G. (eds.). 1998. Safety of Diagnostic Ultrasound. Progress inObstetric and Gynecological Sonography Series. Parthenon.Barnett, S. B. (ed.). 1998. WFUMB Symposium on Safety of Ultrasound in Medicine. Conclusionsand Recommendations on Thermal and Mechanical Mechanisms for BiologicalEffects of Ultrasound. Ultrasound in Medicine & Biology 24, Supplement 1.Baum, G., Cruz, B., and Rosenblatt, R. 1980. Advantages of ultrasound mammography.Ultrasound in Medicine & Biology. Amsterdam: Excerota Medica.Bruining, N., Burgelen, C. V., De Feyter, J., et al. 1998. Dynamic imaging of coronarystructures: an ECG-gated three-dimensional intercoronary ultrasound in humans. Ultrasoundin Medicine & Biology 24: 631–637.Brown, B. and Goodman, J. E. 1965. High Intensity Ultrasonics. London: Iliffe.Crum, Lawrence A. 2004. Therapeutic Ultrasound. Proceedings of the Advanced Metrologyfor Ultrasound in Medicine Meeting, April 27–28, 2004. Teddington, UK.Chen, L., Ter Haar, Gail, Hill, C. R. et al. 1998. Treatment of implanted livertumors with focused ultrasound. Ultrasound in Medicine & Biology 24: 1475–1488.Dalke, H. E. and Welkowitz, W. J. 1960. Journal of the Instrument Society of America7(10): 60–63.Degenhardt, F., Jibril, S., and Eisenhauer, B. 1996. Hysterosalpingo-contrast sonography(HyCoSy) for determining tubal patency. Clinical Radiology 51(s1): 15–18.Ensminger, Dale. 1988. Ultrasonics: Fundamental, Technology, Applications, 2nd ed. NewYork: Marcel Dekker.EPSUMB Study Group. 2004. Guidelines for the use of contrast agents in ultrasound.Ultraschall in Medizine 25: 249–256.

Problems for Chapter 17 507Hope, Simpson D., Chin, C. T., and Burns, P. N. 1999. Pulse inversion Doppler: a newmethod for detection of nonlinear echoes from microbubble contrast agents. IEEE TransactionsUFFC 46: 372–382.Jacobs, J. E. 1962. Proceeding On Physics and Nondestructive Testing. October 2–4, 1962.San Antonio, TX: Southwest Research Institute: 59–74.Jacobs, J. E., Reimann, K., and Buss, L. 1968. Materials Evaluation 26(8): 155–158.Joyner, Claude R, (ed.). 1974. Ultrasound in the Diagnosis of Cardiovascular-PulmonaryDisease. Chicago: Year Book Medical Publishers, Inc. (written by clinicians primarilyfor clinicians.)Kaspersen, J. H. 2003. Clinical trials with new navigation system: Custus X. SINTEF [Online, available at: http://www.sintef.no/static/UM/UL/cx/cx.html].Krüger, J. F. von and Evans, David H. Doppler ultrasound tracking instrument for monitoringblood flow velocity. Ultrasound in Medicine & Biology 28: 1499–1508.Needleman, L. and Forsberg, F. 1996. Contrast agents in ultrasound. Ultrasound Quarterly13: 121–138.Ouellette, Jennifer. 1998. New ultrasound therapies emerge. The Industrial Physicist,pp. 30–34.Shahbender, R. A. 1961. Transactions of I. R. E., UE-8: 19.Smythe, C. N., Poynton, F. Y., and Sayers, J. F. 1953. Proceedings of IEEE 110(1): 16–23.Shi, William T., Forsberg, Flemming, Liu, Ji-Bin, et al. 2005. Orlando, FL: Annual Meetingof the Academy of Molecular Imaging.Sokolov, S. Ya. 1937. U.S. Patent No. 2,164,185.State, Andrei, Livington, Mark A., Hirota, Gentaro, et al. 1996. Technologies for augmentedreality systems realizing ultrasound-guided needle biopsies. Proceedings of SIGGRAPH96 (New Orleans, August 4–9, 1996). In: Computer Graphics Proceedings, AnnualConference Series 1996. ACM SIGGRAPH: 439–446.Sternberg, M. S. 1958. Physics Review 110: 772.Ter, Haar G. and Duck, F.A. 2000. The Safe Use of Ultrasound in Medical Diagnosis.London: British Medical Ultrasound Society/British Institute of Radiology.Waddell, J. H. and Waddell, J. W. 1970. Research and Development 30: 32.Woo, Joseph. 2002. A short history of the development of ultrasound in obstetrics andgynecology. [On line, available at: http://www.ob-ultrasound.net]. (An excellent threepartnot-so-short compendium of the history of the use of ultrasound in these medicalfields.)Worlton, D. C. 1957. Nondestructive Testing 15: 218.Problems for Chapter 171. Can you think of a household application for ultrasonic cleaning?2. A “sing around” flowmeter is used to measure the flow of a chemical slurryinside a pipe. The two transducers are placed 10 cm apart. A 25,000 Hz pulseis initiated and the received frequency is 24,500 Hz. Find the velocity of theslurry.3. If the fluid went the other way, what would be the receiving frequency for thesame setup of Problem 2 emitting 25,000 Hz?4. A beam-deflection flowmeter is used to measure flow inside a nuclear reactorheat exchanger pipe. The velocity of sound in the liquid coolant is known to

508 17. Commercial and Medical Ultrasound Applicationsbe 2550 m/s, and the deflection of the beam is 8.58 ◦ . Find the velocity of thefluid.5. We can also measure the velocity of sound in a fluid by the beam-deflectionmethod. It is known that a gas is passing through a conduit at the rate of120 m/s. The deflection was measured to be 0.056 radian. Estimate the speedof sound in the gas.6. Two transducers are set apart 21.2 cm in a nondispersive medium. The tworesonance frequencies are 2164 Hz and 4617 Hz. Find the sound propagationspeed of the medium.7. How could motion sensing be useful in a home garage? Where else in a homewould you use a sensor?8. Why is ultrasound considered to be safer than X-rays? Would you expect theultrasound equipment to be more expensive or cheaper than X-ray gear?9. Why is it preferable to use ultrasound to examine the fetus in the womb toX-rays? How would the images compare in quality?10. What is the principal difference between high-intensity and low-intensity ultrasoundas used in the human body or on a live animal?11. What are the two primary factors in controlling ultrasound dosage?

18Music and Musical Instruments18.1 IntroductionFrom time immemorial music has impacted humanity in many ways. In momentsof sadness, music provides solace; in happier times, enhances exhilaration; duringstressful periods, a greater sense of calm intertwined with an intensified feelingof purpose; and when diversion is needed, entertainment. Music reached itsgreatest heights through the evolution of primitive contraptions into more elegantinstruments and the emergence of great composers such as Monteverdi,Vivaldi, Bach, Handel, Haydn. Mozart, Beethoven, Verdi, and in more moderntimes, Rimsky-Korsakov, Stravinsky, Mahler, Ravel, Gershwin, Schoenberg, andEllington.Music has given rise to a whole slew of industries: the manufacture and marketingof instruments; staging of performances which can range from solo appearancesto a lavish operatic production or a frenzied giant rock concert, on stage orthrough electronic transmission (radio, television, the Internet); and distributionof recorded media (tape, CD, or DVD) and playback equipment.In surveying the musical scene, we must realize the manner in which musicinfluences people cannot possibly be adequately gauged, and this constitutes asituation that provides a fertile field of research in psychology and anthropology,to say nothing of musicology and music theory. In the medical field, classical musicserves as a valuable tool in psychotherapy. Musical acoustics is an extremely broadinterdisciplinary field—a field that deals with the production of musical sound andthe transmission of musical sound to the listener as well (Rossing, 1990). Thestudy of music from a physical approach provides the opportunity to bridge thegap between art and science.In this chapter the structure of music is examined from an acoustical approachand the principles of generating musical tones with instruments, which also includesthe human voice, are outlined. Many but not all musical instruments aredescribed herein, as well as the makeup of orchestras and bands.509

510 18. Music and Musical Instruments18.2 Musical NotationAs pointed out by Olson, music can be memorized and passed from one personto another by the direct conveyance of the sound, but this does not constitute asatisfactory, nor an efficient method of communicating music to performers andpreserving the music for future performances (Olson, 1967). Accordingly, musicalnotations were developed to use symbols on paper to denote frequency, duration,quality, intensity, and other tonal characteristics. A human can typically distinguish1400 discrete frequencies, but in the equally tempered musical scale, thereare only 120 discrete tones ranging from 16 Hz to 16,000 Hz. Pitch is an attributeof aural sensation, dependent on the frequency of the sound. Musical tones are assignedspecific values, allowing for specific frequency values, and leaving out the“in-between” values. A principal reason that we can identify a musical instrumentis that the musical instruments are essentially resonant instruments and thereforeresponse only to certain frequencies. These resonant frequencies are fixed andcannot be altered, except for certain instruments such as those members of theviolin family and the trombone. Moreover, when same notes are played on differentinstruments, the overtones differentiate one instrument from another. With therelatively small number of fundamental frequencies designated in Western music,matters are greatly simplified in designating the discrete frequency characteristicsof tones.In the five-line staff of Figure 18.1, the pitch of a tone is denoted by placingnotes, , , on the lines and in the spaces between the lines. The pitch rangeof a set of lines is designated by the clef ( or ) which is placed at the left ofthe staff. The most common clefs are the treble or G clef and the bass or F clef,both of which are shown in Figure 18.1. The notes are designated alphabeticallyfrom A to G. The interval in pitch between two notes with the same letter is theoctave. As explained earlier in this text, the two sounds separated by an octavehave a fundamental frequency ratio of 2. The pitch interval between adjacent notes(i.e., between a note on a line and a note in the adjacent space, as designated bysuccessive letters) is a whole tone in the equally tempered scale. Pitches that arehigher and below the staff are designated by notes written upon and between shortlines called leger lines, as shown in Figure 18.1. The number of leger lines canhypothetically be extended without limit. The sign 8va above the staff denotes thatall tones are to be played an octave higher than their placement and, conversely, thesign 8vs placed below the staff indicates that all tones are to be played an octavelower. This is shown in Figure 18.2.There is also another clef—the movable or C clef—which was meant to accommodatemusic instruments with extended range (e.g., the bassoon, cello, or viola).In older musical manuscripts there are C-clefs for the soprano, alto, and tenor parts,Figure 18.3 shows the three different symbols to indicate the C-clef. The C-clefis placed on the middle C line. The old C-clef positions for the soprano, alto, andtenor are given in Figure 18.4. The position of the C clef always corresponds tothe middle C.

Figure 18.1. The five-line musical staff, the notes of the most common bass, and the trebleclefs and the leger lines (the lines above and below the staff) for the treble and bass clefs.Figure 18.2. The position of 8va above the staff indicates that all notes are sounded an octavehigher. When 8va is located below the staff, the notes are to be sounded an octave lower.

512 18. Music and Musical InstrumentsFigure 18.3. Three symbols used to indicate the C, or movable, clef.Figure 18.4. The positions of the C-clef for the soprano, alto, and tenor clefs.Figure 18.5. Sharp, flat, and natural designations: the sharp raises a pitch by a semitone,the flat lowers a pitch by a semitone, and the natural nullifies a sharp or flat to restore anote to normal.A note can be moved up in its pitch a half step or semitone; this is labeled asharp. The sharp is designated by the natural note preceded by the sign, as shownin Figure 18.5. A note can also be moved down a semitone, thus rendering that notea flat with the symbol , which is also shown in Figure 18.5. A natural designationnullifies a sharp or a flat, returning the note to normal. A note may be movedup by a whole step by a double sharp designation or ×. A note may be moveddown a whole note by a double flat . A standard system for the identificationof tones used in music is given in line 1 of Figure 18.6 (Young, 1939). The otherseven lines represent the various systems for identifying the musical tones withoutthe benefit of using the staff, but the system represented by the first line is themost logical one to use and understand. The reference standard frequency C 0 is16.352 Hz, which just about constitutes the lowest frequency that a human ear candetect. It is customary to consider C as the point to begin counting whole octaves.Figure 18.7 displays the frequencies of the notes in equally tempered scale in thekey of C from 16 Hz to 16 kHz.

18.3 Duration of Musical Notes 513Figure 18.6. Eight systems used for tone identification in music (Young, 1939).18.3 Duration of Musical NotesThe duration of a musical tone is the length of time assigned to it in the musicalcomposition. Figure 18.8 displays the symbols used to indicate duration. Whilethe pitch of a tone is given by its position on the staff, its length is assigned bythe choice of one of the symbols of Figure 18.8. However, the magnitude (i.e.,its duration) of a tone is not rigidly fixed and it may vary from composition tocomposition. But nevertheless, in a particular composition the duration of eachtone is kept in proportion to the magnitude of a whole note.In the traditional musical notation a vertical bar is drawn across the staff. Thetime interval between two vertical bars in a staff is called a measure, or the lessprecise but more commonly used bar. The time intervals of all measures withina composition are usually equal. If two whole notes constitute a measure, thenthe measure will need four half notes, or eight quarter notes, or any combinationthat adds up to two whole notes in time interval. A double bar that consists of twovertical bars across the staff denotes the end of a division, movement, or an entirecomposition.To indicate periods of silence in a composition, one or more rest symbols ofFigure 18.9 is used to indicate the duration of the silence. The duration of a wholerest is equal to that of a whole note, the duration of a half rest is equal to that of ahalf note, and so forth.The duration of a tone represented by a note or a rest of a certain denominationcan be modified by the addition of a dot to the note. The effect of the dot is to

Figure 18.7. The frequencies of the notes in the C scale of equal temperament from 16 Hzto 16 kHz.

Figure 18.8. Note values which indicate duration.Figure 18.9. Rest symbols.

516 18. Music and Musical Instrumentslengthen the duration of the preceding note by half as much, i.e., a whole notebecomes equal in duration to a whole note plus a half note, a dotted half noteequals the duration of a half note plus a quarter note.There is no absolute time-interval standard for the duration of a tone representedby a note, and it generally depends on the performer’s interpretation of the musicwith respect to its tempo. Some compositions carry an indication of the setting ofa metronome for a quarter note. A metronome is a mechanical device that consistsof a pendulum activated by a clock-type of mechanism driven by a spring motor.At the extremities of the pendulum swing an audible tick is produced. The intervalbetween ticks can be adjusted by moving a bob on the pendulum arm: the furtherthe bob is located from the fulcrum, the longer the duration between ticks, andvice versa. The pendulum itself is graduated in ticks per minute. The numbersusually indicate the number of ticks per minute, the interval between ticks usuallyspecified as that of a quarter note or, in some cases, half notes. Modern versions ofmetronomes use electronic means to generate ticks. The metronome setting setsthe rate of movement or tempo of the music. Instead of metronome settings, thecomposer may specify one of a number of terms to designate tempos. Commonlyused terms to describe tempos are as follows: Largo: Slow tempo Andante: Moderately slow tempo Moderato: Moderate tempo Allegro: Moderately quick tempo Vivo: Rapid tempo Presto: Very rapid tempo18.4 Time Signature NotationA musical selection’s time signature is specified at the beginning of the staff bya fraction, as illustrated in Figure 18.10. Common time signatures include 2/4,3/4, 4/4, and 6/8. The denominator indicates the unit of measure (i.e., the noteused to define a pulse). The numerator stands for the number of these units or theirequivalents included in a measure (i.e., the interval between two vertical linesacross the staff).Figure 18.10. Time signatures for 2/4, 3/4,Cor4/4, 6/8, and 9/16 times.

18.4 Time Signature Notation 517Figure 18.11. Notes and beats for 2/4 and 3/4 times.In upper portion of Figure 18.11 for 2/4 time, each measure contains one halfnote, or two quarter notes of four eighth notes. Each measure contains two beats,so when a musician plays, the count is one, two. In 2/4 time, a stressed pulseis followed by a relaxed pulse, a sequence used for marches. In the 3/4 time asshown in Figure 18.11, each measure equals three quarter notes or 1 half note plusa quarter note, and so on. Each measure carries three beats, and usually there isone stressed pulse followed by two relaxed pulses, which yields a time used forwaltzes. In the 4/4 (or common) time, each measure contains the equivalent offour quarter notes, with the performer counting one, two, three, four. In 4/4 time, astressed pulse is followed by three relaxed pulses, and this time is used for dances.In 6/8 time, each measure contains six eighth notes or a combination of notesequaling the same duration. There are then six beats to each measure, and in 6/8time the stressed pulses are one and four of six beats.Listeners mentally arrange the regular repetition of sounds into groups ofstressed and relaxed pulses. These groups are called meters. The meter is assignedby the numerator of the time signature, and the most common ones are 2, 3, 4, 6,9, and 12. Each measure contains a certain number of beats or pulses according tothe meter. Meters are classified in terms of the numerators of the time signaturesin the following manner:1. Duple meter: Two beats comprise each measure, with the first beat stressed andthe second beat relaxed. Example signatures are 2/2 and 2/4 times.2. Triple meter: Three beats occur in each measure, with the first one stressedand the following two relaxed. Example time signatures are 3/8, 3/4, and4/8.3. Quadruple meter: Four beats occur in each measure, with the first beat stressedand the remainder relaxed. Occasionally the third beat carries a secondary stress.Examples include 4/2, 4/4, and 4/8 time signatures.4. Sextuple meter: Six beats occur each measure, with the first and fourth beatsstressed. The 6/8 time signature is such an example.Rhythm is the repetition of accents in equal intervals of time.

518 18. Music and Musical Instruments18.5 Key NotationThe keynote denotes the note with which any given scale begins. The tonic is thekeynote of the scale, whether the latter is a major or a minor scale. Many shortcompositions are written in one key only, but more elaborate musical pieces mayshift from one key to another. The key signature of a musical piece is denotedby the number and arrangement of flats and sharps following the clef sign at thebeginning of each staff, or it may appear only once at the beginning. Figure 18.12shows some of the most common key signatures for different major and minorkeys.Major and minor keys play a role in determining the mood of music. In earliertimes, a key may have been selected by a composer because a number of windinstruments were able to play only in certain keys. Certain desired effects may bebetter achieved on more flexible instruments in a specific key than another key.As Machlin pointed out, romantic composers developed affinities for certain keys,for example, Mendelsohn preferred E-minor, Chopin leaned toward C-sharp, andWagner made use of D-flat major for majestic effects.Whether it starts with C, D, E, or any other tone, a major scale follows the samearrangement of whole and half steps. Such an arrangement is known as a mode.All major scales typify the arrangement of whole and half steps.The minor mode serves as a foil to the major. The principal difference from themajor is that its third degree is flatted. For example, in the scale of C, the thirddegree is E rather than E. In a natural minor scale, the sixth and seventh steps arealso flatted (i.e., C-D-E -F-G-A -B -C). The minor differs considerably from themajor in coloring and mood. It should not be inferred that the minor is deemed“inferior”—the nomenclature simply refers to the fact that the interval C-E issmaller (hence minor, the Latin word) than the corresponding interval in the majorscale.If a mode is not specified, the major is implied. For example a Minuet in Gindicates the G-major. The minor is always specified (e.g., Mozart’s SymphonyNo. 40 in G minor).To classical composers the tonal qualities of the minor key assumes a moresomber aspect (e.g., the funeral music of Beethoven and Mendelssohn) than thetriumphal portions of symphonies and chorales which are generally played in majorkeys. Also, the minor mode carries a certain exotic tinge to Western ears, and thus inthe popular view it was associated with oriental and Eastern European music. Thiswas reflected in such works as Mozart’s Turkish Rondo, a number of Hungarianstyleworks by Schubert, Liszt, and Brahms, the main theme of Rimsky-Korsakov’sScheherazade, and other musical pieces that passed for exotica.18.6 Loudness NotationLoudness depends upon the intensity of the musical signal. Although loudnesscan be measured objectively with the use of a sound-level meter, a conductor or

18.6 Loudness Notation 519Figure 18.12. Key signatures for number of major and minor keys.

520 18. Music and Musical Instrumentsa musician depends on his or her own sense of subjectivity to obtain the properintensity or range of intensities. The common notations and abbreviations forloudness are as follows: Pianissimo (ppp): softly as possible Pianissimo (pp): very soft Piano (p): soft Mezzo piano (mp): half soft Mezzo forte (mf ): half loud Forte ( f ): loud Fortissimo (ff ): very loud Fortisissimo (fff ): extremely loudLoudness can vary in musical passages. An increase in loudness can be indicatedby the term crescendo or the abbreviation cres or the sign . A crescendoconnotes a gradual increase in the intensity of the music. A decrescendo is adecrease in loudness and is thus the converse of a crescendo. It is denoted by theword decrescendo (also diminuendo) or the abbreviation decresc or the symbol.18.7 Harmony and DiscordConsider two tuning forks being sounded together. Let us keep the pitch of one forkfixed at 261 Hz, while the pitch of the other begins at 262 and is gradually raised.As the pitch raises, beats can be heard, due to the difference in the frequenciesfor a time and then can longer be discerned. The sound of the combined tonesstarts out by sounding pleasant to the ear and then it becomes gradually moreunpleasant. The unpleasantness reaches a maximum at about 23 beats per second,and then begins to abate. This unpleasantness or discord declines only slightly,and the discord remains at a fairly uniform level until the octave-marking value of522 Hz is reached, at which point the unpleasantness disappears.If this experiment is repeated with violin strings, radically different results willbe obtained. The discord does not stay at a uniform level but fluctuates erratically.It almost vanishes at the interval of a major third, and again at the intervals of thefifth and octave. At the precise points at which the minimums of the unpleasantnessoccur, the frequency ratios of the variable to the fixed tone are found to have thevalues: 5/4, 4/3, 3/2, and 2/1.It has been observed that tone tones sound well together when the ratio of theirfrequencies can be expressed in terms of small numbers. The smaller the numbers,the better is the consonance. Table 18.1 lists the intervals in order of increasingdissonance.The further away from small numbers, the more we encroach into the realm ofdiscord. Pythagoras knew this fact more than 2500 years ago when he associatedconsonance with the ratios of small numbers. The premise of the Pythagoreandoctrine “all nature consists of harmony arising out of number,” may be somewhat

18.8 Musical Instruments 521Table 18.1. Interval Nomenclature and Frequency Ratios.Largest Integer OccurringInterval (second) Frequency Ratio in the RatioUnison 1:1 1Octave 2:1 2Fifth 3:2 3Fourth 4:3 4Major third 5:4 5Major sixth 5:3 5Minor third 6:5 6Minor sixth 8:5 8Second 9:8 9simplistic, but the Chinese philosophers in Confucius’s time also regarded smallnumbers 1, 2, 3, 4 as the source of all perfection.The Swiss mathematician Leonhard Euler adopted the psychological approachin declaring that the human mind takes pleasure in law and order, particular innatural phenomena. His theory of harmony is this: the smaller the numbers requiredto express the ratio of two frequencies, the easier it is to find this law and order,thus making it more pleasant to hear the combined sounds. Euler went so far as topropose a definitive measure of the dissonance of a chord. His idea was to expressthe frequency ratio of a specific chord by the smallest number possible and then tofind the common denominator for these frequencies. For example, the frequencyratio of the common chord CEG c ′ is 4:5:6:8. The least common denominator is120, since it is the smallest number of which 4, 5, 6, and 8 are all factors. Butthis theory falls apart when the same denominator is assigned to the chord ofthe seventh CEFGB (frequency ratios 8:10:12:15) that turned out to be far moreunpleasant to listen to.18.8 Musical InstrumentsMusical instruments fall into four categories: string, wind, percussion, and electricalinstruments. A string instrument may have its strings struck, bowed, orplucked. Wind instruments can be sub-classified as single-mechanical reed,double-mechanical reed, lip reed, air reed, and vocal-cord reed. Percussion instrumentsare classified as being either definite pitch or indefinite pitch. The adventof electronics has given rise to a whole new class of instruments, such as synthesizerswhich can effectively replicate the sounds of conventional strings, winds,and percussion instruments as well as generate unusual sounds not heard from anyother instruments. Even personal computers can function as musical instrumentsprovided they are equipped with special soundboards and speakers and they areprogrammed to simulate various types of instruments. Table 18.2 lists a numberof musical instruments and their respective classifications.

522 18. Music and Musical InstrumentsTable 18.2. Classification of Musical Instruments.String InstrumentsPlucked Strings Bowed Strings Struck StringsLyre Violin PianoLute Viola DulcimerHarpVioloncelloZitherDouble bassGuitarUkuleleMandolinBanjoSitarHarpsichordWind InstrumentsAir Reed Single-Mechanical Reed Double-Mechanical Reed Lip ReedWhistle Free-reed organ Oboe BugleFlue Organ Pipe Reed organ pipe English horn TrumpetRecorder Accordion Oboe d’amore CornetFlageolet Harmonica Bassoon French hornOcarina Clarinet Contra bassoon TromboneFlute Bass clarinet Sarrusophone Bass trombonePiccolo Saxophone (soprano, TubaFifealto, tenor, and bass)BagpipeOrgan (combination air reed and mechanical reed)Percussion InstrumentsDefinite PitchIndefinite PitchTuning fork Celesta Snare or side drum TriangleXylophone Kettledrums (tympani) Military drum Steel drumMarimba Bell Bass drum CymbalsChimes Carillon Gong TambourineGlockenspielCastenetsElectrical or Electronic InstrumentsSiren Electrical piano Electrical carillon Computer (specially configured)Automobile horn Electric guitar SynthesizerElectric organ Music box Metronome18.9 StringsPossibly the oldest method of creating music is vibrating string under tension,which is capable of producing a full range overtones of that are harmonics of thefundamental. As was explained in Chapter 4, the presence and the amplitudes ofthese harmonics depend upon the manner the string is excited (namely, by plucking,striking, or bowing) and where the excitation is applied. Because the string projectsa small area, it is not an efficient producer of sound as it is not by itself capable

18.9 Strings 523of moving very much air. It is for this reason that strings are coupled to a largemultiresonant surface (or soundboard) to increase the sound output.All string-musical instruments make use of a soundboard or a combination ofa ported hollow body and a soundboard to couple the string to the air. The largerthe radiating surface, the greater the acoustic impedance, and this provides theincreased coupling to enhance the sound output of the strings. Soundboards exhibitthe complex modes of vibrations that have been described in Chapter 6 that dealswith membranes and plates. In a number of string instruments such as the lute,lyre, zither, guitar, mandolin, and the violin family, a hollow body with portholesis coupled to the strings. The vibration imparted by the strings to the body exteriorproduces radiation in a fashion similar to that of the soundboard. The hollow bodywith a hole(s) coupled to the outside air comprises a Helmholtz resonator, andthe fundamental frequency can be established from the theory of Section 7.11.The dimensions of the resonator may equal or be larger than the wavelength of thesound in air for some of the tones or overtones produced in the instrument, so thehollow body with its holes can manifest other resonant frequencies in addition tothe fundamental resonant frequency. A hollow body is a quite complex resonantsystem.Some examples of a plucked-string instrument include the lyre, lute, zither, andharp, three of which are illustrated in Figure 18.13.The lyre traces its origin to ancient Greece, and it consists of a frame, fingerboard,and a hollow body with sound holes. The lyre is played by plucking the stringswith fingers, and the length of the string (and hence the resonant frequency) isvaried by pressing the finger against the fingerboard. The hollow body serves as asoundboard to increase the sound output of the string.The lute, developed more than a thousand years ago, is the precursor of theinstruments of the guitar class. It comprises a pear-shaped hollow body, a neckwith frets (projecting ridges across the neck), and a head equipped with pegs totune the strings.The zither, similar to the lyre in appearance but having its soundboard underneaththe entire length of strings, consists of two sets of strings stretched across a flathollow body featuring a large round holes. One set of steel strings, which passover a fretted fingerboard, is used to play the melody while a set of gut strings isused for accompaniment. The modern zither now consists of 32 strings, of whichfour are assigned to the fretted fingerboard. Each of the strings can be tuned byturning pegs of pins at one end of the instrument. The plucking of the zither occursin the following way: A ring-type plectrum is used on the thumb of the right handto play the melody. The left hand is used to stop the melody strings by pressingthe strings against the frets of the fingerboard. The spacing of the frets are suchthat the sounds produced by stopping the strings on any two adjacent frets areone semitone apart. The first, second, and third fingers of the right hand play theaccompaniment. Zithers are available in three sizes, called bass, bow, and concerttypes. The open melody strings of the concert zither are tuned to C 3 ,G 3 ,D 4 , andA 4 . The accompaniment strings provide the fundamental-frequency range of C 2to A 4 .

524 18. Music and Musical InstrumentsFigure 18.13. Plucked-string instruments: lyre, lute, and harp.The harp, often portrayed as being the instrument of angels, consists of stringsstretched vertically upon a triangular frame and connected to a soundboard constitutingthe lower leg of the triangular frame. The soundboard is quite small, andso the strings are not highly damped and the sound from each string persists for arather long time, yielding a rather mellow tone. The modern harp is usually providedwith seven pedals that actuate a transposing mechanism for shortening thestrings in two stages.The mechanism for shortening the strings is illustrated in Figure 18.14. In panelA of the figure the string is positioned at its maximum length. Depressing the pedalhalfway causes disk 1 to rotate along with the pins attached thereto, so the stringis shortened by a semitone as shown in panel B. When the pedal is depressed allthe way, disk 2 is rotated along with its attached pin, thus shortening the string

18.9 Strings 525Figure 18.14. Mechanism for shortening strings in the harp for transposing tones.even further so that the string is a whole tone higher, as shown in panel C. Eachpedal operates on strings with notes having the same letter notation, i.e., the Cpedal controls all the C strings; the D pedal all the D strings, and so on. The harpis generally tuned in the key of C flat. The pillar of the harp serves to bear thestresses produced by the stretched strings and to serve as a housing for the rodsconnected to the pedals and for the transposing mechanism of Figure 18.14. Theoverall height of the standard harp is approximately 173 cm. The harp is playedeither by plucking the strings with the fingers or by sliding the fingers over thestrings in a manner referred to as a glissando. There are 44 strings, usually madeof gut (some bass strings may be constructed of silk overlaid with metal wire forgreater density); portable models have as few as 30 strings.Figure 18.15 shows views of a ukulele, guitar, mandolin, and banjo. The ukuleleis a small version of the guitar. The former consists of four strings, tuned to D 4 ,F 4 ,A 4 , and B 4 , stretched between a combined bridge and tailpiece attached to thetop flat surface of the body and the end of a fretted fingerboard. The body itselfconsists of two flat surfaces fastened together by a contoured panel at their outsideedges. The bottom of the body is mechanically attached to the top by a post, andthe cavity of the body, coupled to the external atmosphere, acts as a resonator.The body of the ukulele may be made of wood or steel or plastic. The 61-cm longinstrument is played by strumming the strings with the fingers, and the resonantfrequency of each string is varied by the fingers pressing against the frets that arespaced so the sounds produced by stopping a string on any two adjacent frets are

526 18. Music and Musical InstrumentsFigure 18.15. Additional plucked-string instruments from top to bottom: ukulele, guitar,banjo, and mandolin (from Olson, 1967).one semitone apart. The guitar is similarly constructed except it is larger than theukulele by nearly twice the length, and it consists of six strings, tuned to E 2 ,A 2 ,D 3 ,G 3 ,B 3 , and E 4 . The guitar is played by either plucking with fingers or with apick or plectrum (a flat piece of metal or plastic) held firmly between the thumband the first finger. The adjacent frets also result in notes that are one semitoneapart.The body of a mandolin consists of a flat top attached to a hollow semiellipsoidalbody. This combination of the body and the body cavity coupled to the externalair through a hole constitutes a complex resonator. A bridge mounted at the centerof the flat surface of the body couples the vibrating strings to the body.

18.9 Strings 527The difference between the construction of a banjo and other string instrumentsis that its body consists of a skin membrane stretched over one end of a truncatedcylinder, thus making the body drumlike. The other end of the cylinder is open. Thebridge is supported by the stretched skin and it couples the strings to the stretchedmembrane, which, in turn, provides a large resonant area that is coupled to theair. Four long strings and a short string are stretched over the bridge between thetailpiece and the fingerboard. The relatively long neck is fretted; the short string isreferred to as the melody string. A more modern version called the tenor banjo isequipped with four strings of equal length, and it has supplanted the older modelwith the one short string. The four open strings are tuned to C 3 ,G 3 ,D 4 , and A 4 .This instrument can be played by either plucking the strings with the fingers orwith a pick or plectrum that can be made of a flat piece of tortoise shell. The noteof a string and hence the resonant frequency can be varied by pressing it againstthe frets. The spacing between a pair of adjacent frets corresponds to a differenceof one semitone. The overall length of a banjo is approximately 86 cm.Over the past three decades, Europeans and Americans have become familiarwith structured musical compositions called ragas through concert performancesby Ravi Shankar who performed them on the sitar which is northern India’s predominantstring instrument. The sitar’s seven main strings are tuned in fourths,fifths, and octaves to approximately F # 3 , C# 2 , G# 2 , G# 3 , C# 3 , C# 4 , and C# 5. In additionthere are 11 sympathetic strings tuned to the notes of the raga. The inharmonicitiesare quite small; the high-curved frets permit the player to execute with vibrato andglissando. The curved bridge allows for both amplitude and frequency modulationfrom the rolling and sliding of the string.The harpsichord (or cembalo) and its older cousin, the clavichord, both of whichresemble shrunken baby grand pianos (the clavichord is more boxlike in configuration),trace their common origin as far back as the twelfth century. The harpsichordwas the mainstay of chamber music during the baroque and classical period untilthe advent of the more versatile and louder pianoforte, the immediate precursorof the modern piano. Because so many excellent examples exist to this day, theart of constructing harpsichords have been revived so that modern audiences cantoday enjoy the music that have been composed expressly for this medium. Duringthe twentieth century, a number of excellent performers revived audience interestin the instrument and the works written for it, among them the great WandaLandowska and later on Igor Kipnis (the son of the great Ukrainian operatic basso,Alexander Kipnis).Figure 18.16(a) illustrates the structure of a harpsichord, which consists of a largenumber of steel strings stretched over a rather triangular steel frame. The keyboardranges about 4 1 / 2 octaves from A 1 to F 6 , but different versions of harpsichordshave been built to cover both larger and smaller ranges. The strings are excited bybeing plucked by a key-actuated mechanism shown in Figure 18.16(b). The keyis coupled through a level system to a short plectrum of leather, fiber, or tortoiseshell that plucks the string which deflects to let the plectrum slip past. When thekey is released, the plectrum, which is attached to a short spring-loaded level, slipsback under the string. A damping pad also mounted on the jack stops the sound.

528 18. Music and Musical InstrumentsFigure 18.16. The harpsichord: (a) its construction and (b) the string-plucking mechanismof a harpsichord.Bowed-string instruments are played by exciting the strings with a bow. Fourmodern instruments of this type, in order of increasing size, are: the violin, viola,violoncello, and double bass (cf. Table 18.3). Figure 18.17 is illustrative of thecomparative size of these instruments. Details and nomenclature of the violin’scomponents (the nomenclature also fairly applies to the other three instruments)are shown in Figure 18.18. The two smaller instruments, the violin and the viola,are generally played with the chin rest tucked under the chin, with the fingerboardcradled in the arch formed between the thumb and the index finger of the lefthand; and the fingers of that hand vary the notes of the strings by pressing themagainst the fingerboard surface. The bow is handled by the right hand. The largervioloncello and the double bass, being much larger instruments, are played heldin a tilted, almost vertical position, between the knees of the seated players, withendpins elevating the instruments from the floor.Table 18.3. Typical Characteristics of Standard Bowed-String Instruments.Typical OverallBowed-String Notes of Tuned Length of Full-Fize Typical Overall Range, NumberInstrument Open Strings Instrument, cm Length of Bow, cm of OctavesViolin a G 3 ,D 4 ,A 4 ,E 5 60 75 >4Viola C 3 ,G 3 ,D 4 ,A 4 70 75 >4Violoncello C 2 ,G 2 ,D 3 ,A 3 124 72 3Contrabass E 1 ,A 1 ,D 2 ,G 2 198 66 3a Smaller violins, as small as quarter size, have been constructed to accommodate small children learningto play. Even smaller ones have been constructed for 2-year-olds learning under the Suzuki method.

18.9 Strings 529Figure 18.17. Modern bowed-string instruments: (a) violin, (b) viola, (c) violoncello, and(d) double bass.

530 18. Music and Musical InstrumentsBOWSTICKHEADHAIRFROGSCREWSCROLLPEGSPEGBOXNECKBACK PLATEFINGERBOARDPURLINGBASS BARRIBSTRINGSSOUND POSTBRIDGETOP PLATE(BELLY)TAILPIECEF-HOLES (SOUND HOLES)CHIN RESTBUTTONATUNERFigure 18.18. Construction details of a violin and nomenclature of its structural elements.These instruments developed in Italy during the sixteenth and seventeenth centuriesrequire extremely sophisticated skills to construct. Their quality of craftsmanshipreached a peak during the eighteenth century in Cremona, Italy, particularlyunder the skilled hands of Antonio Stradivari (1644–1737) and GuiseppeGuarneri del Gesù (1698–1744). Because of the complexity of their constructionthat affects the quality of their tones, the violin family has been the object ofmuch acoustical research. Savart, Helmholtz, and the Nobel Laureate C. V. Raman(1888–1970) contributed to the understanding of the generation of sound with theseinstruments (Savart, 1840; Raman, 1918; Helmholtz, 1954). In more recent times,considerable work has been conducted in Germany through the efforts of Werner

18.9 Strings 531Lottomoser, Jürgen Meyer, and Frieder Eggers. Especially noteworthy is the workof Lothar Cremer (1905–1990) and his colleagues, which culminated in Cremer’sclassic text The Physics of Violins. In the United States, Frederick Saunders (1875–1963), best known for his work in spectroscopy, investigated many violins, makingmany acoustical comparisons between the old and the new. He, Carleen Hutchins(1911–), John C. Schelleng (1892–1979), and Robert Fryxell (1924–1986) establishedthe Catgut Acoustical Society, an organization that promotes researchon the acoustics of the violin family. In 2004, the society combined with thenonprofit Violin Association of America, becoming a sector known as the CASForum.Violin makers (or luthiers) are especially concerned with the vibration of thefree top and back plates. They tested these plates by tapping and listening totones. Modern technology is now being used to visually observe mode shapes(Figure 18.19) through the use of holographic interferometry. The finished violin’svibrational modes are considerably different from those of the free plates formingthe top and bottom of their bodies.The bow for any of these four instruments consists of horsetail hair stretchedbetween the two ends of a thin wood, one end constituting the head and the otherpoint of attachment being a movable frog that is connected at the other end of thebow to a screw inside the bow wood (cf. Figure 18.18). The screw can be turned tomove the frog, thus adjusting the tension of the stretched horsehair. The horsehairis rubbed with rosin to provide friction between the bow and the strings. When the(a) 369 Hz(b) 459 Hz (c) 503 Hz (d) 335 Hz(e) 739 Hz (f) 852 Hz (g) 880 HzFigure 18.19. Holographic interferograms of a top plate of an instrument of the violinfamily (Ågren and Stetson, 1972).

532 18. Music and Musical Instrumentsbow is drawn across the strings, the string vibrates as the result of its being draggedwith the bow and then springing back under the impetus of the restoring force.The normal range of the bow’s action on the strings depends on its force actingon the strings and the position of the bow relative to the bridge. Too heavy a forceresults in raucousness of the tone. Too little force results in instability of the stringdisplacement’s saw-toothed curve. According to Schelleng (1974), the maximumbow force depends primarily on the string and on its coefficient of friction. Thisbow force is inversely proportional to the distance of the bow from the bridge; theminimum bow force, on the other hand, is inversely proportional to the square ofthe distance of the bow from the bridge. The maximum and minimum bow forcesare equal when the bow is placed at a point very close to the bridge, and theydiverge as the bow moves away from the bridge. The disparity between these twolimits, as the bow is positioned further away from the bridge, provides the widetolerance that makes fiddle playing possible. As a far distance from the bridge,for a given bow velocity, the volume of the sound is less, the amount of highfrequencycontent lessens, and the timbre possesses a gentle character designatedby composers as sul tasto (“bow over the fingerboard”). In placing the bow nearthe bridge, the required bow force soars dramatically to almost prohibitive levelsand the solidity of the fundamental tone disappears, leaving little more than theremnants of high harmonics to suggest the fundamental tone: this is the eeriersounding sul ponticello (“bow over the little bridge”). In the normal playing area,the tone becomes more brilliant (i.e., the relative harmonic content increases) as thebow moves toward the bridge or as the bow force increases toward the maximum.A vibrating string alone produces almost no sound, because it is so slender thatalmost no air is displaced by its vibration and, moreover, the two diametricallyopposite sides of the string are so close together that when the air on one side iscompressed, the air on the other side is rarified. These two effects are so close toeach other that they effectively cancel each other. In order to avoid the cancellation,one vibration must follow another by a substantial fraction of a phase.In the bowed instruments, under optimal conditions, the top and the back platescan move inward and outward at a given moment that nearly the entire surfaces ofthe instrument act to change the volume of displaced air, thus acting as a “simplesource” in the lower frequency range (Cremer, 1971). In the violin family, thisvolume change, essential to generating sounds in the lower octaves, is renderedpossible by the asymmetric layout of the bass bar and the sound post inside thebody. When the bow is pulled across the string, a rocking motion is set up inthe bridge so that the two feet of the bridge are in a “push–pull” mode, with onefoot of the bridge pressing down and the other foot going up in opposition. Ifthe box (i.e., the body of the instrument) has total bilateral symmetry, the motionof one foot of the bridge would cancel that of the other foot of the bridge, andno volume displacement would occur. But the sound post, essentially a dowelfirmly coupling the back plate to the top plate tends to immobilize the right (withrespect to the player holding the instrument) foot while allowing the left foot tomore freely and causing the body of the instrument to vibrate under the influenceof the left foot. Thus, the sound post’s chief role can be considered as creating

18.9 Strings 533acoustical asymmetry inside the body. The shape, positioning, wood quality, andfitting of the sound post constitute highly critical factors in performance qualityof the instrument.The bass bar runs along the length of the interior surface of the top plate approximatelyunder the lowest-tuned string. Its function is to keep the vibrations of theupper and lower areas of the top plate in phase with the left foot of the bridge. Thebar is glued to the top plate in such a manner so as to provide structural strength tothe thin wood forming the top plate. Both the sound post and the bass bar enable thetop plate to withstand downward forces of 70–90 N from the strings. The contourof the bass bar is extremely important in the proper “thinning” or “tuning” of thetop plate.The f-holes serves two acoustical purposes: (a) reduction of the stiffness of thesurface on which the bridge stands in order to provide suitably tapered transitionbetween the bridge and the radiating area of the top plate and (b) formation of aHelmholtz resonator with the body of the instrument, thereby strengthening soundin the lowest octaves. The purling are inlays of thin wood placed in grooves alongthe edges of the plates. This allows the plates to vibrate more as if they were hingedrather than being clamped at the edges.In the violin, the top plate is generally constructed of softer Norway spruce(Picco abies or Picca excelsis), and back plate and the ribs are carved from (harder)curly maple (Acer platanoides). Fingerboard, tailpiece, and pegs are usually madeof ebony. The bow stick is made from pernambuco.The strings of the violin were originally all gut. Since about the seventiethcentury, the lowest (G) string was commonly wound with silver to improve theresponse. Present-day violinists use wound strings (i.e., metal wound over coresof gut, nylon, or metal) for D and A strings and steel E-string. The latter string canbe finely adjusted with the use of a fine tuner.In addition to the four principal members of the violin family, the aforementionedCatgut Acoustical Society was responsible for the development of the violin octet,an ensemble of eight specially scaled new violin family instruments (Hutchins,1967). The octet of instruments, illustrated in the photograph of Figure 18.20 anddeveloped by the society to meet the challenge of encompassing the entire rangeif orchestral music, consists of the treble violin, the soprano violin, the mezzoviolin, the alto violin, the tenor violin, the baritone violin, the small bass, and thecontrabass. Although these instruments form a family with basic traits in common,each member has its own individual personality. The homogeneity of sound arisesfrom adjustments in body length and other physical characteristics so that eachinstrument has its own main wood resonance and main air resonance near the twoopen middle strings. The musical ranges of the strings on these instruments areshown in Figure 18.21, as they relate to the corresponding notes on a standardpiano. These eight instruments of the new violin family range in overall lengthfrom 48 cm to 214 cm.The principal modern struck-string instrument is the piano that are availablein several models, ranging in size from the more modest upright or spinet pianoto the concert grand piano. The heart of the piano consists of a large number of

Figure 18.20. The violin octet developed by the Catgut Acoustical Society. (Courtesy ofthe New Violin Family Association.)Figure 18.21. The musical ranges of the of the new violin octet developed by theCatgut Acoustical Society as related to piano notes. (Courtesy of the New Violin FamilyAssociation.)

18.9 Strings 535Figure 18.22. Schematic of the mechanism of a grand piano (from Olson, 1967).steel strings stretched on a metal frame. The strings couple through a bridge toa large soundboard. The strings are activated by being struck by hammers thatare connected to keys forming a keyboard. Pressing down on a key actuates thehammer, which, in turn, strikes the string. The conventional piano is equippedwith 88 keys, and the piano covers a wide frequency range of more than sevenoctaves, from A 0 to C 8 (27.5–4186 Hz). The Bosendorfer 244-cm concert grandpiano features 97 keys, but not much music has been written for these extra keys.The extra keys are mainly there because of the additional resonances produced bythe extra strings and the larger soundboard.Figure 18.22 illustrates a schematic of the piano mechanism for a grand piano.The strings stretch from the pin block across the bridge to the hitch-pin rail at theother end. When a key is pressed downward, the damper rises and the hammerimpacts the string, causing it to vibrate. The string’s vibrations are transmitted to thesoundboard through the bridge. The hammer rebounds, remaining about 1.25 cmfrom the string as long as key remains pressed. When the key is depressed, thedamping pad does not engage the string. When the key is released, the dampingpad engages the string to speed up the decay of the sounding note.The largest version of the piano, the concert grand, has 243 strings that varyin length from about 200 cm at the bass end to approximately 5 cm at the trebleend. In this group there are 8 single strings wrapped with one or two layers ofwires, 5 pairs of strings also wrapped, 7 sets of 3 wrapped strings, and 68 setsof 3 unwrapped steel strings. The smaller pianos may contain fewer strings butthey still play the same number of notes. A small grand piano may carry 226strings. The arrangement of the strings is such that the bass strings may overlaythe middle strings, so that they can function nearer the soundboard. The soundboarditself is usually made of spruce and it is up to 1 cm thick; it acts as theprincipal source of radiated sound just as the top plate of a violin does. Becausethe tension forces in the strings are so high, the frames are fabricated of cast

536 18. Music and Musical Instrumentsiron, which also provides dimensional stability necessary to maintain the stateof tune.Three pedals are provided on the conventional piano. The right or sustainingpedal removes all dampers from the strings so that the strings become damped onlyby the soundboards and end supports. The center or bass sustaining pedal removesthe dampers from all the bass strings. The left pedal, or the soft pedal, reduces thesound output by lessening the length of the stroke of the hammers or by shiftingthe hammers so that fewer strings are struck or by permitting the dampers to act.The dulcimer, considered by some to be the forerunner of the piano, consists ofa large number of strings stretched over a frame mount in an oblong box. Thesestrings pass over bridges that are coupled to a soundboard. The oblong box ismounted on legs, causing the instrument to resemble a square piano without keys.The instrument is played by striking the strings with two hammers, one in eachhand. Dampers controlled by a foot pedal are provided. The range of the dulcimeris from D 2 to E 6 .18.10 Wind InstrumentsA wind musical instrument is a device that generates sound by (a) blowing a jetstream of air across some type of opening, as in whistles, flutes, or piccolos, fifeand flue organ pipe; or (b) by buzzing of lips (acting as reeds) of a bugle, Frenchhorn, trumpet, tuba, or trombone; or (c) by vibrating a reed (or a double reed, i.e., aset of two reeds) through the means of airflow in an accordion, clarinet, saxophone,oboe, bagpipe, English horn, bassoon, sarrusophone, and the human voice.The reed instruments fall into two sub-categories: one category entails instrumentsin which air pressure tends to force the reed valve open (the human larynx,buzzing lips of brass instruments, and harmonium reeds); the other category includesinstruments in which the air pressure forces the reed valve to close (e.g., clarinets,oboes, similar woodwinds, and organ reed pipes). The first category tends toact as a sound generator over a relatively narrow fundamental frequency range, justabove the fundamental frequency of the reed. The other category serves as a soundgenerator over a wider range of frequencies, just below the resonant frequency ofthe reed—but some type of coupling to a pipe resonator must be provided.Figure 18.23 illustrates two possible configurations of a vibrating reed generator.The reed generator is really is a pressure-controlled device for cutting off andreinstating airflow at selected frequencies. In both cases of Figure 18.23, the blowair pressure p b (gauge pressure, relative to the atmospheric pressure) is appliedfrom the left. If p b > 0 (i.e., the blow pressure is above atmospheric pressure), thevalve in (a) is forced closed by the positive pressure and the valve in (b) is forcedopen. The opposite situation occurs when p b < 0. In general terms, the motion ofa reed can be described by (Fletcher and Rossing, 2004).[ d 2 xm r = + 1 ]dxωdt 2 r + ωr 2 Q r dt(x − x U 20) = γ gr (p b − p) + γ be|x| 2

18.10 Wind Instruments 537Figure 18.23. Two types of vibrating reed generator.wherem r = mass of reedQ r = quality factor of reed resonancex = position of valve relative to its seatx 0 = equilibrium position of valveω r = angular frequency of reedp b = blow pressurep = acoustic pressureγ gr = geometric factor for the exposed reed facesγ be = Bernoulli force factor based on internal flowin the narrow part of valve gapU = flow velocity.In the air-reed instruments, a steady stream of air does the activation by flippingin and out of the pipe or cavity at the resonant frequency of the system, as shownin Figure 18.24. This steady stream of air thus becomes an alternating flow, andbecause of the nonlinear nature of the exciting force, a number of the resonantelements in the system are excited, thus yielding a series of overtones that add tothe fundamental tone. The whistle and flue organ pipe, each consists of a cavity,a closed or open pipe coupled to an air reed that is activated by a steady airstream. The air stream entering the pipe vacillates between moving to the insideof the pipe and moving to outside of the pipe. Upon entering the pipe, the airstream compresses the air in front of it, as depicted in Figure 18.24 at θ = 0 ◦ . Thepressure inside the pipe builds up to the equilibrium point, and no more air will

538 18. Music and Musical InstrumentsFigure 18.24. The mechanism of the flue pipe organ, an air-reed instrument. The magnitudeof the pressures are indicated by the diameter of the circles. A dark circle denotesa pressure above atmospheric, and a white circle indicates a pressure below atmospheric.The direction of the arrows represents the direction of airflow, and the magnitude of particlevelocities are indicated by the length of arrows.then enter the pipe, as indicated in Figure 18.24 at θ = 90 ◦ . The excess pressurewill now direct the incoming air stream to the outside at θ = 180 ◦ . This resultsin the excess pressure being relieved and a rarefaction owing to the inertia of theoutgoing air, as shown for θ = 270 ◦ .The decreased pressure pulls in the air to renew the cycle at 0 ◦ . This cycle consistingof the four phases repeats itself at the resonant frequency of the system.The frequency of the complete cycle occurs at the resonant frequency of the closedpipe. Odd harmonics are also produced, because the closed pipe resonates at thesefrequencies as well as the fundamental; this action steers the air stream vacillatingfrom the interior and the exterior. In the case of the open pipe, both odd and evenharmonics are produced. Sectional views of the open-flue pipe (made of metal) anda stopped-flue pipe are shown in Figure 18.25. A whistle, such as that used by policeofficers and referees of athletic contests, operates on the same principal as the organflue pipe except that the resonating chamber is a Helmholtz resonator, which isessentially a chamber with a narrow neck. The resonance of a whistle can be foundfrom application of Equation (7.66), on the basis of the volume of the whistle chamberand the area of the sound-radiating hole coupling the chamber to the outside air.The overtones due to resonances within the small chamber occur at relatively highfrequencies and are effectively suppressed by the inertance of the sound-radiatinghole, thus resulting in a nearly pure tone of the whistle. A calliope is a groupof whistles with frequencies corresponding to the notes of a musical scale. Eachwhistle is controlled by a valve connected to a key that is part of a keyboard similarto that of a piano. Either steam or compressed air is used to actuate the whistles.

18.10 Wind Instruments 539Figure 18.25. Sectional views of the open flue pipe (left) and a stopped-flue pipe (right).The recorder and the flageolet can be categorized as instruments of the whistleclass. Each is equipped with a mouthpiece, a fipple hole, and a cylindrical tubebearing a set of fingerholes. The recorder has eight fingerholes plus one thumbholeon the opposite side of the tube to alter the resonant frequency of the air column.Different types of recorders have been constructed ranging in length from 30 cm toalmost 900 cm. The flageolet features a set of four fingerholes plus two thumbholes.The ocarina’s construction differs from that of a flageolet in that its resonatingsystem is a cavity and hole combination, not a pipe. The resonator is coupled tothe air through several holes, including the fipple hole. The resonance frequencyis increased as the number of holes is increased, because the inertance decreaseswith the number of holes. Ocarinas, which may be made of metal, ceramic, orplastic, cover about a range of an octave and a half.The flute illustrated in Figure 18.26 is a cylindrical main tube with a slightlytapered head-joint. One end of the flute is open and the other is closed. The embouchure(blowhole) is located a short distance from the closed end. The holes arecontrolled by closing or opening them in order to vary the resonant frequencies

540 18. Music and Musical InstrumentsFigure 18.26. The modern flute.corresponding to the musical scale, either by the fingers directly or through actuationof keys. This system of keys and connecting shafts and levels constitutea mechanism that actuates valves that are basically disks that cover fingerholes.Springs are utilized to keep the valves in the unactuated position, which may beeither open or closed. This system renders it possible to open or close fingerholesthat are too far apart or are too large to be stopped by the fingers alone.The sound of the flute is generated in this manner: In Figure 18.27 an air streamfrom the lips impinges upon the embouchure of the flute. Resonant frequencies aregenerated by the air stream slipping back and forth between entering the flute bodyand flowing past the embouchure. A stream of air enters the blowhole (ϑ = 0 ◦ ),causing a pressure to build up to the extent that it stops air from entering (ϑ = 90 ◦ ).The excess pressure (ϑ = 180 ◦ ) then forces the air out of the blowhole until it stopsleaving the blowhole (ϑ = 270 ◦ ). The cycle then repeats itself.The piccolo can be considered a smaller version of the flute that operates oneoctave higher. The operational principle of the piccolo is fundamentally the sameFigure 18.27. The action of the flute: the directions of the arrows indicate airflow direction.

18.10 Wind Instruments 541Figure 18.28. Reed position and particle velocities (indicated by arrows) in a mechanicalreed instrument for a complete cycle (after Olson, 1967).as that of the flute, but the fundamental range is from D 5 to B 7 instead of the rangefrom C 4 to C 7 of the latter instrument. The fife is a somewhat simpler instrument,consisting of a tube of metal or wood that is closed at one end and open at theother. It is equipped with six fingerholes distributed along the tube and a blowholenear the closed end.Mechanical reed instruments are those that use a steady air stream to actuate amechanical reed to throttle the airflow at the resonant frequency of the reed andthe associated acoustical system. These instruments may be subdivided into twocategories: single- and double-reed instruments.Single-reed instruments use a single-mechanical reed to throttle a steady airstream to generate a musical sound. A free-reed instrument is one that radiatesits sound directly into the air. Modern examples of the free-reed instrument arethe free-reed organ (harmonium), accordion, and harmonica. In other instruments(e.g., reed organ pipe, clarinet, saxophone, or bagpipe), the reed couples to aresonant air column. The action of the reed is that of a free-end (cantilever) vibratingbar described in Chapter 5. The mechanical reed in its quiescent stateis positioned so that it forms an opening with the structure of the instrument asshown in Figure 18.28. The velocity of the air moving past the reed through theopening results in a lower pressure on the flow side, due to the Bernoulli effect, 1which causes the reed to flex toward the lower pressure, causing the opening to1 Bernoulli’s principle states that the dynamic pressure of a nonviscous fluid flow will drop with anincrease in the velocity of the flow and vice-versa according to the relation:p + 1 2 ρu2 = constantwhere p is the air pressure, ρ the air density, and u the air velocity.

542 18. Music and Musical InstrumentsFigure 18.29. Two accordion versions: (a) button keyboard accordion and (b) piano keyboardaccordion. (Photographs of the Hohner Corona II diatonic accordion and HohnerAtlantic IV piano keyboard accordion courtesy of Hohner, Inc.)become smaller. The airflow then becomes reduced by the constricted passage,which causes the pressure on the flow side to increase, and the reed springs backto its original position. It then moves beyond its original position, under the effectof inertial energy, and the airflow becomes larger. The pressure again drops andthe reed returns to the normal position it had at the beginning, and the cycle ofevents repeats at the resonant frequency of the system. This action has the effectof converting a steady air stream into a saw-toothed pulsation that contains thefundamental and its harmonics.The harmonium or the free-reed organ operates through a series of air-actuatedfree reeds tuned to specific notes. The air supply driving the reeds is providedby two pedal-operated pedals connected to bellows that connects to a wind cheston the top of which reeds are mounted. This type of organ features a piano-likekeyboard. Each key operates a valve that controls the air supply to the reed, withone key for each reed. Reed organs are usually equipped with stops for connectingbanks of reeds. Thus, a single key can activate a number of reeds.The accordion shown in Figure 18.29 functions by the player’s arms working abellows which provides the air supply to the reeds, alternatively creating pressure(when the bellows is compressed) and a partial vacuum (when the bellows isexpanded). The air-actuated reeds, tuned to notes of a musical scale, are controlledthrough a keyboard. Each key connects to a valve that controls two separate reeds.One reed operates under pressure and the other under a vacuum. In some versionsof the instrument, two differently tuned reeds are assigned to each key that one toneis produced with expansion of the bellows and another tone with the compression.The vibrational action of the reed is that of a free-end bar. Accordions, which arereally portable versions of reed organs, are usually equipped with stops to connectindividual reeds so as to form banks of reeds. This enables operation of several

18.10 Wind Instruments 543reeds from a single key. In playing the accordion, the performer uses the left handto move the bellows and to play the bass parts and the accompaniment and usesthe right hand to play the melody.The simple accordion shown in Figure 18.29 contains 10 melody buttons orkeys and tow bass keys. Each key produces two different tones, one tone uponbellows expansion and the other upon bellows compression. The piano accordionof Figure 18.29(b) is equipped with a two-octave piano-style keyboard (with blackand white keys) and 12 bass and chord buttons. The same tone is produced onboth compression and expansion cycles. There is no standard size for accordions.More elaborate accordions have been constructed with melody keyboards coveringup to four octaves, and some of the larger accordions contain as many as 120bass buttons arranged in six rows rather than the two rows of six buttons each asshown in the figure. The first and the second rows provide the bass notes whilethe other four rows produce, in the following order: the major, minor, dominantseventh,and diminished-seventh chords. A medium-sized accordion with a pianostylekeyboard can cover in its melody section a frequency range from F 3to A 6 .The mouth organ or harmonica, which comes in number of versions, is playedby a performer’s breath providing the air supply through both exhalation andinhalation. The instrument consists of a set of tuned free reeds mounted on a wood,metal, or plastic box, with channels leading from orifices to the reeds along one sideof the box. Each channel connects to two differently tuned reeds, one reed operatingunder pressure (exhalation) and the other under a partial vacuum (inhalation). Eachreed behaves as a cantilever bar in operation. To play a melody the mouth coversone or more holes, with the tip of the tongue providing some additional control.Harmonicas fall into three principal categories: the simple harmonica (a single rowof 10 holes, 20 different notes), the concert harmonica (two rows, one row tunedan octave higher than the other), and the chromatic harmonica. The chromaticharmonica, although similar in appearance to the concert type, is really a set oftwo harmonicas, one placed above the other. The bottom instrument is tuned onesemitone above the upper instrument. This instrument contains a slide which ismoved by a knob to the right. When the slide is in, the upper holes are closed andthe lower holes are open; the converse occurs when the slide is out. A fairly typicalsimple 10-hole harmonica can cover 2 1 / 2 octaves from C 3 to F 5 .The reed organ pipe contains an air-actuated reed coupled to a conical pipe,with the fundamental frequencies of the reed and the pipe itself fairly matching.In some installations, a combination of a conical pipe and cylindrical pipe is used.Because the pipe and the reed are intimately coupled, the resonant frequency isestablished by the combination of these two components. The resonant frequencycan be changed by altering the resonant frequency of either the reed or the pipe.A reed organ pipe is customarily tuned by changing the effective stiffness of thereed by moving a tuning spring that is in contact with the reed. A cross-sectionalview of the organ pipe mechanism is shown in Figure 18.30. The organ mayalso be tuned by changing the effective length of the pipe by a metal rollbackin the side of the pipe near the open end. Both the shape of the reed and that of

544 18. Music and Musical InstrumentsFigure 18.30. Reed organ pipe and reed mechanism.the pipe determine the timbre. Voicing is the process of selecting and adjustingthese components to yield the proper timbre. Reed organ pipes are generally designedto imitate orchestral instruments such as the trumpet, tuba, oboe, clarinet,the human voice, and so on. These pipes are classified as chorus and orchestralreeds.The clarinet consists of a single-mechanical reed coupled to a cylindrical tubewith a flared open end. Control over the effective length of the resonating air columnis provided by a number of holes that may be opened or closed by fingers, eitherdirectly or through keys. Any change in the effective length of the air columnresults in a change of the resonant frequency. The reed position in the bottomof the mouthpiece functions in the manner described above for Figure 18.28.The throttling action of the reed changes a steady air stream into a saw-toothedpulsation that contains the fundamental and its harmonics. When sounded alone,the reed produces a sound rich in harmonics. The quality of the tone is improvedwhen the reed is coupled to the cylinder air column. The clarinet produces a rangecovering more than three octaves, from D 3 to F 6 . The clarinet’s overall lengthis about 63 cm. The bass clarinet is a larger instrument, being 94 cm in overalllength, which produces tones in a lower frequency range, from D 2 to F 5 . This lowerfrequency range is obtained by doubling the tube on itself.The saxophone operates in a similar fashion to the clarinet. The principal differencein its construction is that the diameter of the tube at the reed end is smaller,thus resulting in a lower acoustic impedance. Also, the coupling between the reedand the pipe is not as intimate as in the case of the clarinet. The buildup of the reedvibration is extremely quick, producing a sharp attack that is characteristic of thesaxophone. Saxophones come in various sizes, including soprano, alto, tenor, baritone,and bass. The smallest of these, the soprano saxophone, employs a straight

18.10 Wind Instruments 545Figure 18.31. The bagpipe.tube with a slightly flared open end; all the other saxophones use a curved mouthpieceand an upturned bell at the other end. Each of these types of saxophonescovers a fundamental range of about two and a half octaves. The soprano producesA 3 to E 6 ; the alto from D 3 to A 5 ; the tenor from A 2 to E 5 ; the baritone fromD 2 to A 4 ; and the bass from A 4 to D 4 . Their overall lengths range from 40 cmfor the soprano version to nearly 100 cm for the bass saxophone.The bagpipe of Figure 18.31, the instrument so symbolically evocative ofScotland, contains one or more combinations of air-actuated reed coupled to aresonating pipe. A leather bag serves as a reservoir and air supply for actuatingthe reeds. Air is supplied to the bag by blowing with the breath. Because the reedsare supplied with a steady stream of air, the reeds sound continuously withoutinterruption, distinguishing the bagpipe from other breath-blown instruments thatdo not provide such steady sounds. There are usually two or three fixed-frequencyreed–pipe combinations that are called drones. In addition there is a variable pitchreed–pipe combination, or the chanter. On the chanter pipe, eight fingerholes areprovided so that discrete frequencies can be achieved over the range of an octave.The chanter supplies the melody, and the drones produce a harmonious steadytone. The reed and pipe mechanism is similar to that of the reed–organ pipe.However, in the modern version of the bagpipe, the reeds in the drone are of thesingle-mechanical type and the reed in the chanter is a double-mechanical type.Double mechanical-reed instruments use two mechanical reeds for throttling asteady stream of air to produce a musical sound. All of these instruments couplethe double-mechanical reeds to a resonant air column. Such instruments includethe oboe, English horn, oboe d’amore, bassoon, and sarrusophone.Figure 18.32 depicts the operational principal of the double reed. When thereeds are in the normal position, i.e., they are slightly apart, air is forced through

546 18. Music and Musical InstrumentsFigure 18.32. The action of a double reed. Air particle velocities are indicated by arrows.the opening between the reeds. The high velocity of the air reduces the pressurebetween the reeds, in accordance with the effect of Bernoulli’s principle.This causes the two reeds to be forced closer to each other, thereby constrictingthe airflow. With the airflow now reduced by the constriction, the pressureincreases, causing the reeds to spring back toward their original position, but owingto their momentum, they moved beyond their original positions. The openingis now at its largest, and the airflow is increased accordingly. The internal pressureon these reeds are now quite small, so they return to their original position,and the cycle begins again with this sequence of events repeating at the rate ofthe resonant frequency. Thus, a steady air stream undergoes a throttling actionthat generates a saw-toothed signal that contains the fundamental and all of itsharmonics.The oboe consists of a double-mechanical reed such as that shown in Figure18.32, coupled to a conical tube with a slight flared mouth. The effective length ofthe resonating air column is controlled by the number of holes that are opened orclosed by the fingers either directly or through keys. The oboe covers three octavesfrom B to G 6 ; the overall length of this instrument is 62 cm. The English hornresembles the oboe in most respects, especially in the key and fingering system,with the principal difference being that the double-mechanical reed is coupled toa tapered conical tube that terminates in a hollow spherical bulb with a relativelysmall mouth opening, thus producing a unique timbre. The fundamental frequencyrange of this 90-cm long instrument covers less than three octaves, from E 3 to B 5 .Approximately 70 cm long, the oboe d’amore is a smaller twin of the English horn,and it ranges over nearly three octaves from G# 3 to C# 6 .The bassoon is a noble-sounding instrument that consists of a double-mechanicalreed coupled to a conical tube that is doubled back on itself so that a lower frequencyrange is provided without compromising the portability of the instrument itself.There is no appreciable flare at the mouth. A set of holes on the side can be opened

18.10 Wind Instruments 547Figure 18.33. A contemporary organ console (from Olson, 1967).or closed by fingers either directly or through the use of keys to determine theeffective length of the resonating air column. The overall length of the bassoonis about 123 cm, but the doubled conical air column is about 245 cm long. Thebassoon covers roughly three octaves from E 1 to E 3 . The contra bassoon can beconsidered the bigger brother of the bassoon, but its tube is folded several timesto yield an air column as long as about 480 cm while keeping the overall lengthof the instrument to a manageable 127 cm. The fundamental frequency range runsfrom B 0 to F 3 .The sarrusophone, essentially a double-mechanical reed coupled to a foldedbrass tube of conical bore with a flare at the open end, comes in several sizescovering different fundamental ranges. The effective length of the air column isvaried by a number of holes that are opened and closed by cover valves operatedby keys. The most common type of sarrusophone is the contrabass type, whosefundamental frequency range from D 1 to B 3 .The modern organ illustrated in Figure 18.33 is really more than one type ofinstrument. It is considered to be a combination mechanical-reed and air-reedinstrument. It consists of a large number of flue- and reed-type pipes controlleddirectly by manual and pedal keyboards and less directly by stops, couplers, andpistons. The organ console in Figure 18.33 is shown to contain three manuals, apedal keyboard, tablet couplers, thumb and toe pistons, and swell pedals. Organs

548 18. Music and Musical InstrumentsFigure 18.34. Schematic of the elements of the modern organ (from Olson, 1967).and organ consoles, it should be mentioned here, are not standardized. The organmay be constructed with one or more manuals and with or without a pedal keyboard.In a five-manual organ, the functions of the manuals are as follows: The first (or thelowest) controls the pipes in the choir organ; the second controls the pipes of thegreat organ; the third controls the pipes of the swell organ; the fourth controlsthe pipes of the solo organ; and the fifth controls the pipes in the echo organ.The pedals control the pipes in the bass organ. In a number or more elaborateinstallations, other instruments have been added to pipes (e.g., gongs, cymbals,gongs, drums, etc.). Yet other installations contain as many as seven manuals. Thetypical pipe organ in use today has two to five manual keyboards and a pedalkeyboard.It can be seen from the schematic of Figure 18.34 that the organ is a combinationof several organs, each organ having a different kind of pipes. We have earlierdescribed the flue organ pipe and the reed organ pipe. Pipes can be classified intofollowing basic groups: the diapasons, flutes, strings, and reeds. The more commonflue and reed pipes are illustrated in Figure 18.35.In the evolution of the organ, it was found that the tones could be improved bygrouping a set of pipes to sound simultaneously by depressing a single key. Thuseach key can represent any number of pipes, and each combination of pipes iscalled a stop, a term which is also applied to a knob along the sides of the console.Each stop knob controls a specific group of pipes.

18.10 Wind Instruments 549Figure 18.35. Different types of organ pipes (from Olson, 1967).The pedal organ features the largest flue and reed stops. The great organ, thechoir organ, and the swell organ feature decreasing steps of flue and reed stops.The swell organ is placed in an enclosed space equipped with shutters in the wallbetween the wall and the audience. The opening and closing of these shutters arecontrolled by the swell pedals in the organ console. The solo organ, if there is one,is also enclosed in a swell box.Just above the top manual, a row of tilting tablets serve as couplers. This couplingsystem allows the actuation of a mechanism associated with more than one keyby simply pressing one key. The different manuals can be interconnected by thiscoupling arrangement; and the pedal keyboard, for example, can be tied to anymanual by this coupling procedure.In the early organs, keys were connected directly to valves that controlled theairflow to the pipes. This required considerable strength on the organist’s part topress the keys. The action was improved later by the introduction of a pneumaticsystem, in which the key operates a small pneumatic valve requiring a lighter force,which in turn operates a valve connected with the pipes. But this resulted in slowaction owing to the slowness of the propagation rate of air impulses from the key tothe valves at the organ pipes. As a result the console had to be located very near thepipes. Except for small organs, modern organs now use either electropneumatic orelectric action to actuate the pipes. In the electropneumatic action, pressing downon the key energizes an electromagnet to move a set of valves in an air chamber,causing bellows to expand and move a linkage that opens up a larger valve whichin turn lets in the air supply into the wind chest to activate the organ pipes. Theadvantages of the electropneumatic system are: a light force is required to pressthe key, the action is swift, the keys can be interconnected by merely flipping aswitch, and the console can be located almost any distance from the organ and evenmoved about, because the wiring connecting the console to the different organscan be grouped together inside a reasonably small, flexible cable. The all-electrical

550 18. Music and Musical Instrumentsaction essentially constitutes a servomechanism setup, where the key activates arelay that opens the wind chest to the organ pipes.Because of the large size and number of pipes, organs require large amountsof air for actuating the pipes. The air pressure required is quite small, rangingfrom about 1 kPa gauge pressure for the early hand-powered organs to as much as12 kPa air gauge pressure in some modern units. The air supply for modern organsis provided by an electric fan or an electrically driven centrifugal pump.Lip-Reed InstrumentsIn these types of instruments, the lips serve as the reeds and the air supply isprovided by the lungs. The puckered upper and lower lips can be imagined as apair of double-mechanical reeds of Figure 18.33, with the operational principlebeing virtually the same. The frequency of the pulses, which occur as a resultof the lips throttling the airflow, corresponds to the resonant frequency of thelips combined with the associated instrument. Because the coupling between thelips and the instrument’s mouthpiece is fairly loose, the fundamental resonantfrequency of the lips must correspond to that of the instrument. However, thecombined resonant frequency of the horn and the lips can be varied slightly up ordown by changing the tension of the lips, thus changing the resonant frequencyof the system. The modern lip–reed instruments are the bugle, trumpet, cornet,trombone, French horn, and tuba—collectively referred to as the brasses.The simplest of the brass instruments is the bugle, which consists of a cuppedmouthpiece attached to a coiled tube with a low rate of taper, terminating in abell-shaped mouth. The length of the air column is fixed; there are no valves,and the number of notes that can be played is limited. The overall length of thestandard bugle is about 57 cm, and the notes it can play are C 4 ,G 4 ,C 5 ,E 5 ,G 5 ,B 5 ,and C 6 .As it is equipped with three sets of keys that operate piston valves, the trumpetis a more versatile instrument than the bugle. The trumpet is constructed of acoiled tube almost 3 m in length with a slight taper, terminating in a bell-shapedmouth. The first third of the tube is almost circular, with the remainder slightlyconical except for the last 30 cm that flares into a bell-shaped mouth. Pushingdown on the keys of the piston valves adds to the effective length of the tube.With three valves closed and opening them singly, opening in three different combinationsof pairs, or with all three valves opened, it becomes possible to obtaineight different lengths of the resonating tube. A tuning slide or bit is provided sothat the resonant frequencies can be matched with other instruments. The trumpet,which has an overall length of 57 cm, covers about three octaves, from E 3to B 5 . A mute in the form of a pear-shaped piece of metal or plastic can be insertedinto the bell in order to change the quality of the tone and attenuate theoutput.Only about 36 cm in overall length, the cornet is a smaller version of thetrumpet. In addition to its size, the cornet has a bore that is tapered rather than

18.10 Wind Instruments 551cylindrical. It covers the frequency range from G 3 to B 5 . The French horn, arguablythe most beautiful appearing brass instrument, consists of a mouthpiececoupled to a slightly tapered coiled tube of 365 cm length, which terminates in alarge bell-shaped mouth. The modern version of the horn is equipped with threesets of rotary valves controlled by three keys. This valve system provides eightdifferent lengths for the resonating tube, and hence a series of different resonantfrequencies corresponding to the notes of the musical scale. The large size of themouth of the French horn renders it possible for the player to inert his hand andraise or lower the pitch or to produce a sound (e.g., muted) effect. The Frenchhorn can sound over three octaves, from B 1 to F 5 . Its overall length is nearly58 cm.The trombone and the bass trombone differ from the other brass instruments inthat they each feature a telescoping section of the tube that can be moved by theperformer to vary the length of the tube and hence the resonant frequency. Thetrombones are the only instruments besides the violin family that can provide acontinuous glide through the musical scale. The pitch is determined by the positionof the slide as well as the performer’s lips and the applied lung pressure. As with thecase of performers of the violin family, the player must possess an accurate sense ofpitch and the ability to produce the correct note. The trombone’s U-shaped tube isnearly 3 m long, with a slight conical taper that culminates in a bell-shaped mouth.It is capable of covering nearly 2 1 / 2 octaves, from E 2 to B 4 . The bass trombone islarger, covering three octaves from A 1 to G 4 .The tuba’s mouthpiece is attached to a coiled tube nearly 6minlength thatgradually increases in its cross-sectional area until nearly the end when it terminatesin a large bell-shaped flare. Three piston valves are provided, each addinga different length to the tubing. Thus, eight different lengths are provided, in thesame basic manner as the trumpet or cornet. Some versions of the tuba include afourth valve, with a corresponding increase in the number of different resonant frequencies.Largest of the brass instruments, a typical tuba, measures approximately1 m. The tuba covers three octaves from F 1 to F 4 . Different versions of the tubaexist, with a variety of sizes and forms. The sousaphone (named after band conductor/composerJohn Sousa of the “March King” fame) is the largest of the tubafamily.In concluding this section on wind instruments, it might also be mentioned thatthe human voice also classifies wind instruments, i.e., it is a reed instrument inwhich the vocal cords serve as the reeds. The mechanism of the voice consistsof three sections: (1) the lungs and associated muscles to serve as the air supply;(2) the larynx bearing the vocal cords for converting the airflow into a periodicmodulation; and (3) the vocal cavities of the pharynx, mouth, and nose, all ofwhich help vary the tonal content of the output of the larynx. A sectional viewof the voice system in the human head is shown in Figure 18.36. The frequencyof the vibration is governed by the tension of the vocal cords, the inertance, andthe combined acoustical impedance of the vocal cords and the vocal cavities. Thevocal mechanism is a complex one, entailing a number of acoustical elements

552 18. Music and Musical InstrumentsFigure 18.36. The human voice mechanism in a sectional view of the head.that can be varied by the singer to yield a wide variety of tones, which differ inharmonic content, quality, loudness, duration, growth, and decay. According toMachlis,the oldest and still the most popular of all (musical) instruments is the human voice. In noother instrument is contact between the performer and medium so intimate. In none otheris expression so personal and direct. The voice is the ideal exponent of lyric melody andhas consistently been the model for those who made instruments as well as for those whoplayed them.A human voice can range over two octaves, but variations do occur amongdifferent individuals as to the range, to say nothing of the beauty or quality oftones. It is a rare voice such as Yma Sumac’s that can cover four octaves or more.An effortless coloratura such as Joan Sutherland, a powerful heldentenor such asLauritz Melchior, or a sonorous basso such as Alexander Kipnis or Martti Tevalacomes along once a generation. On the average, a soprano can range from C 4 toC 6 ; an alto, from G 3 to F 5 ; a tenor, from D 3 to C 5 ; a baritone, from A 2 to G 4 ; andthe basso, from E 2 to D 4 .

18.11 Percussion Instruments 55318.11 Percussion InstrumentsPercussion instruments are made to sound by striking or shaking. The vibrationsystem that is excited by an impact may be a bar, rod, plate, membrane, or a bell.Two classes of percussion instruments exist: viz. definite pitch, in which the tonehas an identifiable pitch or fundamental frequency; and the indefinite pitch, inwhich the musical sound does not have a definite pitch or fundamental frequency.The former category includes the tuning fork, xylophone, celesta, glockenspiel,bells, and chimes. Drums, the triangle, tambourine, castanets, and cymbals fall intothe latter category. The tuning fork, discussed in Chapter 5, is used as a standardfrequency sound source. The resonant frequency of a tuning fork is determined byits dimensions and its material.The xylophone consists of a number of metal or wood bars, having specificresonant frequencies, mounted horizontally and supported on soft material at twonodal points. The vibration of each bar is that of a free bar, with the resonantfrequency dependent on the dimensions of the bar. A pipe acts as a resonatorby being coupled to each bar to improve the coupling between the bar and theair, thus yielding greater sound output. Felt-covered hammers are used to strikethe bars. Differential models of xylophones are available to provide fundamentalfrequency ranges of two to four octaves, generally from C 3 to E 7 . The marimba,which has African and South American roots, is basically an enlarged version ofthe xylophone that usually covers about five octaves, from F 2 to F 7 .The glockenspiel (also called orchestral bells) resembles a small xylophonebut small wooden hammers are used to strike the notes. The bell lyre is anotherversion of the glockenspiel. A set of steel bars is mounted on a lyre-shaped frame.A ball-shaped hammer is used to strike the bars. The bell lyre is used in marchingbands as it is small enough to be hold almost vertically by one hand while the otherhand holds the hammer. The glockenspiel can be constructed to cover up to threeoctaves. The frequency range usually falls between C 3 and C 6 .The celesta contains a series of resonant steel bars with specific resonant frequencies.The bars are actuated by hammers linked to keys forming a keyboard.Thus, the celesta resembles a small upright piano. The steel bars are suspendedabove wood resonating boxes, which are designed to improve coupling of the barsto the air, thus improving the sound output. Dampers are provided, and a singlepedal, called a sustaining pedal, removes the dampers so that the energy of thevibrating bar may be dissipated over a longer period. The celesta is a four-octaveinstrument that ranges from C 4 to C 8 .Chimes or tubular bells are a series of resonant brass tubes suspended verticallyfrom a wooden frame, and a foot pedal controls a damping bar. The tubularbells, whose resonant frequencies correspond to the notes of the musical scaleand are played by being struck by a wooden mallet, range up to two octaves,generally between G 1 and G 3 . The tubular bells are intended as substitute forreal bells. Bells themselves also constitute musical instruments. They are actuatedthrough side-to-side movements by clappers that hang loosely inside the bell. Thefundamental frequency of a bell depends on the geometry of the bell, the wall

554 18. Music and Musical Instrumentsthickness, and the density and moduli of the metal. A carillon is a set of fixedbells tuned to a musical scale. A number of arrangements have been devised toplay the bells: the bells may be struck directly by hammers held in the hand ofthe carillonneur, or they may be struck by a clapper through linkages to a keyboardthat is played by a carillonneur. With a mechanical setup, fists are used withconsiderable force to operate the keys. In electrified carillons, the clappers areactivated by solenoids activated by the keys of a keyboard. In this setup the forcerequired to move the keys are greatly decreased to a level no greater than that forplaying a piano or an organ. A carillon usually covers the range of three or moreoctaves.The kettledrum or timpani consists of a large hemispherical bowl over whicha specially treated leather skin is stretched. Several types of sticks are used tostrike the membrane, according to the percussion effect desired. These sticks mayhave striking surfaces made of sponge, felt, rubber, or wood. The kettledrum maybe tuned and changed in its fundamental frequency when so desired by varyingthe tension of the membrane through adjustment of head screws and by movementof a tuning pedal. The kettledrum emits a low-frequency sound of a definite pitch.The pedal provides a quick yet accurate variation of the pitch that a melody canbe played on the kettledrum. Two versions of the timpani are standard, the smallerone having a diameter of 58.5 cm and the larger one, a diameter of about 76 cm.The smaller unit covers the frequency range from B 2 to F 3 , and the larger fromF 3 to C 3 .An interesting development of rather recent past is that of the steel drum, whichis the principal instrument of Trinidad, where it originated, and of other Caribbeannations. Steel drums developed in Trinidad in the 1940s because the use of “bambootamboo” sticks, which supplied rhythmic cadences at the annual Carnival festivities,were proscribed after matters got out of hand and these sticks were usedas weapons in melees between rival bands and fights with the police. But thesemusicians were determined to continue their musical ways, and their resourcefulnessturned them to buckets, garbage cans, brake drums, and whatever wasavailable. The first steel drums were rhythmic rather than melodic. With the availabilityof the 55-gallon steel drums used by the petroleum industry after WorldWar II, the continued development of tuned steel drums continued apace to thepoint that they must now be considered rather sophisticated examples of musicalinstruments and intricate workmanship. These instruments are now achieving evergreaterpopularity in North American and in Europe, to the point where an aspiringmusician can study steel drums in a well-developed curriculum at Northern IllinoisUniversity.A variety of steel drums or pans are available. Steel drums are generally fabricatedfrom 55-gallon oil drums. The drums in a steel band may consist of a varietyof drums that have been termed soprano, ping pong, double tenor, guitar, cello,and bass. The soprano or ping pong can have anywhere from 26 to 36 differentnotes, but a bass drum may have only three or four notes. Because of the relativepaucity of notes on a single drum, the bass drummer is likely to play on a half adozen drums in the same manner as a timpanist in a symphonic orchestra.

18.11 Percussion Instruments 555Figure 18.37. Layout and tonal segmentation of different steel drums: (a) lead pan (soprano),(b) double tenor (alto), (c) double second (tenor), (d) cello, and (e) bass (fromFletcher and Rossing, 1998 with acknowledgment to Clifford Alexis).These drums are fabricated by first hammering the head of an oil barrel into theshape of a shallow basin. A pattern of grooves is cut with a nail to delineate sectionsof different notes. Each section is “ponged up” or shaped with a hammer. Afterthe drum is heat-tempered, each section is tuned by the adept use of a hammer.Figure 18.37 shows the layout and the tonal segmentation of different types ofsteel drums.Indefinite pitch instruments include the triangle, which is a steel rod bent intoa triangular shape that is struck by a metal beater. Because the triangle undergoesa complex vibration when struck, the fundamentals and the overture forman indefinite noise mixture. The bass drum, a hollow cylinder of wood or metal

556 18. Music and Musical Instrumentscovered at each end by a stretched membrane or parchment or skin, is struck witha softheaded stick; its vibration is quite complex, owing to nodal contributionsby the air column and the stretched membrane. Thumbscrews supply the meansof adjusting the stretch of the drumskin. The bass drums are constructed in sizesranging from 61 cm to as much as 3 m, with approximately 78 cm being the mostcommon diameter. The military drum is a smaller rendition of the bass drum, beingabout 40 cm in diameter and about 30 cm in depth. At nearly 36 cm in diameter,the snare drum is even smaller than the military drum. Across its lower head, cordsof catgut are stretched so these cords vibrate when struck by the membrane. Thisresults in a sound output that is buzzing and rattling.The tambourine consists of a hoop of wood or metal with a single membranestretched over one end. Smaller circular metal disks are inserted in pair in thehoop and loosely strung on wires. During performance, the tambourine is heldin one hand and the membrane struck with the fingers or palms of the otherhand.A cymbal is a brass circular disk that is concave at its center. Its vibration is thatof a circular plate supported at its center. Cymbals equipped with handles can behold, one in each hand, and struck together. A single cymbal can be supported ina fixed horizontal position on a stand or on some other support and struck with adrumstick. A sock cymbal consists of a pair of horizontally positioned cymbals,one being fixed and the other being moved by a pedal. Cymbals are made in varioussizes, from as little as 5 cm in diameter to 51 cm.Castanets, used to accentuate Spanish-style music, are hollow shells of hardwood. These are hold in the hand and clapped together. They can also be mountedon a handle in such a manner that shaking the handle clacks the castanets.The gong is a round plate of hammered bronze, with the edges curved up so thatthe shape takes on the appearance of a shallow pan. The gong is usually suspendedby strings attached to a frame. It is excited by striking with felt-covered hammer,with the sound output resembling a heavy roar. The diameter of the gong rangesfrom about 45 cm to 90 cm.18.12 Electrical and Electronic InstrumentsA tone can be generated by a number of electrical means. The means of producing atone can be achieved by interruption of an air stream (as with a siren that consists ofa rotating perforated wheel), an electrically driven diaphragm (as in the automotivehorn), and electronic alternator and loudspeaker combination (electric organs),an electronic oscillator–amplifier–loudspeaker combination (the electric piano isan example). The electric guitar is an adaptation, where an electromechanicaltransducer is attached to the bridge of the instrument. The transducer convertsthe mechanical vibrations into a correspondingly varying electrical signal that istransmitted to an amplifier. The amplifier increases the strength of the signal andsends it on to a loudspeaker that converts the electrical signal into the correspondingacoustical output. A volume control in the amplifier or integrated into the guitar

18.12 Electrical and Electronic Instruments 557Figure 18.38. Schematic of a transistor oscillator network.makes it possible to adjust the output sound level. The sound of the guitar resemblesthat of the conventional instrument.Generating music by electronic means has a history older than most people realize.Early electronic instruments include the Theremin (1919), the Ondes Martenot(1928), the Trautonium (1928), and the Hammond organ (1929). 2 The early meansof electrically generating music were based on the technology that prevailed in therecording and sound reproduction industry of the time. Electronics progressedrapidly during World War II and even more rapidly with the development of transistorsand eventually with the advent of microcircuitry.Because of the rising costs of traditional pipe organs and the desire then prevailingfor creating sustained tones, much attention has been conferred on developingand marketing the electronic organ. Vacuum-tube oscillators and, later on,solid state oscillators were used to generate the tones in an electronic organ. Theschematic of a transistor oscillator network is shown in Figure 18.38. Power fromthe output network is fed back to the input network. Oscillations occur when morepower is developed in the output than is necessary for the loss in the input circuitry,combined with appropriate phase relations between the current and voltages in theinput, feedback, and output networks. Under these conditions, the reactions consistof regular surges of power at a frequency that depends on the constants of theresonant elements in the input or output networks. The resonant elements includequartz, crystal, tuning fork, inductance–capacitance, and so on. These electronicsystems are capable of simulating the wave shape of almost any musical instrument.One type of electrical organ makes use of air-driven reeds and electrostaticpickups. Most of the electronic organs consist of two manual keyboards and a2 An interesting review of these instruments and their history may be found in texts by Rossing (1990)and Strong and Plitnik (1983).

558 18. Music and Musical Instrumentspedal keyboard and incorporate a system of stops and couplers. Loudspeakersmay be housed in a separate cabinet; and the driving electronics housed in theconsole.The electronic organ has been displaced in the 1980s by the growing popularityof synthesizers. Modular synthesizers made their appearance in the1960s in both analog and digital versions. The analog synthesizer consists ofa group of signal generating and signal processing modules operating on electricalvoltages. The modular approach to analog synthesis is embodied in threefundamental voltage-controlled modules: (1) the voltage-controlled oscillator(VCO), (2) voltage-controlled amplifier (VCA), and (3) voltage-controlled filter(VCF).The VCO is really a function generator that produces a periodic voltage signaland is the initial source of pitched sounds. The frequency of the periodic signalis determined by a control voltage. Usually when the control voltage increases byone volt, the frequency doubles (i.e., it goes up one octave). The control mode istherefore exponential, which fits beautifully into the scheme of things in music andin electronics. Controllers can be fabricated in the form of keyboards, and musicaleffects of modulation (trill, vibrato, or glissando) are independent of the DC valueof the control voltage. Switching keys of a melody require only the addition of aconstant to the series of control voltages representing the notes of a melody. Theexponential feature inherent in the voltage control is appropriate electronically,given the fact that the collector current in a bipolar transistor is an exponentialfunction of the base-emitter voltage. In addition to being stable and accurate, theVCO needs a subsequent filter to shape the waveforms for the desired tone color. Ahigh-purity sine wave output is normally needed, particularly if the VCO output isprocessed by a nonlinear waveform shaper, distorter, or a ring modulator or an FMsynthesizer. Otherwise an excessively dense spectrum that is downright unmusicalcan be created.The voltage-controlled amplifier is a voltage amplifier with a gain that dependslinearly upon a control voltage. An exponential control is generally available, butit is more usual for the amplifier gain to grow linearly with the control voltage witha slope of unity gain for a 5-V control input and a maximum gain of 2. Ideallythe gain should be zero when the control voltage is zero or negative. It is the taskof the VCA to turn on tones. Since the oscillators run continuously, the VCAbears the responsibility of shaping the amplitude envelope of the tone. Here thecontrol voltage comes from an envelope generator. When a note is begun froma keyboard (or sequencer), the envelope generator begins a transient phase. Theenvelope generator sustains an appropriate level for the duration of the tone andends the tone with an exponential decay. As with the tone control parameters ona synthesizer, the performer can set the envelope transients, the sustain level, anddecay time by adjusting individual potentiometers on a modular analog instrumentand through programming on a digitally controlled keyboard. Special timbres canbe created by adding two sounds, one of them with a delayed onset. To achievethis the output of two VCAs can be added, where one of them is controlled by

18.12 Electrical and Electronic Instruments 559a delayed envelope. A tremolo can be added by a VCA controlled by subsonicperiodic waveform.A VCA can be utilized to multiply two separate audio signals, producing amplitudemodulation, with one signal serving as the carrier and the other servingas a modulator. The output consists of the spectral components of the carrierplus sidebands. More commonly used than amplitude modulation is the balancedmodulation or ring modulation. While the VCA is a two quadrant multiplier (i.e.,the control voltage must be positive for a nonzero output), the balanced modulatoris a four-quadrant multiplier.As the terminology implies, the definitive frequency of a voltage-controlledfilter (VCF) is controlled by a control voltage. In a low-pass VCF, for example, thecutoff frequency increases from a low value to a high value as the control voltageincreases. A scaling of 1 V per octave is the standard. There is a large number ofdifferent filtering circuits available, but predominant design seems to be the statevariablefilter that is constructed from two integrators and a summer. This designbears the advantage of incorporating (a) three simultaneous outputs (low-pass,bandpass, high-pass), (b) a constant value of Q independent of band frequency,(c) adjustability of Q through controlled feedback, and (d) stability of the sineoscillation mode. This type of filter features asymptotic slopes of ±12 dB/octavein the low-pass and high-pass outputs and ±6 dB/octave for the band-pass output.A four-pole low-pass filter with an asymptotic slope of ±24 dB/octave is oftenpreferred by musicians to deal with the more subdued sounds, particularly in thelower registers.Voltage control is applied in filters to generate dynamic effects, particularlymusical attacks. As an example of this application, consider the fact that theharmonics of a brass instrument entering the attack phase are in the order ofascending frequencies (Risset and Mathews, 1965). This effect is electronicallysimulated by raising the cutoff frequency of a low-pass VCF with an envelopegenerator at the onset of the tone. On the other end of the tone, the decay of anenvelope generator is utilized to decrease the cutoff frequency of the low-passVCF to simulate the proclivity of high-frequency modes to become damped morerapidly than the low-frequency modes in a free vibrator (such as a percussioninstrument).A number of other musical effects can be generated. For example, in a delay-andaddapplication for a digital delay line, specifically the generation of reverberation,an input signal is passed through a delay line that is tapped at different delayintervals. Signals at these taps are fed back into the input as a weighted sum.This weight determines the reverberation time of the system. In the flanger use ismade of comb filtering. The flanger uses a single delay-and-add. The delay time isslowly modulated so that the comb filtering changes with time, with typical delaysin the order of 2 ms. Chorusing, which is the attempt to make a single-voice soundlike a group of many, functions in the same manner, except several modulateddelay-and-add circuits are used to transmit to different channels, and the delaysare considerably longer, typically in the order of 10 ms.

560 18. Music and Musical InstrumentsFigure 18.39. Arrangement of modules for an analog synthesizer. The signals from thetwo keyboard-manipulated voltage-controlled amplifier (VCO) are summed into a voltagecontrolledamplifier (VCA) and then relayed to a voltage-controlled filter (VCF). Both VCAand VCF are controlled by the envelope generators (EG). A third oscillator operating atlow frequency (LFO) is on hand for modulation of any voltage-controlled module.In the early years of the synthesizer, control was a cumbersome matter, involvingpatch cords for connecting analog synthesizers. The early digital synthesizers wereeven more cumbersome, none of them operating in real time. In the early 1970s,prepatched analog synthesizers were introduced, primarily for creating specialeffects in a live performance. The setup of the modules is given in Figure 18.39.In the mid 1970s, the modules of Figure 18.39 were combined into a singleintegratedchip. One such chip was placed under each key of the keyboard, so eachkey essentially constituted the control for a miniature synthesizer. The PolyMoogwas the first and the last instrument to be constructed this way. Subsequent designsnow employ digital scanning keyboards and assign a specific sound synthesis chainto each key. The number of chains (generally 8, 12, or 16) establishes the maximumnumber of simultaneous notes.The sequencer was introduced to provide additional automated control. Theearlier sequencers generated several channels of control voltages in repetitivesequences. The analog sequencer could even replace a keyboard for generatinga repetitive bass line. The sequencers have evolved into small computers whichhave almost unlimited musical capabilities. Thanks to computer memory, they

18.13 The MIDI Interface for Synthesizers and Digital Synthesis 561can recall and reproduce the control sequences for multiple parameters for allthe voices in the entire performance. Data entry is achieved through the use ofan organ-type keyboard, and the programming allows for editing the data anddisplaying or printing the data in the traditional musical notation.18.13 The MIDI Interface for Synthesizers andDigital SynthesisWith rapid advances in the development and manufacture of microprocessors andequally rapid drop in the cost of these microprocessors, the digital control of synthesizerfunctions became inevitable. In 1983 the manufacturers of commercialsynthesizers adopted the Musical Instrument Digital Interface (MIDI) standard,which expands the range of control possibilities for the musical performer by theadoption of an interface that makes it possible to transfer control data among differentsynthesizers or other processors as well as computers (International MIDIAssociation, 1983). The interface is a unidirectional asynchronous serial line operatingat 31.25 kbaud (1 kbaud = 1000 bits/s). An MIDI word is defined as 10 bitslong, 8 data plus start and stop bits. To enable long interconnections, the interfaceuses a 5-mA current loop. Repeaters with each MIDI-specification instrument allowfor daisy chaining. Sixteen logical channels exist for data transmission, andinstruments may be set to obey data on one channel or data on all channels.Some musical tradition remains in the way MIDI commands are defined toemphasize the use of the keyboard in electronic instruments. Specific data words aredefined for key on/off, key velocity, key aftertouch, and for encoding the positionof modulation wheels (this device was originally introduced in the MiniMoog andit remains a standard). The performers are free to use the parameters as they like,and the MIDI has even been used to control stage lighting. Data of arbitrary lengthcan be transmitted by concaternating MIDI words. Owing to its high data rate andflexible control structure, the MIDI control has proven to be very powerful andwell accepted in the industry and by performers of popular music.With the affordability and availability of digital processors, it becomes inevitablethat digital oscillators can become precise enough to replace VCOs for improvedflexibility and stability. Moreover, a single digital oscillator can be constructed tocreate several simultaneous voices.A cost-effective digital technique is that of FM synthesis, which involves thegeneration of a frequency-modulated waveform given byx(t) = sin[ω c t + β sin(ω m t)]where ω c denotes the carrier frequency and ω m the modulating frequency; β isthe modulating index which is given by β = ω/ω m , with ω being the maximumfrequency excursion. This FM algorithm provides for flexible addressing. Aspectrum of the FM signal contains a component at the carrier frequency and acomponent that contains sidebands displaced in frequency by ±nω m , where n is

562 18. Music and Musical Instrumentsthe sideband order. Sidebands of order n have amplitudes that are proportional toBessel functions J n (β). As the modulation index is increased, the bandwidth ofthe signal increases. A dynamically varying spectrum is achieved by varying themodulation index as the tone is initiated. When ω c and ω m are commensurate, thespectrum may be harmonic; otherwise, it may be inharmonic.The FM algorithm has been proven highly successful in creating both sustainedand percussive musical sounds, but the principal disadvantage of the FM techniqueis that the control is unintuitive. Bessel functions are not monotonic functions oftheir arguments, so it becomes difficult or almost impossible for the performersense what will happen if the modulation index is changed one way or the other.The Yamaha musical instrument company installed FM oscillators on large-scaleintegrated circuits in its DX series of instruments, the most popular electronicinstruments ever. These instruments contain six or four dynamically controlledoscillators in a voice, together with possible feedback among the oscillators.A more recent methodology of creating music electronically is that of physicalmodeling (Computer Music Journal, 1992). A mathematical model is developedfor the production of a tone by a traditional instrument or by some other mechanicaldevice. The model may result in a set of coupled, usually nonlinear differentialequations that can be solved by computerized means. The solution to these equationsis played through a digital analog computer. This is a most time consumingprocess and has been applied commercially for only a few years. It is hoped withthis methodology that even greater subtlety can be captured in the reproduction ofactual instruments.18.14 The Orchestra and the BandThe orchestra is a fairly full-fledged group of musicians playing on string, wind,and percussion instruments under the direction of a conductor. An orchestra differsfrom a band in that the main body of the music is generated by string instruments,whereas the band is generally made up of musicians playing on wind and percussioninstruments. Orchestras differ according to the demands of the music, asthe requirements differ for playing symphonies and overtures, or providing accompanimentfor operas, musicals, oratorios, or incidental music for theatricalperformances, or ballet or dance music.Four groups usually constitute a symphony orchestra, namely, the strings, woodwinds,brass, and percussion. Violins, viols, violoncellos, contrabasses, harps, and(usually one) piano comprise the strings section. Woodwinds include the flutes,piccolos, oboes, bassoons, English horns, contra bassoons, clarinets, and bass clarinets.Brass instruments include trumpets, trombones, French horns, and tubas. Inthe percussion group there may be included timpani, bass, military and snaredrums, tambourines, gongs, celestas, glockenspiels, tubular chimes, xylophones,castanets, and triangles. In order to achieve sufficient sound output for suitableartistic effects in large halls, the orchestra may have to consist of 80–120 players.An organ may be added to the performance where required. Some performances

18.14 The Orchestra and the Band 563may require fewer players, and in some situations, a limited-size orchestra canachieve a dramatic effect through the use of sound reinforcement systems. Figure18.40(a) shows a plan view of the arrangement of the instruments of a symphonyorchestra and the placement of its conductor. Figure 18.40(b) illustrates the layoutof a smaller orchestra, and 18.40(c) that of a small dance band. An orchestra forperformances of popular and dance music may number anywhere from 5 to 25.A band consists of musicians playing wind and percussion instruments. Theyare generally more suited for outdoor performances, since the main body of thetone is produced by the brass and woodwind sections, both of which can produceconsiderable acoustic power in free space. Bands help to sustain unison in marchinggroups and can whip up enthusiasm at athletic meets. A standard band generallyfeatures a complete range of woodwind, brass, and percussion instruments. Thenumber of performers can vary from 25 to 50 or even more. The military bandresembles the standard band except that it may include a fife, drum, or bugle corps.A concert band may include contrabasses, timpani, harps, and other instrumentsthat lack the portability for marching. There is virtually no difference between thedance band and the dance orchestra. Table 18.4 shows the components of a typicalsymphony orchestra and those of a standard band.Table 18.4. The Elements of a Typical Orchestra and of a Typical Band.Orchestra aStandard Band bInstruments Number of Players Instruments Number of PlayersFirst violins 10 to 20 Flutes 4Second violins 14 to 18 Piccolo 1Violas 10 to 14 Clarinets 14Violoncellos 8 to 12 Oboes 2Contrabasses 8 to 10 Bassoons 2Flutes 2 to 3 Sarrusophones 2Piccolos 1 to 2 Saxophones 4Oboes 3 Cornets 4English horn 1 Trumpets 2Bassoon 3 French horns 4Contra bassoon 1 Trombones 4Clarinets 3 Tubas 6Bass Clarinet 1 Snare drum 1Trumpets 4 Bass drum 1Trombones 4 Percussion 1 to 5French horns 4 to 12Tuba 1Timpani 1Harp 1Percussion 1 to 5a In addition, other instruments such as a piano or an organ may be included. Percussionincludes bass, snare and military drums, gongs, cymbals, tambourines, celestas,glockenspiels, tubular chimes, xylophones, castanets, and triangles.b Percussion includes triangles, bells, cymbals, castanets, and xylophones.

564 18. Music and Musical InstrumentsFigure 18.40. Arrangement of players for (a) symphony orchestra, (b) small orchestra, and (c) small dance band.

References 565Figure 18.41. The fundamental frequency ranges of various musical instruments, includingthe human voice.The frequency ranges of the fundamental frequencies of various musical instruments,including the human singing voices, are displayed in Figure 18.41.ReferencesÅgren, C. H. and Stetson, K. A. 1972. Measuring the resonances of treble viol plates byhologram interferometry and designing an improved instrument. Journal of the AcousticalSociety of America 51:1971–1983.

566 18. Music and Musical InstrumentsArnold, Denis (ed.). 1984. The New Oxford Companion to Music. New York: OxfordUniversity Press, pp. 1925–1933.Cremer, Lothar. 1984. The Physics of the Violin. Translated from the German by J. S.Allen. Cambridge, MA: MIT Press. (Originally published as Physik der Geige, Stuttgart:S. Hirzel Verlag, 1981.)Computer Music Journal 16, 4. 1992. (A special issue on physical modeling. Of interestare articles by Woodhouse, Keefe, and Smith.)Fletcher, Neville H. and Rossing, Thomas D. 1998. The Physics of Musical Instruments,2nd ed. New York: Springer-Verlag. (A modern classic in its own right, it is probablythe best scientific text devoted to the physics of nearly the entire gamut of musicalinstruments, with some interesting insights into musical history.)Hartmann, William Morris. 1997. Electronic and Computer Music. In: Encyclopedia ofAcoustics, Crocker, Malcolm J. (ed.), Vol. 4, Chapter 138, pp. 1679–1685.Helmholtz, Herman L. F. 1954. On the Sensations of Tone, 4th ed. Translated by A. J. Ellis.New York: Dover.Hutchins, Carleen M. (ed.). 1967. Founding a family of fiddles. Physics Today 20:23–27.Hutchins, Carleen M. (ed.). 1975. Musical Acoustics, Part I. Violin Family Components.Stroudsburg, PA: Dowden, Hutchinson and Ross.Hutchins, Carleen M. 1976. Musical Acoustics, Part II. Violin Family Functions. Stroudsburg,PA: Dowden, Hutchinson and Ross.Hutchins, Carleen M. 1981. The acoustics of violin plates. Scientific America 245(4):170–186.Hutchins, Carleen M. 1983. A history of violin research. Journal of the Acoustical Societyof America 73:1421–1440.Hutchins, C. M., Stetson, K. A., and Taylor, P. A. Clarification of ‘free plate tap tones’ byholographic interferometry. Journal of the Catgut Society 16:12–23.International MIDI Association. 1983. MIDI Musical Instrument Digital Interface Specification1.0. Sun Valley, CA.Jeans, James. 1968. Science and Music. New York: Dover Publications. Originally publishedby Cambridge University Press in 1937. (A clear exposition on the fundamentalsof musical acoustics and music itself.)Machlis, Joseph and Forney, Kristine (contributor). 1995. The Enjoyment of Music: AnIntroduction to Perceptive Listening, 7th ed. New York: Norton and Company. (Thestandard text in music appreciation courses in universities and colleges worldwide, itcontains a most fascinating history of the evolution of music to the modern times andan overview of the technical aspects of music and musical instruments.)Mathews, Max V., Miller, J.V., and David, E. E. Jr. 1961. Pitch synchronous analysisof voiced Sounds. Journal of the Acoustical Society of America 33:1725–1736. (Thepioneering paper on digitization of audio signals.)Mathews, Max V. 1969. The Technology of Computer Music. Cambridge, MA: MIT Press.(Written by a pioneer in the field.)Meyer, Jürgen. 1975. Akustische Untersuchungen zur Klangqualität von Geigen. Instrumentbau29(2):2–8.Meyer, Jürgen. 1985. The tonal quality of violins. Proceedings SMAC 83. Stockholm:Royal Swedish Academy of Music.Moog, R. V. 1965. Journal of the Audio Engineering Society 13:200–206. (By the manwho started it all—the father of the Moog synthesizer.)

Problems for Chapter 18 567Olson, Harry F. 1967. Music, Physics and Engineering, 2nd ed. New York: Dover Publications.(A classic in the field of musical acoustics written by a major researcher in thefield of sound reproduction.)Raman, C. V. 1918. On the mechanical theory of the vibrations of bowed strings and ofmusical instruments of the violin family, with experimental verification of the results.Indian Association for the Cultivation of Science Bulletin 15:1–18. Excerpted 1975 inMusical Acoustics, Part II. Violin Family Functions. C. M. Hutchins (ed.). Stroudsburg,PA: Dowden, Hutchinson and Ross.Risset, J. C. and Mathews, M. V. 1965. Journal of the Audio Engineering Society 13:200–206.Roederer, Juan G. 1995. The Physics and Psychophysics of Music, An Introduction, 3rded. New York: Springer-Verlag. (An interesting combined approach to the physical processesof producing music and human perception of music. While intended for musictheorists, composers, performers, music psychologists, and therapists, this text includessome materials on recent findings that may prove useful to acousticians, psychoacousticians,audiologists, and neuropsychologists.)Rossing, T. D. 1990. The Science of Sound. New York: Addison Wesley. Chapters 27–29.Rossing, Thomas D. and Fletcher, Neville H. 2004. Principles of Vibration and Sound,2nd ed. New York: Springer Verlag. (An excellent text that concentrates on the physicsunderlying the acoustical and vibrational aspects of musical instruments and includeschapters on underwater acoustics, noise, architectural acoustics, and nonlinear acoustics.)Sadie, Stanley (ed.). 1984. The New Grove Dictionary of Musical Instruments, Vols. 1and 3. New York: Macmillan Press Grove’s Dictionary of Music, pp. 8–22, 736–814.(Dictionary is really an encyclopedia dealing with music, music history, musical instruments,and performance techniques. A real treasure trove of information.)Savart, F. 1840. Des instruments de musique. Translated by D. H. Fletcher, in Hutchins(1976):15–18.Schelleng, John C. 1974. The physics of the bowed string. Scientific American: pp. 87–95.Strong, W. J. and Plitnik, G. R. 1983. Music, Speech and High Fidelity. Salt Lake City:Soundprint.Young, Robert W. 1939. Journal of the Acoustical Society of America 11(1):134.Problems for Chapter 181. Why does a middle C from an oboe and a piano sound different, even thoughit is the same note?2. Although the piano is classified as a string instrument, in what way it acts asa percussion instrument?3. What is the difference in the manner a piano and a harpsichord sound a note?4. What acoustical theory (e.g., vibrating bar) would you apply to the study of(a) piano(b) bass drum(c) xylophone(d) cello?

568 18. Music and Musical Instruments5. Give examples of the use of the Helmholtz resonance principle in the designof music instruments.6. Why is it necessary to use rosin on bows used for playing a member of theviolin family?7. Why does a cello produce lower notes than, say, a violin or a viola?8. What part of the human ear can be compared to a music instrument?9. Why does a grand piano sound “grander” than a spinet?10. Give examples of tunable and nontunable music instruments.11. Why does a concertmaster sound the A-note prior to the commencement of aconcert program?12. What is the role of a conductor in directing an orchestra?

19Sound Reproduction19.1 Historical OverviewUntil 1877 when Thomas Edison (1847–1931) developed the phonograph, therewas no way to record and reproduce sounds. While working to improve theefficiency of a telegraph transmitter, Edison noted that the noise emanating fromthat type of machine resembled spoken words when operating at high speed.This caused him to wonder if he could record a telephone message. He beganexperimenting with the diaphragm of a telephone receiver by attaching a needleto it. He reasoned that the needle could prick a paper tape to record a message.He continued to experiment and he tried a stylus on a tinfoil cylinder, which tohis surprise, played back the short message he had recorded, “Mary had a littlelamb.” The word phonograph was Edison’s trade name for his device that playedcylinders rather than disks.Edison’s phonograph was followed by Alexander Graham Bell’s (1847–1922)gramophone that played a wax cylinder that could be replayed many times, but theprogram content had to be recorded separately for each copy. No mass reproductionof the same music or sounds was possible.In 1877, a German immigrant Emile Berliner (1851–1929) working inWashington, DC, invented a system of recording that could be used over and overagain. Berliner switched from a cylinder-type medium to a flat disk. He patented thegramophone (the true precursor of the modern phonograph), enhanced by a springmotor developed by Elridge Johnson (1867–1945), which allowed the turntable torevolve at a steady speed without the need for hand cranking of the gramophone.Berliner also invented the carbon microphone that became part of the first Belltelephones and founded the Gramophone Company to mass-produce his sounddisks and his gramophone for playing them. He made two smart marketing moves:he persuaded popular artists, among them Enrico Caruso and Dame Nellie Melba,to record their music; and in 1908 he used Francis Barraud’s painting of “HisMaster’s Voice” as the company logo.Until the late 1920s, motion pictures were silent except for the music accompanimentprovided by the theater management in the form of a piano player orlive orchestra. All this changed in 1926 when the Warner Brothers (Jack, Harry,569

570 19. Sound ReproductionAlbert, and Sam) in collaboration with Western Electric introduced a new soundon disk system that worked in synchronization with the film. In order to exhibitthis new technology, the Warner Brothers Studios released Don Juan that provedto be a box office hit, but many studios still refused to adapt to talking picturetechnology. However, in October 1927, the premiere of The Jazz Singer starringAl Jolson really triggered the talking picture revolution. Even though The JazzSinger was not the first movie to use sound, it was the first film to use spokendialog and music as part of the action. The advent of cinema sound was furtheredby the introduction of photographic sound tracks that vary in transparency andimprovements in playback equipment.The genesis of the tape recorder occurred when a German chemist Fritz Pfleumerreceived a patent in Germany in 1928 for application of magnetic powders to stripsof paper or film. Three years later he and the German company AEG began to constructthe first magnetic tape recorder, and in 1935 the first public demonstrationof the BASF/AEG “Magnetophone” was given at the Berlin Radio Fair. The followingyear, the first BASF/AEG tape recording of a live concert was made, withSir Thomas Beecham conducting. In the United States, Marvin Camras developedindependently the wire recorder in 1939 at the Armour Foundation. The inventionwas sold to the military during World War II, and wire recorders were popular withamateurs until the late 1950s. In 1945, after the German surrender, U.S. SignalCorps Captain John Mullin found Magnetophones at Radio Frankfurt and 100-mreels of 6.5-mm ferric-coated BASF tape with 20-min capacity per reel. He mailedhome two machines with 50 reels of tape and worked on them to improve theelectronics. Alexander M. Poniatoff (1892–1980) learned about them and beganwork on developing a U.S.-made magnetic tape recorder. In 1948, his first model,the Ampex 200, was used to record a Bing Crosby radio show on 3M Scotch 111gamma ferric oxide-coated acetate tape.Since then, the magnetic tape recorder has evolved into other formats, such as8-track tape (now defunct), DAT, and the cassette. While the cassette is still inuse, it has given way to the compact disk (CD) and, later on, the digital versatiledisk (originally called digital video disk and abbreviated as DVD). The CDs andDVDs also come in recordable formats, thus enabling the average person who hasa personal computer and/or recording/playback equipment to operate a recordingstudio in the comfort of his/her home.The term high fidelity refers to sound recording and reproduction that result inlow harmonic and intermodulation distortions and a frequency response coveringmost, if not all, of the entire audio range of 20 Hz–20 kHz. A monophonic, stereo,or a multichannel system cannot be termed high fidelity unless it reproduces soundfaithfully. The high fidelity (or audiophile, using the more modern terminology)industry got under way when Avery Fisher (1906–1994) introduced high-qualityequipment in the late 1930s. After World War II, the industry truly began to flourishwith Sidney Harmon (1920–) developing the first high-fidelity receiver; Sonyco-founders Akio Moria (1921–1999) and Masara Ibuka (1908–1997) introducingconsumer-type tape recorders, James B. Lansing (1902–1949) building qualityspeaker systems, and Saul Marantz (1902–1997) hand constructing quality amplifiersand preamplifiers (that remain classics to this day).

19.2 Recording Equipment 571In the following sections of this chapter, we outline the principles of soundrecording and playback, concluding with a prognosis of future developments.19.2 Recording EquipmentRecording of sound occurs in different ways and it can be defined as the storingof information of how the amplitude of sound varies with time. The stored informationcan be retrieved by playback at a later time through control of a sourceof sound waves. The twentieth century has seen the evolution of the phonographcylinder into the 78-rpm recordings which in turn gave way to the 45 rpm and33-1/3 rpm vinyl disks. These mechanical types of recordings use wavy grooveson a surface that guide a stylus to replicate the signals that were recorded. Reelto-reeltape recorders have been used since the 1940s to make recordings, andmany great performances have been archived on this medium. These master tapesare used to make vinyl disks and eventually CDs. While tapes carry the advantageof being relatively easy to edit, they do lack the random accessibility of anytype of disks. Tapes solely for playback are now principally in the compact cassetteformat.Magnetic RecordingFigure 19.1 shows a fundamental record-play system using magnetic recording.A close-up of the interior construction of the magnetic head is also included inthe figure. Tape D is made of a smooth, durable plastic such as polyester (Mylar)and is coated on one side with a magnetizable material, most commonly a driedsuspension of acicular gamma ferrite particles about 0.25 μm long in a lacquerbinder. 1 The thickness of the magnetic layer is of the order of 12 μm or less. Thetape unreels from supply reel A and threads over a magnetic head B that suppliesa magnetic field at recording gap C. The magnetic field varies in its strengthaccording to the sound being recorded. The now magnetized tape is pulled pastthe head by capstan roller E and pinch roller F and spooled onto a take-up reel G.In rewinding the tape, the recording head is deenergized. For playback, the tapeleaves the supply reel, as before, but the winding on head B is connected to anamplifier that, in turn, connects to a loudspeaker. The magnetized elements inducea voltage in the head winding, which translates into a replication of the gap fieldvariations during recording. The recording may be erased by an additional headH that produces a strong, steady AC field. It is not necessary to erase the tape ifit is being recorded over, because the erase head is automatically energized whenrecording, thus removing previously recorded material.1 After the tape has been coated and while the suspension is still damp, it is subjected to a magnetic fieldthat aligns the magnetic particles in order to increase recording signal strength and minimize noise.Other alternative magnetic materials include chromium dioxide and cobalt-coated gamma ferrite. Itis obvious that tapes have to be manufactured under stringent clean-room conditions because theslightest contaminants can cause impaired recordings.

572 19. Sound ReproductionD. Magnetic tapeC. Record playback gapE. Capstan rollerA. Supply reelH. EraseheadF. PinchrollerB. RecordplayheadG. Take-up reelMAGNETIC RECORD/PLAYBACK SYSTEMDirection oftape travelGapMagneticpole pieceTapeElectricwindingCLOSE-UP SHOWING MAGNETIC HEAD CONSTRUCTIONFigure 19.1. The elements of a magnetic tape recording and playback system, with detailsof the interior construction of a magnetic head.A magnetic head is designed to concentrate the magnetic field in the possiblesmallest region of tape, thus giving a high-recording density of maximum numberof wavelengths possible. To achieve these high densities, the head gap must bemade very small, in the order of 1 or 2 μm, which yields low output voltage andrequires very smooth tape surface and good contact between the head and thetape. The head cores are usually fabricated of Permalloy (80% Ni, 20% Fe) forhigh permeability and low-magnetic retention. In more recent times ceramic-likemagnets have been used for better wear and they do not require lamination.

19.2 Recording Equipment 573Tape speeds have been standardized at 30, 15, 7.5, 3.75, and 1.875 in./s foranalog recording. All of these speeds are much lower than that required for digitalrecording, which necessitates bandwidths in the megahertz range. Digital audiorecorders make use of the video recording techniques that entail rotating heads.These heads scan the tape at 228 in./s while the tape itself moves at only 0.66 in./s.The highest speed is used for master recordings, in situations where the highestquality is demanded in covering the 20 Hz–20 kHz range. Cassettes use the1.875 in./s speed.The Dolby r○ systems for noise reduction employ circuitry that pre-emphasizeshigh frequencies before they are recorded on tape in order to make them louder thanthe tape hiss with which they compete. The circuit for recording tape is amplitudesensitive with the result that only soft, high-frequency sounds are emphasized.Emphasis of loud, high-frequency sounds might drive the tape into its distortionlevels. Upon playback, a matching de-emphasis circuit is employed to restore thehigh frequencies to their proper balance with the other parts of the recorded signal.Digital RecordingThe capabilities of analog recording pale in comparison with the advantages ofdigital recording. Zero wow and flutter, more than 90 dB signal-to-noise ratio,and incredibly low-distortion levels are the most outstanding attributes of digitalrecording. To achieve digital recording, the analog signal is sampled at regularintervals. Sampling of the amplitude is done with almost absolute accuracy (onepart per 10 9 in a 96-dB signal-to-noise ratio system). The amplitude is recorded asa number, say, 59,959,498. The sampling should be done at more than twice theNyquist number (the highest frequency in the audio signal). For a signal with amaximum of 22,000 Hz, a rate of 44,000 samplings per second is quite practicable(20,000 × 2 + 10%). Because the series of digital samples occur in the megahertzrange, they are recorded and played back on videotape recorders. The audio programin digital form can be processed with digital features, error correctors, andother techniques that yield results that cannot be achieved with analog means. Thenumbers comprising the signal can be stored successively in buffers and read outin perfect crystal-controlled time intervals, even if there exist erratic mechanicalfluctuations in the tape drive. Thus, wow and flutter are eliminated. The digitalsignals are restored to analog format through digital/analog (D/A) converters.Experiments on the digitization of sound were conducted during the late 1950sand early 1960s for the purpose of computer analysis, speech synthesis, simulationof music, and simulation of reverberation, mostly at the Bell TelephoneLaboratories in Murray Hill, New Jersey (Mathews et al., 1961). In the early1970s, commercial digital recordings were commonly used by recording studiosfor master recordings, but the results were distributed by analog means.A significant breakthrough occurred in the early 1980s, when Sony Corporationof Japan and Koninklijke Philips Electronics N.V. of the Netherlands came out withthe compact disk (CD) that quickly replaced the analog long-play (LP) record andthe cassette as the most common medium for playback of recorded music. Later

574 19. Sound Reproductionon, machines for digital recording on cassette videotape and then the rotary-headdigital audio tape (R-DAT) came in the market. In the past few years, the digitalcompact cassette (DCC developed by Philips-Matsushita) and the mini-disk (MD)by Sony arrived on the recording scene, but they have not gained much in the wayof general acceptance.The DVD is a multimedia device for playback through video monitors as well asthrough audio channels. A DVD player can also function as an audio-only playbackunit for compact disks, but CD players cannot play the audio portion of DVDs. Astandard CD has a capacity of about 74 min of standard CD audio music. Thereare extended CDs that can actually exceed this limit and pack more than 80 minon a disk, but these are nonstandard. Regular CD-ROM media hold about 650 MBof data, but the actual storage capacity depends on the particular CD format used.The DVD uses smaller tracks, 0.74 μm versus 1.6 μm for the CD. Thus, DVDtechnology writes in smaller “pits” to the recordable media than is the case withCD. Smaller pits require that the drive’s laser produce a smaller scanning spot.This is achieved by reducing the wavelength of the laser from 780 nm infraredlight used in standard CD drives to 625 nm–650 nm red light. This helps to yielda storage capacity of 4.7 GB. Smaller data pits allow more pits per data track.The minimum pit length of a single-layer DVD-RAM is 0.4 μm as compared to0.834 μm for a CD.The various DVD formats are as follows:1. DVD-R: The most universal of recordable DVD formats used by DVD burnersand many DVRs (digital video records). DVD-R is a write-once format, muchlike CD-R, and disks made in this format can be played in most current DVDplayers.2. DVD-RW: Recordable and rewriteable format (like CD-RW). Disks are playablein most DVD players, provided they are recorded in the straight video modeand finalized.3. DVD+RW: Recordable and rewriteable format. This format is claimed to providea greater degree of compatibility in current DVD technology than DVD-RW.4. DVD+R: A record-once format that is claimed to be easier to use than DVD-Rwhile still playable in most DVD players.5. DVD-RAM: Recordable and rewriteable format that is not compatible withcurrent DVD technology.6. DVD-Audio: Intended to play in audio playback players (such as in-car players),it can offer slideshows and texts that can be displayed. The bit rate of this formatis 9.6 Mb/s as compared to 6.14 Mb/s of a video DVD. DVD Audio titles beingreleased include a DVD-Video compatible zone using Dolby or DTS coding.This means that the DVD-Audio is playable in tens of millions existing DVDplayers, but not with the maximum quality that a DVD-Audio player affords.There are three optical formats that offer more than 8.5 GB storage. These arePlasmon’s UDO (intended for business data storage), HD-DVD, and Blu-ray. Allthree systems use blue laser light that affords a narrower beam than red laser

19.2 Recording Equipment 575CD 0.7 GB DVD 4.7 GB Blu-ray Disc 25 GB1.6 μmPit Length0.74 μm0.32μmTrack Pitch: 1.8 micronMinimum Pit Length: 0.8 μmStorage Density: 0.0636 GB/cm 2Track Pitch: 0.74 micronMinimum Pit Length: 0.4 μmStorage Density: 0.429 GB/cm 2Track pitch: 0.32 micronMinimum Pit Length: 0.15μmStorage Density: 2.283 GB/cm 2Figure 19.2. Data layers in 0.7 GB CD disk, 4.7 GB DVD disk, and 25 GB Blu-ray disk.Note the pit lengths generally decrease with increasing disk capacities.light, hence yielding smaller data areas and more densely packed data deliveringhigh capacity. The Blu-ray is a Sony development that holds 25 GB in singlelayerformat, but will have two layers and a double-sided recording holding asmuch as 100 GB. Such capacities can hold high-definition movies and in theaudio mode could presumably easily contain the entire Wagner’s Ring Cycle. Itneeds a hard coating to protect it and requires a special player. The HD-DVD,also intended for playback of high-definition movies, features a 20-GB capacitybut could possibly be expanded to 50 GB via multiple layers. The HD-DVD doesnot require a protective cartridge or coating but it also does require a specialplayer. Figure 19.2 compares the data layers of the 0.7-GB CD, the 4.7-GB DVD,and the 25-GB Blu-ray disks. As of this writing, there is a formidable conflictraging as to which of these two high-definition formats will gain ascendancy. Somemovie studios and developers (NEC and Toshiba) have advocated HD-DVD; otherstudios, major computers manufacturers (Dell and Hewlett Packard), and the BlurayDisc Founders (Sony, Philips, LG, Matsushita, Pioneer, Samsung, Sharp, andThomson) are boosting Blu-ray.Voice RecognitionVoice or speech recognition is the ability of a machine or device to receive andinterpret dictation or to comprehend and carry out oral commands. In use withcomputers, analog audio must be converted into digital signals through an analogto-digital(A/D) converter. In order that it can decipher the signal, a computer musthave a digital database, or vocabulary, of words or syllables, and a rapid meansof comparing this data with signals in the format of speech patterns stored in thehard drive and loaded into memory when the program is operating. A comparatorcompares these stored patterns against the output of the A/D converter. Thus, theprogram’s vocabulary constitutes a recording of sorts.

576 19. Sound ReproductionIt is fairly obvious that the size of a voice-recognition program’s effective vocabularyand its speed of execution directly relates to the random access memory(RAM) capacity of the computer in which it is installed. A voice-recognition programruns many times quicker if the entire vocabulary can be loaded into the RAMinstead of searching the hard drive for some of the matches. Processing speed isalso critical, because it affects how rapidly the computer can search the RAM formatches.In the present state of the art, all voice-recognition programs make errors. Backgroundnoise, such as barking dogs, loud conversation, and noisy children, canproduce false input. There is also a problem with words that sound alike but arespelled differently and have different meanings—for example, “know” and “no”or “dear” and “deer.” This problem may someday be solved by using stored contextualinformation, but this requires more RAM and even faster processors thanare currently available in personal computers.Voice recognition programs are now applied to dictation in conjunction withword processing programs and in voice-response customer servicing on the telephone.19.3 Playback Audio EquipmentFigure 19.3 shows a schematic of a fairly complete playback system for multichannelsound reproduction, in this case a “5.1 surround sound” system. Whetherthe system is monophonic (single-channel), stereo (two-channel), or multichannel(more than two channels 2 ), there are three stages to sound reproduction: the firststage consists of the source or sources of program material; the second stage is thepreamplification and amplification. The third stage consists of converting the electricalsignals into acoustic signals, and this occurs in loudspeakers or headphones.Two-channel (stereo) setups reproduce the spatiality of the original program.When properly recorded and reproduced, the strings located on one side of theorchestra get reproduced more strongly in the same side of loudspeaker than in theloudspeaker on the other side, giving the listener the illusion of strings’ location.Multichanneling is intended to reproduce not only the spatiality of the program butalso provide some degree of the acoustic ambience of the hall where the programwas recorded or to reproduce the effects of multichannel “surround sound.” Theacoustics of the listening room, of course, affects the reproduced sound, whichmay even assume additional characteristics that may not be so desirable.2 A 5.1-multichannel system consists of right, left, and a center-channel speaker in the front and a rightanda left-rear speakers, plus a subwoofer (the “.1” of the 5.1 designation) that theoretically couldbe placed anywhere in the room. A 7.1-channel system contains the five sets of speakers and thesubwoofer of the 5.1-system plus two more sets of speakers placed to the right and left in the middleof the room. In either case, additional amplifiers are required for each of the additional speakers.A multichannel system also would normally incorporate an audio encoding/decoding system, suchas the Dolby Digital r○ , Dolby Digital EX r○ , or Dolby Digital Surround EX r○ to deliver discretemultichannel audio in an optimal fashion to the appropriate channels.

19.3 Playback Audio Equipment 577AM/FM AND/ORSATELLITERADIO TUNERTAPE PLAYERAND RECORDER(REEL-TO-REEL,CASSETTE, DCC,DAT, ETC.)CD AND/ORDVD PLAYERREMOTECONTROLMICROPHONEAUDIO FROMTV TUNERAND/OR VCRPHONO-GRAPHPREAMPLIFIER/EQUALIZATIONAND CONTROLCENTERHEADPHONEAMPLIFIERHEADPHONESAMPLIFIER AMPLIFIER AMPLIFIER AMPLIFIER AMPLIFIERAMPLIFIERLEFT REARSPEAKERLEFT FRONTSPEAKERCENTERSPEAKERRIGHT FRONTSPEAKERRIGHT REARSPEAKERSUBWOOFERFigure 19.3. Playback system for multichannel sound reproduction.The sources constituting the first stage can include one or more of the following:microphone, AM/FM and/or satellite radio tuner, phonograph (consisting of aturntable, arm, and cartridge 3 ), reel-to-reel or cassette player, compact disk (CD)player, digital versatile disk (DVD) player, and the audio output of a televisionreceiver. The second stage consists of a preamplifier that provides the means toselect signal sources, the proper equalization for program sources, and means ofcontrolling the volume of the program material. After the initial preamplificationstage, the program material is amplified in the amplifier. An amplifier may bemultichanneled in that there is a separate amplifier for each channel, and a numberof these amplifiers may be mounted on a single chassis. 4 An integrated amplifier3 Magnetic cartridges convert stylus movements into electrical signals. Ceramic cartridges are essentiallypiezoelectric converters that yield stronger electrical signals that need less preamplification butdo not have the range and subtlety of magnetic cartridges.4 In extremely elaborate (and expensive!) systems, an electronic active crossover system may be insertedbetween the preamplifier and the amplifier, and the customary passive crossover system built insidethe loudspeaker system is bypassed. Each channel’s input into the electronic crossover separates intoappropriate frequency bands, which are fed into separate amplifiers. Each of the amplifiers linksdirectly to the individual drivers. A single channel would require three amplifiers, one to feed thewoofer, another to feed the midrange, and the third to feed the tweeter.

578 19. Sound ReproductionINNER SUSPENSION(SPIDER)OUTER SUSPENSIONCENTER DOME(DUSTCAP)FRAMEVOICE COILPOLE PIECESCONEMAGNETMAGNETIC GAPFigure 19.4. Elements of an electrodynamic speaker. (Courtesy of JBL Professional.)combines on a single chassis the preamplifier (which acts as a controller) andthe amplifier. A stand-alone amplifier is referred to as a power amplifier and itfunctions in conjunction with a separate preamplifier. A chassis that combines aradio tuner with a preamplifier and a power amplifier is called a receiver. This typeof construction makes for more economical production, with a single-power supplyserving all of these sub-components, and conservation of rack or shelf space; itsprincipal disadvantage is potentially increased heat output from the electronicsand the lack of flexibility in upgrading individual sub-components.The third stage consists of loudspeakers and or headphones. A tremendous varietyof loudspeakers are available in the market. The most common type is theelectrodynamic speaker, which can range from a single-cone type to more elaboratemultidriver units consisting of woofers to reproduce low-frequency sounds,midrange-drivers to handle the frequencies between the low frequencies producedby the woofers and the high frequencies produced by tweeters. A two-way loudspeakersystem divides its program material between a woofer and a tweeter. Athree-way loudspeaker system incorporates mid-range drivers, and four and fivewayunits have also been constructed. In order that the individual drivers receivethe proper frequencies to the exclusion of the program contents outside of theirrespective optimal operating ranges, special types of bandpass filters, or crossovernetworks, are employed to separate out the high frequencies from the signals beingfed into the woofers, to band-pass the mid-frequencies into the mid-range drives,and to channel only the high-frequency portion of the signals into the tweeters.Because much of the acoustical energy is contained within the low-frequency portionof the signals, passage of unfiltered signals into the tweeters can destroy thesedrivers.Figure 19.4 shows the construction of an electrodynamic driver. Three separatebut interrelated subsystems constitute the driver. The motor system consists ofthe magnet, pole piece, front plate, and voice coil. The diaphragm, generally a

19.3 Playback Audio Equipment 579cone and a dust cap or a one-piece dome, constitutes the second subsystem; thesuspension system, which includes the spider and the surround, constitutes theremaining subsystem.In the motor assembly, the back and front plates and the pole pieces are madefrom a highly permeable material, such as iron, which provides a path for the magneticfield of the ring-shaped magnet that is usually constructed of ceramic–ferritematerial. The magnetic circuit is completed at the gap, with a strong magneticfield existing in the air space between the pole piece and the front plate. The coil,which connects to a pair of input terminals, is wound around a thin cylinder thatis attached to the speaker cone that, in turn, is mounted at its outer edge througha flexible surround to the frame. A spider, essentially a movable membrane alsoattached to the frame, positions centrally the diaphragm at its inner edge and thevoice coil. The signal containing the program material feeds into the voice coil,which generates a change in the magnetic field imposed by the permanent magnetsurrounding the coil. If an alternating current is fed into the coil, the flow of thecurrent in one direction will cause the voice coil to move in one direction, and thereverse flow will cause the coil to move in the opposite direction. The cone underthe impetus of the moving voice coil acts as a piston in moving the air in frontof it.The ideal cone would act as a perfectly rigid piston pushing against the air. Thetransfer of the motion from the piston to the air is bound in terms of frequency byresonance frequency of the cone at the low end (here the ability to transfer energyto the air is limited by mechanical constraints) and by the radiation impedance atthe upper limit. This upper frequency limit occurs from the fact that it is a functionof both the nature of radiation impedance of the air and the radius of the radiatingsurface. Smaller radiating surfaces can reproduce higher frequencies moreeffectively than larger surfaces, which accounts for the smaller sizes of tweeters.Real-world cones, however, are not perfectly rigid and will flex depending on thetraits of the materials they are constructed from. Cone flexure has a critical effecton the high-frequency efficiency, the sound–pressure level output, and driver–polarresponse. Driver materials may differ in degrees of stiffness and transmit vibrationsat different speeds internally, but they tend to produce the same sort of flexuresor modes. In Chapter 6, these modes have been discussed for circular membranesfixed at their outer edges.The most important function of the speaker enclosure is to control transmissionof the driver’s rear-radiated sound energy in order to avoid its mutual cancellationwith forward-radiated energy at low frequencies. The enclosure also acoustically“loads” the driver by providing a suitable acoustic impedance to match the characteristicsof the driver and the requirements of the speaker system with regard tolow frequency and large-signal performance. The enclosure must be made sufficientlyrigid so that the cabinet vibrations and resonances do not add appreciablyto the program material. Two major classes of electrodynamic-driver enclosuresare (1) the infinite baffle/closed box in which the sole radiation source is the driverdiaphragm and (2) the bass reflex that is vented to augment the driver’s radiationat low frequencies. The bass reflex may be a vented system that incorporates a

580 19. Sound Reproductiontuned aperture of a specified cross-sectional area and port length in which theenclosed air mass resonates with the enclosure’s air spring. This causes a woofer’sback wave to communicate with the external acoustic space with the enclosureeffectively acting as a Helmholtz resonator.Another version of the bass reflex system is the port/passive radiator (PR). Thevent is replaced with an acoustically driven diaphragm (aptly called a drone cone).Another type of enclosure is the transmission line (TL) that effectively functionsas a low-pass filter with a 90 ◦ phase shift, absorbing all of the rear wave energyof the woofer except for frequencies below 75 Hz. The TL enclosure provides afolded path or a labyrinth equal in length to the 25% of wavelength at or just abovethe resonance of the woofer. Damping material such as Dacron r○ , fiber wool,or fiberglass fills much of the labyrinth. The TL enclosures generally provideexcellent, clean bass response from relatively compact floor-standing cabinets.The principal disadvantage of TL is the complexity of the cabinet structure.Other types of speaker systems are available in the market. Among these are theelectrostatic speakers, the ribbon-type speaker system, and there are others thatconsist of hybrids, e.g., a system that combines the electrodynamic drivers withan electrostatic or ribbon tweeter. A ribbon type of drive essentially consists of avoice coil patterned (like the conductive pattern on a printed circuit board) onto amagnetizable diaphragm (i.e., the “ribbon”) made of strong, flexible materials suchas Mylar. The diaphragm is suspended in the front of a flat magnet that matchesthe area of the diaphragm. A varying signal sent into the coil causes a change inthe magnetic field that causes the ribbon to move with respect to the magnet, thuscausing sound to be radiated. If the magnet is perforated in one way or the other,the ribbon acts as a dipole sound generator.In the electrostatic speaker (which is similar operationally to the condensermicrophone but functioning in reverse), a voltage is maintained between a thindiaphragm and another surface. In effect the two thinly separated surfaces areacting as a large condenser, with the varying voltages fed to one of the elements,causing the diaphragm to move. Because of the small excursions of the diaphragm,the electrostatic driver does not function well in the low-frequency range butit is capable of providing excellent quality high-frequency signals. The highlycapacitive loading of the electrostatic speaker may be problematic for some poweramplifiers that are usually designed to handle primarily resistive impedances.Headphones are essentially miniaturized speakers mounted in earcups, whichare in turn attached to headbands that serve the purpose of holding these driversagainst the ears. There are also a number of headphone models that use electrostaticelements either by themselves or as supplemental components to electrodynamicelements.19.4 The Iosono SystemOn the basis of research conducted for more than 15 years at the Technical Universityin Delft, the Netherlands, Karlheinz Brandenburg, the director of Fraunhofer

19.5 Portable Audio Playback Equipment 581Institute of Digital Media in Germany who is known as the “father of MP3 describedin the next section, developed a spatial sound technology called Iosono r○ .It is intended to carry sound reproduction beyond the 5.1- and 7.1-channel systemsand is based on the principle that sound waves can be reproduced using secondarysources at the perimeter of the original sound field. This process is called wavefield synthesis (WFS).In the Iosono system, speaker arrays ring the listening area and function in acoordinated, phased fashion to reproduce each individual sound wave. An exampleof an acoustical illusion that can be created is that of a helicopter that slowlyapproaches the audience, flies through the middle of the theater, and disappearsinto the distance. The audience hears the helicopter noise that would be generatedif the helicopter were actually flying this path. It is also claimed for this system,aimed at cinemas and eventually the home theater, that the whole room can act asa listening “sweet spot.”An Iosono system consists of a continuous ring of speakers configured in panelsof 8 two-way speakers, mounted on the peripheral walls of the space and connectedvia fiber-optic cables to a central Iosono processing unit. The number of speakerpanels used depends on the size of the space. In February 2003, a 100-seat movietheater in Ilmenau, Germany, received the first commercial Iosono Cinema system.The installation used 198 speaker systems. Because of the fact that commercialmovie sound has yet to be mixed, the Iosono system is also capable of playback ofother formats, such as 5.1, DTS, SDDS as well as stereo. However, the audiencenoticed the enlargement of “sweet spot” and was treated to a 90-s trailer thatdemonstrated the effect of a full Iosono system.19.5 Portable Audio Playback EquipmentIn this section reference is made to handheld devices, not to playback units thatare transportable but too large to be held conveniently. The first units of this typewas the Sony Walkman r○ , a cassette player, which was to be joined by competingmodels, and subsequently the Sony Discman r○ that played compact disks. Thenewer units we describe here can download music from the Internet as well asfrom other sources and the storage is usually done on rewriteable nonvolatilememories.MP3MP3, developed by Germany’s Frauenhofer research institute, denotes MPEGAudio Layer 3. It is an audio compression technology that constitutes part of theMPEG-1 and MPEG-2 specifications. The company Thomson Media patented thistechnology in the United States and in Germany.Uncompressed audio, such as that found on audio compact disks store moredata than human hearing can process. Music on CDs has a bandwidth of 1.4 MB/s.This means that 1 min of music on a CD takes up to 10 MB of data. Through the

582 19. Sound Reproductionuse of MP3 compression, this bitstream is greatly reduced by a factor of 8–12. Atypical MP3 file will require 0.828 MB/s, and hence 1 min of music is reducedfrom 10 MB to about only 1 MB. Even greater compression is possible for use onthe Internet, for example, but at the expense of a decrease in sound quality.MP3 uses two compression techniques to achieve size reduction from uncompressedaudio: one lossy and the other lossless.1. The process begins by breaking the signal into smaller components calledframes, each frame typically lasting a fraction of a second. The signal is analyzedto determine its spectral energy distribution. Because different portionsof the frequency spectrum are most efficiently encoded via slight variant ofthe same algorithm, the signal can be broken into sub-bands, which can beprocessed independently for optimal results.2. The encoding bitrate comes into play, and the maximum number of bits that canbe allocated to each frame is calculated. For example, if the encoding is set at128 kbps, there is an upper limit on how much data can be stored in each frame.This step establishes how much available audio data will be stored and howmuch will be discarded. This constitutes a lossy phase of the MP3 procedure.3. The frequency spread for each frame is compared to mathematical modelsof human psychoacoustics, which are stored in the codec 5 as a reference table.From these models, it can be determined which frequencies need to be renderedaccurately, because they are most perceptible to humans, and which ones canbe dropped or allocated fewer bits, owing to the fact that they are less likely tobe perceived by human hearing. This stage is also a lossy procedure.4. The bitstream is run through the process of Huffman coding that compressesredundant information throughout the sample. The Huffman coding does notwork on the basis of a psychoacoustic model but achieves additional compressionthrough more traditional means so that even less space is required forstorage. The action is quite lossless, similar to the “zipping” and “unzipping”of files on a computer.5. The collection of frames is now assembled into a serial bitstream, with headerinformation preceding each data frame. (The header contains extra informationabout the file to come, such as the name of the artist, the track tile, the nameof the album from which the recording came from, the recording year, genre,and personal comments that may have been added—this is referred to as anID3 tag).iPodThe iPod r○ is Apple Computer’s multiuse record/playback ultraportable unit thatmakes use of a mini disk drive to store data. It plays music and other audio that5 The word “codec,” a shortening of the words compress and decompress, refers to any of a class ofprocesses that allow for the systematic compression and decompression of data. Various codecs arefundamental to many formats and transmission methods, for example, image and video compressionformats. Here we are concerned with the audio MP3 codec.

References 583has undergone MP3, AAC, or Apple Lossless compression programs. One versionweighs less than 160 g and it is available with a choice of 20 GB or 40 GB storage,resulting in a capability of storing as many as 10,000 songs. Even more compactversions with less storage and other models with more storage are emerging fromApple, and competitive units are being issued by other manufacturers. The iPod canalso be used to record meetings, enter voice memos, and its speed can be adjustedto accommodate audio books. Synchronization with a home computer for downoruploading data can be achieved via FireWire r○ or USB 2.0 cables. The iPod canalso be used as a portable hard drive, which would allow the user to carry computerfiles from one place to another. It can even used to store photographs, function asan organizer with contact listings, calendars, and to-do lists; and, with the use ofadapters, can be played through home and auto sound systems.19.6 The Future of Sound ReproductionThere is no question that nanotechnology, still in its infancy, will result in morecompact recording media, more powerful electronic processing units, and morerealistic audiovisual imaging. Digitization of acoustic information will be refinedto an even greater extent.Loudspeakers have been the weak link in the audiophile chain of playback ofrecorded sound. Amplifiers and preamplifiers have reached the point of diminishingreturns with respect to low distortions and noise. The principal deviation from theideal sound reproduction occurs in transducers, hence it is the loudspeaker that isthe component that needs the most improvement. It is apparent that new technologywill be needed to develop new types of transducers, most likely resulting from acollaboration of the best minds in the fields of acoustics, materials science, solidstate physics, electronic and mechanical sciences, and even nanotechnology. Notonly loudspeaker distortion would be reduced to theoretical minimum (Raichel,1979), but also the sound sourcing would be considerably more sharply focused toenhance the listener’s feeling of actually “being there” at the concert venue wherethe music was recorded.ReferencesBallou, Glenn (ed.). 2005. The Handbook For Sound Engineers, 3rd ed. New York: FocalPress. (A comprehensive work on sound engineering. Includes coverage of MIDI, cinemasound, consoles, and other topics.)Borwick, John (ed.). 2001. Loudspeaker and Headphone Handbook, 3rd ed. New York:Focal Press.Cooke, Raymond E. (ed.). 1980 (Vol. 1), 1984 (Vol. 2), 1992 (Vol. 3). Loudspeakers, Vols.1–3. New York: Audio Engineering Society. (A treasure trove of technical papers writtenduring 1980–1992 on the design, construction, and operation of loudspeakers.)Crowhurst, Norman, 1960. The Stereo High Fidelity Handbook. New York: Crown Publishers.(A succinct text by one of the top writers in the field of stereo reproduction.)

584 19. Sound ReproductionDickason, Vance. 2000. Loudspeaker Design Cookbook, 6th ed. Peterborough, NH: AudioAmateur Press. (An excellent text to learn about the design and construction ofloudspeakers.)Hacker, Scot. 2000. MP3: The Definitive Guide. Sebastopol, CA: O’Reilly Media,Chapter 2.Holman, Tomlinson. 1999. 5.1 Surround Sound—Up and Running. New York: Focal Press.(Written by the developer of the THX sound system.)Olson, Harry F. 1967. Music, Physics and Engineering, 2nd ed. New York: DoverPublications.Olson, Harry F. 1991 (reprint). Acoustical Engineering. New York: Professional AudioJournals.Raichel, Daniel R. 1979. Minimum harmonic distortion levels in ideal loudspeakers.Journal of Audio Engineering Society 27(8): 492–495. (Previously presentedNovember 7, 1977 at the 58th Convention of the Audio Engineering Society, NewYork, NY.)Risset, J. C. and Mathews, M. V. 1965. Journal of the Audio Engineering Society 13:200–206.Strong, W. J. and Plitnik, G. R.1983. Music, Speech and High Fidelity. Salt Lake City:Soundprint.Watkinson, John. 2004. The MPEG Handbook. New York: Focal Press. (Probably the mostdefinitive book on the popular MPEG technology.)Problems for Chapter 191. Why was the cassette tape overtaken by the compact disk in popularity?2. What are the principal advantages and disadvantages of data compression insound recordings?3. What are the advantages and disadvantages of having, say, a two-way versusa three-way versus a four-way speaker system?4. What are the dangers of using high-powered audio amplifiers?5. What are the key factors in determining the quality of a playback system?6. Why would a reputable recording studio use higher speeds for taping ratherthan presumably more economical slower speeds?7. Design your “dream” audio system on the basis of your market research.Explain the reasons for your choice of specific components.8. Research loudspeaker systems in the market. What specifications would youlook for in making your selection?9. Why would an MP3 or an iPod system be objectionable to an audiophile foruse in an audio system?10. How is computer technology is being used in the recording stage and playbackstage?11. How does room acoustics affect playback of audio material? What are theincipient problems and how would you deal with them?

20Vibration and Vibration Control20.1 IntroductionNoise often results from vibration. Many sources of vibration exist and they includeimpact processes, such as blasting, pile driving, hammering, and die stamping; machinerysuch as motors, engines, fans, blowers, and pumps; turbulence in fluidssystems; and transportation vehicles. Attenuation of vibration generally cuts downon the noise level and in many cases lengthens the service life of the machinery itself.Damping, correction of imbalances, and configuration of flow paths constitutethe principal measures of cutting down on the deleterious effects of vibration.20.2 Modeling Vibration SystemsWe commence with a basic one-degree-of-freedom system of Figure 20.1, whichconsists of a mass, a spring, and a damper (also referred to as a dashpot). Thesystem has only one degree of freedom because it is constrained to move onlyin the x-direction. Summing up all the forces acting on the mass and applyingNewton’s second law, we obtainwherem d2 xdt + C dx + kx = f (t) (20.1)2 dtm = masst = timeC = coefficient of dampingk = linear elastic constantDamping occurs from energy dissipation due to hysteresis, sliding friction, fluidviscosity, and other causes. The damping force may be proportional to the velocity(as stated in the above equation) or it may even be proportional to some other powerof velocity. Sliding friction is often represented as a constant force in a direction585

586 20. Vibration and Vibration ControlFigure 20.1. Model of a vibrating system with one degree of freedom.opposing the velocity. Viscous damping proportional to and opposite in directionto that of the velocity serves as a reasonable model in many situations. Moreover,this assumption is quite amenable to mathematical treatment. When f (t) is set tozero, Equation (20.1) describes the free, viscous-damped, one-degree-of-freedomsystem,m d2 xdt + C dx + kx = 0 (20.2)2 dtWe shall employ Laplace transforms to treat Equation (20.2). DividingEquation (20.2) by mass m and assuming for the moment that the initial conditionsare zero, we can write Equation (20.2) using the Laplace transform variable s, ass 2 X(s) + C m sX(s) + k m X(s) = sx(0) + ẋ(0) + C m x(0)or(s 2 + C m s + k ) (X(s) = s + C )x(0) + ẋ(0) (20.3)mmHere ẋ ≡ dx/dt. Setting the parenthesized term on the left-hand side ofEquation (20.3) to zero yieldss 2 + C m s + k m = 0which can be rewritten in the forms 2 + 2ξω n s + ωn 2 = 0 (20.4)

20.2 Modeling Vibration Systems 587Figure 20.2. Root locations and nomenclature for a second-order dynamic system. Thes-plane to the right shows the domain of the roots as a function of the damping ratio ξ.where2ξω n = C mω 2 n = k m(20.5)Equation (20.4) is the characteristic equation of which roots essentially determinethe response of the system [i.e., the position of the mass as a function of time orx(t)]. Solving for the roots of Equation (20.4) yieldss 1 , s 2 = −2ξω n ± √ (2ξ 2 ω n ) 2 − 4ωn2 (20.6)2These roots can assumed complex values and so can be plotted on a s-plane,where s = σ + iω and ω = 2π f . The general plot is shown in the complex planeof Figure 20.2(a). For this plot it is assumed that the system is stable and the valueof ξ falls between zero and unity. The term ξ is the damping ratio and ω n is theundamped natural frequency. In Figure 20.2(a) it is noted that ξ = cos θ. Bothξ and ω n constitute the key factors that determine the roots of the characteristicequations and hence the response of the system. The following four cases ofdamping are of interest:Case 1: ξ1(Overdamped system)The roots are s 1 , s 2 =−ξω n ± iω √ ξ 2 − 1 and the system response isgiven by( √ ) ( √ )x(t) = Ae − ξω n +ω n ξ 2 −1 t − ξω+ Ben −ω n ξ 2 −1 t

588 20. Vibration and Vibration ControlCase 3: ξ = 1(Critically damped system)The roots are s 1 , s 2 =−ω n and system response is given byx(t) = Ate −ω nt + Be −ω ntCase 4: ξ = 0(Undamped system)The roots are s 1 , s 2 =±iω n and the system response is given byx(t) = A cos (ω n t + θ)Figure 20.2(b) shows the domains of the roots in the s plane as a function of thedamping ratio ξ for a constant value of ω n .In order for us to appreciate the influence of system damping, consider the systemof Figure 20.1 which is subjected to an initial displacement x 0 and whichhas no external forcing function acting on it. The dependency of the response ofthe system on the root locations is shown in Figure 20.3 for ω n held constant. InFigure 20.3. System responses showing the influence of root location in a second-orderdynamic system.

20.2 Modeling Vibration Systems 589Figure 20.3(a) the system is very sluggish and returns to the equilibrium positionvery slowly. In Figure 20.3(b), the system response is rapid, but it overshoots theequilibrium position and eventually the oscillation dies out due to the effect ofdamping. Figure 20.3(c) represents the critically damped system that promptlyreturns to the equilibrium position with no overshoot. This represents the dividingline between the overdamped system (Case 2) and the underdamped system(Case 1). The roots for the underdamped system occur in complex-conjugate pairs,the roots for the overdamped system are real and unequal, and the roots for thecritically damped system are real and equal. Figures 20.3(b) and 20.3(d) show, respectively,the difference between a lightly damped system and a heavily dampedsystem. The response of the more heavily damped system dies out more quickly.If the roots lie on the iω axis as shown in Figure 20.3(d), no damping occursand the system will oscillate forever. If any roots appear on the right half of thes plane, then the response will increase monotonically with time and the systemwill become unstable, as shown in Figure 20.3(f). Thus the iω axis represents theline of demarcation between stability and instability.From Equation (20.4) the natural frequency of the system is√kω n =m(20.7)and the damping ratio isξ =C2 √ km(20.8)When ξ = 1, critical damping occurs. We set C = C c for this value of ξ = 1. Thenfrom which we obtainξ = 1 =C c2 √ kmC c = 2 √ km (20.9)as the critical damping factor. The damping ratio is often written asξ = C C c(20.10)Example Problem 1The system of Figure 20.1 has the following parameters: the weight W of mass mis 28.5 N, C = 0.0650 N s/cm, k = 0.422 N/cm. Determine the undamped naturalfrequency of the system, its damping ratio, and the type of response the systemwould have if the mass is to be initially displaced and released.

590 20. Vibration and Vibration ControlSolutionThe mass is found fromm = W g = 28.5N = 2.91 kg29.807 m/sThen from Equation (20.7)√42.2 N/mω n == 3.81 rad/s2.91 kgFrom Equation (20.8)ξ =C2 √ km = 6.50 N s/m2 √ (42.2 N/m)(2.91 kg) = 0.294Since the damping ratio is less than unity, the response will be underdamped. Thesystem response is that of the form shown in Figure 20.3(b).Example Problem 2For the system of Example Problem 1, find the amount of additional dampingneeded to have the system become critically damped.SolutionFrom Equation (20.9), the condition for a critically damped system isC c = 2 √ km = 2 √ (42.2 N/m)(2.91 kg) = 22.13 N s/m, = 0.2213 N s/cmThe additional damping required isC = 0.2213 − 0.0650 = 0.1563 N s/cmWith this additional damping, Equation (20.2) becomesẍ(t) + 22.13 42.2 ẋ(t) +2.91 2.91 x(t) = 0The characteristic equation in the Laplace transform variable form becomess 2 + 22.132.91 s + 42.22.91 = 0or(s + 3.81) 2 = 0which demonstrates that Case 3 of Figure 20.3 exists.

20.3 General Solution for the One-Degree Model of Simple System 59120.3 General Solution for the One-Degree Modelof Simple SystemAn initial displacement is introduced to the model system of Figure 20.1, and itis desired to determine the response of this system. The differential equation instandard mathematical shorthand notation isẍ(t) + 2ξω n ẋ(t) + ωn 2 x(t) = 0 (20.11)which undergoes a Laplacean transformation intos 2 X(s) − sx(0) − ẋ(0) + 2ξω n [sX(s) − x(0)] + ωn 2 X(s) = 0Let the initial displacement be represented by x(0) = x 0 and the initial velocityẋ(0) = 0. The preceding equation reduces tos + 2ξω nX(s) = x 0(20.12)s 2 + 2ξω n s + ωn2The graphical residue technique, described by Example Problem 4 in AppendixC, can be used to find the inverse transform for Equation (20.12) and hence thesolution of Equation (20.11):x(0√x(t) = √1 − ξ2 e−ξω nt cos ω n 1 − ξ 2t − π )2 + cos−1 ξwhich reduces tox(0√ )x(t) = √1 − ξ2 e−ξω nt sin ω n 1 − ξ 2t + θ(20.13)from the trigonometric relationship cos (ϕ − π/2) = sin ϕ and by settingθ = cos −1 ξ (20.14)Example Problem 3Consider the system of Figure 20.1 which is initially displaced and then suddenlyreleased. Its resultant motion is described byx(t) = 3.0e −3.43t sin (11.4t + 60 ◦ )Find the system’s damping ratio, natural frequency, and the initial displacement.SolutionApplying Equations (20.13) and (20.14),θ = 60 ◦ = cos −1 ξ, ξ = 0.500, and ξω n = 3.43Hence ω n = 6.86. The initial condition is established fromx 03.0 = √ = x 0√ = 1.155x 01 − ξ2 1 − (0.500)2x 0 = 2.597

592 20. Vibration and Vibration ControlFigure 20.4. Response of a second-order system that is underdamped.Because ξ x 0 , i.e., a slope of x existsinitially, an initial velocity is present. The actual damping is ξω n , and the dampedperiod is defined as2πτ d = √ (20.15)ω n 1 − ξ2In Figure 20.4, it is evident that the decay rate e −ξωnt of the free oscillation dependson the system damping. The greater the damping, the faster is the rate of decay.Let us now establish the relationship between the rates of decay and damping.First, the time response at two distinct points, each of which is a quarter periodfrom the crossover points in the first two lobes, must be determined. These pointsare the points where the sine function equals unity and not the peak points of thedamped response in Figure 20.4. From Equations (20.4) and (20.13), it is notedthatx(0√ )x 1 = √1 − ξ2 e−ξω nt 1sin ω n 1 − ξ 2t 1 + θ(20.16)andx 2 =x(0√ )√1 − ξ2 e−ξω n(t 1 +τ d ) sin ω n 1 − ξ 2(t 1 + τ d ) + θ(20.17)Taking the ratio of the two amplitudes represented by Equations (20.16) and (20.17)and noting that in this case the sine functions equal unity, we obtainx 1x 2= e−ξω nt 1e −ξω n(t 1 +τ d ) = eξω nτ d

20.4 Forced Vibration 593The logarithmic decrement δ is defined at the natural logarithm of the ratio oftwo points such as x 1 and x 2 . Invoking Equation (20.14) and this definition forlogarithmic decrement, we can derive( )x1δ = ln = √ 2πξ(20.18)x 2 1 − ξ2Thus, the logarithmic decrement δ is expressed in terms of the damping ration ξof the system.A negative value of the response function can also be used to find the logarithmicdecrement, but in this case τ d should be replaced by τ d /2, because a half period isused in this evaluation. For this situation,δ = 2ln(x1x 3)The values of the response function at any two points can be used to find the twounknowns ξ and ω n . It is also important to realize that the ratio of the points in theresponse curve, e.g., x 1 /x 2 , x 2 /x 3 ,orx 3 /x 4 , will be identical only in the presenceof viscous damping.20.4 Forced VibrationForced vibration occurs when f (t) ≠ 0 in Equation (20.1). The forcing functioncan be a harmonic excitation, an example of which is the imbalance in rotatingmachinery such as a motor. Or it can be an impulse type of excitation, such as thatproduced by a hammer or it can be simply the weight of the moving part itself.Harmonic ExcitationA harmonic forcing function can be represented by F 0 sin ωt. The differentialequation for the model of Figure 20.1 assumes the following form:mẍ + Cẋ + kx = F 0 sin ωt (20.19)With the assumption of zero initial conditions, the Laplacian transform ofEquation (20.18) yields(ms 2 + Cs + k ) ( )ωX(s) = F 0 (20.20)s 2 + ω 2and solving for X(s)( )F 0 ωX(s) =(20.21)ms 2 + Cs + k s 2 + ω 2As a rule, the solution x(t) will consist of two parts, viz. the complementarysolution and a particular solution. The former corresponds to the transient part ofthe total solution and the latter to the steady-state part. The transient portion of the

594 20. Vibration and Vibration Controlsolution is in the form of Equation (20.13) and it is determined by the residues ofthe complex poles, wherein the poles constitute the solution ofms 2 + Cs + k = 0The steady-state solution of Equation (20.19) is a sinusoidal oscillation expressedasx(t) = x(ω) sin (ωt − φ) (20.22)and this solution is determined by the residues at the complex poles s = s =±iω.While the exact solution can be derived by using the residue method, the solutionof Equation (20.19) can be readily obtained through the standard differentialmethods. The advantage of the Laplace transform method that it facilitates findingthe frequency and stability information.Through Equations (20.21) and (20.22), the magnitude of the steady-state oscillationcan be determined fromF 0X(s) =(20.23)ms 2 + Cs + kFrom Equation (20.23), the magnitude of the oscillation can be determined as afunction of frequency. Because s = σ + iω and ω = 2πf, substituting s = iω intoEquation (20.23) yieldsF 0x(iω) =−mω 2 + iωC + kThe denominator is a complex number, so the magnitude of this number is equalto√(k − mω2 ) 2 + (ωC) 2The magnitude of the oscillation as a function of frequency isx(ω) =F 0√(k − mω2 ) 2 + (Cω) = F 0 /k√ (20.24)2 [1 − (m/k)ω2 ] 2 + (Cω/k) 2From the definitions of Equations (20.7) through (20.10), Equation (20.24) can beexpressed asX(ω)F 0 /k = 1√ (20.25)[1 − (ω/ωn ) 2 ] 2 + [2ξ(ω/ω n )] 2The magnitude is now a function of only two quantities, the ratio (ω/ω n ) and thedamping ratio ξ. The phase angle φ is also a function of these two parameters andit is given bytan φ = 2ξ(ω/ω n)(20.26)1 − (ω/ω n ) 2The system undergoes a resonance when the excitation frequency f = ω/2πequals the natural frequency of the dynamic system f n = ω n /2π. The damping

20.4 Forced Vibration 595Table 20.1. System Response as a Function of Frequency.Frequency Response Controlling Parameterω 2 ≪ ωn 2 x(ω) = F 0kω 2 ≫ ωn 2 x(ω) = F 0mω 2ω 2 = ωn 2 x(ω) = F 0CωStiffness controlledMass controlledDamping controlledratio ξ affects the magnitude of the oscillation peak at resonance and the sharpnessof this resonant peak.In considering the system response as a function of frequency, we observethat the response varies with the frequency as shown in Table 20.1. The relationshipslisted in the table show that each parameter listed—the stiffness, the mass,and damping—effectively controls the response only within a limited region. Forexample, the damping is primarily effective at resonance. The selection of anyvibratory corrective measure depends on whether the excitation frequency is lessthan, greater than, or equal to the resonant frequency of the system.The effect of the amplitude by stiffness, mass, or damping is exemplified by themagnification factor MF, defined asMF = x(ω)F 0 /k = 1√ (20.27)[1 − (ω/ωn ) 2 ] 2 + [2ξ(ω/w n )] 2At resonance ω = ω n , and therefore(MF) resonance = 1(20.28)2ξA measure of the shape of the resonance peak is given by the bandwidth at thehalf-power points, as shown in Figure 20.5. These points are the two points, one tothe right and one to the left of the peak, which have a magnitude equal to (1/ √ 2)of the value of the peak. The square root occurs because power is proportional tothe square of the magnitude.Let us set h ≡ ω n /ω. At half-power points, Equation (20.27) becomes12 √ 2ξ = 1√(1 − h2 ) 2 + (2ξh) 2Solving the preceding equation algebraically for h 2 results inh 2 = 1 − 2ξ ± 2ξ √ 1 + ξ 2We also assume small values of damping (i.e., ξ ≪ 1) and neglect second-orderterms. Then the following result occurs:h 2 = 1 ± 2ξ

596 20. Vibration and Vibration ControlFigure 20.5. Principal features of a resonance peak.ThenHenceWe approximateh 1 =( ωω n) 2= 1 − 2ξ, h 2 =( ωω n) 2= 1 + 2ξThe bandwidth bw is given byω 2 2 − ω2 1ω 2 nh 2 − h 1 = 4ξbw = (ω 2 − ω 1 )ω n≈ 2(ω 2 − ω 1 )ω n= ωω= 2ξ (20.29)The reciprocal of the bandwidth is the quality factor Q, expressed asQ = ω nω = 1(20.30)2ξWhen a frequency response of system is plotted in the format of Figure 20.5,this figure in conjunction with Equation (20.30) can provide the basis for findingthe equivalent viscous damping of the system.

Excitation by Impulse20.4 Forced Vibration 597Impulse occurs very commonly as a cause of vibration excitation in the industrialenvironment. An impulse force is one that acts for a very short time. A hammeris an example that provides an impulse force. Such a force can be represented bythe Dirac delta function or distribution that has the following properties: considera pulse that starts at time t = ε and has width a and height 1/a. Let a approachzero, so that the pulse essentially approaches zero width and infinite height so that∫ ∞0δ(t − ε) dt = 1 0

598 20. Vibration and Vibration ControlFrom Equation (20.27)xF 0 /k = 12ξHence at resonance, the amplitude at resonance isx res =Static Deflection14.25 N/945 N/m0.276= 0.0546 m = 5.46 cmIn Figure 20.1 the deadweight W constitutes a static load that causes a deflectionδ st that is given byδ st = W k = mg(20.33)kwhere g represents the gravitational constant. Because the natural frequency of thesystem is found fromf n = 1√k(20.34)2π mwe can combine Equations (20.33) and (20.34) to obtain the relationship betweenstatic deflection and natural frequency as follows:Example Problem 5f n = 12π√ gδ st(20.35)A large machine weighs 875 N and the static deflection of the springs supportingthe machine is 0.83 cm. Find the undamped natural frequency.SolutionFrom Equation (20.35)√f n = 1 980.7 cm/s 2= 5.47 Hz2π 0.83 cm20.5 Vibration ControlTransmissibilityIn the case of a forcing function being harmonic in nature, two cases of vibrationtransmission can occur. One case occurs when force is transmitted to the supportingstructure, and the opposite case occurs when the motion of the supporting structure

20.5 Vibration Control 599Figure 20.6. A spring-damper system that is subjected to force excitation.is transmitted to the machine. In Figure 20.6, f (t) = F 0 sin ωt represents theharmonic force imparted to the system and f T (t) is the force transmitted throughthe spring/damper system to the supporting structure. The force sent through thespring and damper to the supporting structure is given byf T (t) = kx + CẋThe magnitude of this force is a function of the frequency:F T = √ √( cω[kx(ω)] 2 + [CωX(ω)] 2 = kx(ω) 1 +k) 2(20.36)Inserting Equation (20.24) into Equation (20.36) results inF T =√F 0 1 + ( Cω√ (1 − mω2kk) 2) 2+( Cωk) 2(20.37)Applying the definitions for ω n and ξ, and defining transmissibility T as the ratioof the amplitude of the force transmitted to the supporting structure to that of the

600 20. Vibration and Vibration ControlFigure 20.7. A spring-damper system that is subjected to motion excitation.exciting force, we now have√T = F 1 +(2ξω / ) 2ωTn= √F 0 [ ( ) ](20.38)22 ( ) 2ω1 −ω n+ 2ξ ω ω nMotion ExcitationAs the corollary to the force excitation model of Figure 20.6, the system for motionexcitation is given in Figure 20.7. Variable x represents the motion of the dynamicsystem and variable y represents the harmonic displacement of the supportingbase. The dynamics of this system is characterized by the following equation:mẍ + C(ẋ − ẏ) + k(x − y) = 0 (20.39)The ratio of the magnitudes of the two displacements as a function of frequencydenotes the transmissibility given by√)T = x √ 2y = k2 + (Cω)√ 21 +(2ξ ω(k − mω2 ) 2 + (Cω 2 ) =ω n√ 2 [ ( ) ](20.40)22 ( ) 21 − ωω n+ 2ξ ω ω nThe right-hand side of Equation (20.40), which was expressed in terms of ω nand ξ, is identical to the transmissibility of Equation (20.38). This equality indicatesthat the methodologies employed to protect the supporting structure underforce excitation are also applicable to insulating the dynamic system from motionexcitation.Equations (20.38) and (20.40) are used to plot Figure 20.8 in order to illustratethe interrelation between the damping ratio ξ, the ratio of disturbing frequency to

20.5 Vibration Control 601Figure 20.8. Plot of transmissibility versus frequency ration f/f n as a function of thedamping ratio of a linear system with one degree of freedom.the natural frequency ω/ω n , and transmissibility T . If the ratio ω/ω n < √ 2, thentransmissibility T > 1, which means that the input disturbance is amplified, notdecreased. In the region ω/ω n > √ 2, T decreases with increasing ξ. At resonancecondition ω/ω n , = 1, transmissibility can be quite large. The region ω/ω n > 2isthe only domain where isolation is possible, and there T (which is less than unity)decreases with decreasing ξ, and this indicates that better isolation can be achievedwith very little or no damping.The curves of Figure 20.8 thus demonstrate the effectiveness of an isolator inmitigating vibration. It is also apparent that isolators should be selected to avoidexciting the natural frequencies of the system, and that damping is important inthe range of resonance, when the system is operating near resonance or merelypassing through resonance during startup. From scrutiny of the isolation region it isnoted that the larger the ratio ω/ω n (or the smaller the value of ω n ), the smaller thetransmissibility will be. From the relation of Equation (20.7), ω n can be made quite

602 20. Vibration and Vibration ControlFigure 20.9. Isolation efficiency [(1 − T) × 100] versus natural frequency f n of a linearsingle-degree-of-freedom system with zero damping ratio ξ.small by selecting soft springs, which, in conjunction with light damping, providesgood isolation. The natural frequency can also be reduced by increasing the massbut this also increases the dead weight of the dynamic system, thus imposing agreater load on the supporting structure.A measure of the effectiveness of isolation as a function of frequency can begained by the parameter percent isolation (%I ) which is defined by%I = 100(1 − T )where T is the transmissibility which is ≤ 1. The isolation efficiency is usually plottedwith the disturbing frequency and natural frequencies as shown in Figure 20.9for a damping ratio ξ of zero.Example Problem 6A machine weighs 208.5 N and it is supported on a spring having a spring constantof 935.23 N/cm. A rotating mass within the machine generates a disturbing forceof 51.23 N at 4,800 rpm, owing to imbalance of the rotating mass. Determine theforce transmitted to the mounting base for ξ = 0.12.

Solution20.6 Techniques for Vibration Control 603The static deflection of the spring is first obtained:δ st = W k = 208.5N = 0.223 cm935.23 N/cmwhich, according to Equation (20.35), gives us the natural frequency of the system√f n = 1 980.7 cm/s 2= 10.6 Hz2π 0.223 cmThe disturbing frequency generated by the imbalance is 4800/60 = 80 Hz. Itfollows thatω= f = 80ω n f n 10.6 = 7.58From Equation (20.40) or Figure 20.8 we find√) 21 +(2ξ ω √ω n 1 + (2 × 0.12 × 7.58)2T = √ [ ( ) ]= √22 ( ) [1 ] 2ω− (7.58)1 −ω n+ 2ξ ω 22+ (2 × 0.12 × 7.58)2ω n2.076= √ = 2.0763187.33 + 3.509 56.49 = 0.0367The force transmitted to the mounting structure isF T = (0.0367)(51.23 N) = 1.88 N20.6 Techniques for Vibration ControlModification of SourceIt is often possible to mitigate vibration by modifying the source of the vibration.For example, the vibrating structure may be made more rigid, closer tolerances ofmachining can eliminate or lessen imbalance, or the system mass and stiffness canbe adjusted so that the resonant frequencies of the system do not coincide with theforcing frequency (this procedure is called detuning), or it may be possible to cutdown on the number of coupled resonators between the vibrational source and thecomponent of interest (this procedure is called decoupling).IsolationThere are three principal types of isolators, viz. (1) metal springs, (2) elastomericpads, and (3) resilient pads. Metal springs are generally the best for low-frequencyisolation and they possess the advantage of being impervious to the effects of

604 20. Vibration and Vibration Controltemperature, humidity, corrosion, presence of solvents and they allow maximumdeflections. But springs have almost no damping and hence can lead to very hightransmissibility at resonance. But dampers can be included in parallel with springsto minimize resonance effects.Elastomeric mounts are generally constructed of either natural rubber or syntheticrubber materials such as neoprene. Synthetic rubber is generally far moreimpervious to environmental effects than natural rubber. These types of mounts aregenerally used to isolate relatively small electrical and mechanical devices fromrelatively high-forcing frequencies. Elastomeric compounds inherently containdamping. Rubber can be used in either tension, compression or shear modes, but itis generally used in compression or shear and hardly ever used in tension modes.Isolation or resilient pads include a variety of materials such as cork, felt, fiberglass,and special plastics. These items can be purchased in sheets and cut intogaskets to fit the particular application. Also, they can be stacked to provide differentdegrees of isolation. Cork is available in squares 1 to 2.5 cm thick. Corkcan withstand corrosion and solvents, but felts being made of organic materialscannot be utilized in an environment where solvents are present. Cork and felthave damping ratios that typically range from ξ = 0.05 to 0.06.Inertia BlocksCertain very large machines such as reciprocating compressors generate largeinertia forces that result in unacceptably large motion that can cause the machine tofunction improperly. One means of limiting this motion is to mount the equipmenton an inertia base, which consists of heavy steel or concrete mass. This mass limitsthe motion by the dint of its heaviness overcoming the inertia forces producedby the mounted equipment. Also low natural frequency isolation needs a largedeflection isolator such as a soft spring. But the use of soft springs can lead torocking motions that cannot be tolerated. Hence, an inertia block mounted on theappropriate isolators can serve to effectively limit the motion as well as providethe required isolation. Inertia blocks also help to lower the center of gravity andthus provide an additional degree of stability. The introduction of additional massthrough inertia blocks decreases vibrational amplitudes and minimizes rocking.Alignment errors can be minimized because of the greater stiffness of the base,and moreover, these blocks can serve as a noise barrier between the floor on whichthey are mounted and the mounted equipment itself.Vibration AbsorbersIn order to better understand its operation, let us consider the vibration absorbermodel of Figure 20.10. The components of the vibration absorber are m 2 and k 2 .The applicable equations of motion for the system arem 1 ẍ 1 + k 1 x 1 + k 2 (x 1 − x 2 ) = fm 2 ẍ 2 + k 2 (x 2 − x 1 ) = 0

20.6 Techniques for Vibration Control 605Figure 20.10. The analytical model for a vibration absorber.The frequency response can be gotten from the following transformed equations:Rearranging these last two equations:−m 1 ω 2 X 1 + (k 1 + k 2 )X 1 − k 2 X 2 = F 0−k 2 X 1 + (−m 2 ω 2 + k 2 )X 2 = 0(−m 1 ω 2 + k 1 + k 2 )X 1 − k 2 X 2 = 0 (20.41)−k 2 X 1 + (−m 2 ω 2 + k 2 )X 2 = 0 (20.42)The two simultaneous equations may now be written in the matrix format asfollows:[−m1 ][ ] [ ]ω 2 + k 1 + k 2 −k 2 X1 F0−k 2 −m 2 ω 2 =(20.43)+ k 2 X 2 0Now consider what happens to Equations (20.41)–(20.43) when the forcing frequencyω equals the natural frequency ω 2 = (k 2 /m 2 ) 1/2 of the vibration absorber.This condition leads toX 1 = 0X 2 =− F 0k 2(20.44)

606 20. Vibration and Vibration ControlFrom the Equation set (20.44) it can be deduced that when the natural frequencyof the vibrating absorber is tuned to the vibrational forcing frequency, the motionof the principal mass m 1 is ideally zero and the spring force of the absorber isat all times equal and opposite to the applied force F 0 . It follows that no netforce is transmitted to the supporting structure. Because the natural frequencymust be tuned to the vibration forcing frequency, the absorber is customarily usedfor constant speed machinery. Even though this isolation feature is useful over abroad band of frequencies, absorption is useful principally for very narrow-bandor single-frequency control.Active SystemsActive isolation and absorption systems normally incorporate a feedback control,which may be electromagnetic, electronic, fluidic, pneumatic, mechanical, or acombination thereof. Because the cost of such systems is quite high, they are usedmainly in precision instruments such as electronic microscopes, lasers, stabilizedplatforms, and other devices where a high degree of isolation is required. Activecontrol is also applied where a high degree of isolation is required, for example, inthe case of an air-conditioning system that is mounted on the top floor of a buildingabove work areas and offices. Because the cost of computerization is falling continuously,active control is becoming a more viable option in many applications(including automotive suspensions), particularly for isolating very low-frequencyvibration (

Then the energy dissipated in a cycle is obtained from∫ 2π20.6 Techniques for Vibration Control 607∫ 2πCẋ 2 (t) d(ωt) = CA 2 ω cos 2 (ωt + φ) d(ωt) = πCA 2 ωω 00The energy dissipated per radian is CA 2 ω/2, and the peak potential energy iskA 2 /2. Therefore, the system loss factor isη s = CA2 ω/2kA 2 /2= CωkIntroducing Equations (20.8) and (20.9), Equation (20.45) becomes(20.45)η s = 2ξ ω ω n(20.46)At resonance, η s assumes the value of 2ξ. This indicates that systems withlarge loss factors are highly damped. For small values of damping, applyingEquations (20.18), (20.28), and (20.29):η S = 2ξ = δ π = 1(MF) resonanceIn the analysis of structural damping, complex stiffness and complex moduliare principal parameters. We recall that the steady-state oscillation is described bythe following expression:which can be rewritten aswhere(−ω 2 m + iωC + k)X = F 0(−ω 2 m + k ∗ )X = F 0k ∗ = k + iωCThe imaginary stiffness is the damping. Employing Equation (20.45), we canexpress the complex stiffness ask ∗ = k(1 + iη s )In an analogous manner, the complex Young’s modulus E for damping materialcan be expressed asE ∗ = E(1 + iη M )where E is the real part of the Young’s modulus of the damping material and η Mis the loss factor of the damping material. 1The loss factors and complex moduli vary with frequency and temperature. It hasbeen assumed that the damping force is proportional to velocity and independentof amplitude, but there are situations where the damping of the materials depends1 η s is the loss factor of the system structure and damping material.

608 20. Vibration and Vibration ControlFigure 20.11. Sound transmission characteristics: (a) incident wave resolving into a reflectedwave and a transmitted wave and (b) transmission loss in a structure as a functionof frequency.mainly on vibration amplitude, so the assumption of viscous damping must beapplied judiciously.Damping converts mechanical energy into thermal energy; and while there area number of mechanisms for such energy conversion, we describe here thosethat are the most useful ones. Damping materials also reduce sound transmission.When a sound wave strikes a structure, causing its surface to vibrate, the vibratingsurface produces a reflected wave and a transmitted wave [Figure 20.11(a)].The transmission loss through the structure varies with frequency as shown inFigure 20.11(b) for a given temperature. The region of damping control nestlesbetween the low-frequency region where stiffness reigns as the controlling parameterand the higher-frequency region where mass predominates as the controllingparameter. Between these two regions, many natural vibratory modes of the structureexist, and this region is the only one where transmission loss depends greatlyupon resonance conditions. Here structural damping is the controlling parameter.Internal DampingA material with very high-damping internal properties could be utilized to eliminatenoise emanating from a structure. Ferromagnetic materials and certain magnesium

20.6 Techniques for Vibration Control 609and cobalt alloys exhibit such properties but these materials are generally too costlyto use as structural materials and they may not meet the strength criteria.Damping MechanismsFigure 20.12 shows three types of damping mechanisms, namely, the tuned damper,free viscoelastic layer, and constrained viscoelastic layer.The tuned damper of Figure 20.12(a) consists of a mass attached to a pointof vibration through a spring and dashpot or a viscoelastic spring. This device isnot very useful because it functions well at only a single frequency or in a verynarrow frequency band. Moreover, any change in temperature is likely to changethe tuning frequency of the damper.The other two mechanisms of Figure 20.12 entail adding viscoelastic layers toa structure that is to be damped. When a structure consisting of different layerof materials undergoes bending, the layers will extend or deform in shear. Theresultant deformation causes energy dissipation, a phenomenon that constitutesthe basis for the two viscoelastic mechanisms.In the mechanism of Figure 20.12(b), a layer of free (i.e., uncovered) viscoelasticlayer is bonded to the main structure. The damping material thickness should beFigure 20.12. Different types of damping mechanisms: (a) tuned damper, (b) free viscoelasticlayer, and (c) constrained viscoelastic layer.

610 20. Vibration and Vibration Controlabout one-third the thickness of the structure or wall thickness. One thick layer onone side of the structure is generally more effective than two lesser layers on eitherside of the structure. This technique, which entails extensional damping, results inan economical, highly damped structure that is easy to fabricate.The mechanism of Figure 20.12(c) is essentially a sandwich arrangement, wherea viscoelastic layer is added to the structure. The covered (hence, constrained) viscoelasticlayer provides high extensional damping, but the entire structure becomesharder and more expensive to assemble.In order to be effective, the damping action must store a major portion of theenergy present in the entire system. Damping is best applied to points wherestretching or bending is the maximum, because these are the locations of maximalenergy storage.Viscoelastic layer techniques can be employed in a large variety of applicationsranging from walls, enclosures, barriers, conveyers, chutes, racks, and hoppers tothe most specialized, technologically sophisticated electronic instruments.20.7 Finite Element AnalysisWe have used relatively simple mathematical models to deal with vibration, butthe analysis of plates, shells, and other continuous systems can be difficult toanalyze without the aid of computers. Experimental modal analysis can providethe needed information, but this requires that a real structure be constructed andinstrumented to yield the desired data. Finite-element analysis (FEA), however,allows the problem to be represented with some detail and permits the designer tooptimize a design by investigating the effects of minor changes in the model uponthe static and dynamic states of the structure being evaluated. FEA is also used notonly to determine the statics and dynamics of beams, plates, shells, trusses, andother solid bodies but also to treat problems involving fluids, including airbornenoise propagation. A number of FEA programs can predict stresses and strains,temperature distribution stemming from heat and mass transfer as well as vibratorystates. Transient states are amenable to treatment by FEA. Since FEA is apowerful analytical tool, it no longer became necessary to fabricate a series ofactual structures before freezing a design. The most prominent programs containinggeneral codes for vibrational analysis include NASTRAN, developed by theU.S. National Aeronautics and Space Administration and now in public domain;MSC-NASTRAN r○ , a proprietary code developed from NASTRAN coding availablefrom MacNeal-Schwindler Corp.; and ANSYS r○ , another proprietary codethat is available from Swanson Analysis System, Inc.A structure undergoing analysis is modeled by subdividing it into various typesof finite elements from the finite-element library. The library may include more than100 element types, including beam elements, triangular and rectangular plate andshell elements, conical shell elements, and mass, damping, and stiffness elements.The preprocessing stage in FEA entails the creation of a finite-element mesh todepict the structure being evaluated. The elements may be automatically sized

20.8 Vibration Measurements 611by proportional spacing so that the mesh is denser in areas of greater concern(particularly where sudden changes or discontinuities in geometry can lead tohigher values of stresses or steeper temperature gradients). Constraints and forcingfunctions are specified. The input data are converted into matrix format by the FEAcode. The form of matrix equation iswherema + Cv + kx = Fm, C, and k = the mass, damping, and stiffness matrices, respectively, forthe finite element representation of the structure being analyzeda, v, and x = the acceleration, velocity, and displacement vectors, respectivelyF = the forcing function vectorThe matrices are manipulated to determine natural frequencies, frequencies,mode shapes, response amplitudes, and/or dynamic stresses due to harmonic, random,or transient forcing functions. In the postprocessing phase, the FEA programarranges the output in a convenient format for engineering evaluation. The deformedstructure can be graphically portrayed as an overlay over the undeformedmesh diagram. In order to show the difference, mode shapes or displacement amplitudesare exaggerated in the display. In almost all situations, a great deal ofengineering time must be spent in defining even a simple problem by modelingit and setting up a mesh. But once a solution is obtained, redesign to improvevibrational characteristics can be done quickly and effectively by modifying thestored data. FEA software can be installed and used on personal computers, but formore elaborate designs it may be necessary to use minicomputers or mainframes inorder to deal with much larger matrices (cf. Brooks, 1986; Hughes, 1987; Grandin,1986; Huebner and Thornton, 1982).20.8 Vibration MeasurementsVibration displacement and vibrational velocity can be measured but the mostcommon measurements of vibration are those of acceleration. The basic transducerused to measure vibration and shock is the accelerometer. Most of the othercomponents in vibration measurement systems are similar to those used to measureairborne sound (cf. Chapter 9). Many instruments such as the fast Fouriertransform (FFT) analyzers are designed to be used for both acoustic and vibrationapplications.AccelerometersAccelerometers are usually mounted directly on a vibrating body, using a threadedstud, adhesive, or wax. Other versions mount the accelerometer in a probe thatis held against a vibrating body. A typical accelerometer houses one or more

612 20. Vibration and Vibration Controlpiezoelectric elements against which rest a mass that is preloaded by a stiff spring.When an accelerometer becomes subject to acceleration imparted by the vibrationbeing measured, the mass exerts a force on the piezoelectric element that is proportionalto the acceleration. The charge developed in the piezoelectric elementis, in turn, proportional to the force. The accelerometer may be designed so thatthe piezoelectric element is stressed in compression or in shear. The piezoelectricelements are usually quartz crystals or specially processed ceramic materials.There is more than one type of sensitivity of interest in accelerometer specifications.The charge sensitivity, measured in picocoulombs/g, and voltage sensitivity,measured in terms of mV/g, are important, depending whether the accelerometer isused with charge measuring or voltage measuring equipment. Also, the transversesensitivity, which is the sensitivity to acceleration in a plane normal to the axis tothe principal accelerometer axis should be a low value, preferably less than 3% ofthe main axis sensitivity at low frequencies.The preamplifier, which constitutes the second stage in signal processing, servestwo purposes: one is to amplify the vibration pickup signal that is generally quiteweak and the other purpose is to serve as an impedance transformer between theaccelerometer and the subsequent chain of equipment. A preamplifier may be designedto function as a voltage amplifier in which case the output voltage is directlyproportional to the input voltage or it may function as a charge amplifier in whichcase the output voltage is proportional to the input charge. Each type of amplifierhas its own advantages and disadvantages. When a charge amplifier is used,changes in cable length (which modifies the cable capacitance) have negligibleeffects on the measurements. When a voltage amplifier is employed, the systemwill be extremely sensitive to changes in cable capacitance. Because the input resistanceof a voltage amplifier cannot be disregarded, the extremely low-frequencyresponse of the system may be affected. But voltage amplifiers are usually lessexpensive and may be more reliable because they contain fewer components.20.9 Random VibrationsMost of the vibration problems discussed earlier in this chapter are deterministic,i.e., the forcing function can be described as a function of time. An example ofa deterministic problem is the imbalance of a shaft operating at a known speed.Random vibrations, on the other hand, result from excitation that can be onlydescribed statistically. Structural excitation of an aircraft fuselage due to jet enginenoise or turbulent flow is considered to be a random process. Both frequency andamplitude vary and they do not establish a deterministic pattern.Probability DensityBecause the amplitude or acceleration of random vibration cannot be determinedas a function of time, it is described in terms of its probability density. Onemodel widely used is the Gaussian distribution or normal probability density curve

20.9 Random Vibrations 613expressed as follows in normalized form:p(x) = 1σ √ /σ 2 (20.47)2π e−0.5x2and the probability of the value of x falling between a and b iswhereP(a < x < b) =∫ bap(x) dx (20.48)p(x) = the probability density of the functionx = the amount the function differs from the meanσ = the standard deviationP = the probability of x falling within a particular rangeIt is apparent that the total area under the probability density curve must be unity:P(−∞ < x < ∞) =∫ +∞−∞p(x) dx = 1Root-Mean-Square Value and AutocorrelationThe time-average root-mean-square of a function is defined by√( ∫ 1 T)x rms = lim xT →∞ T2 (t) dt0(20.49)wherex rms = the root-mean-square of function xT = the time intervalThe temporal autocorrelation function describes, on the average, the way in whichthe instantaneous value of a function depends on previous values. It is given by( ∫ 1 T)R(τ) = lim x(t)x(t + τ) dt(20.50)T →∞ T 0whereR(τ) = autocorrelation functionτ = the time interval between measurementst = time

614 20. Vibration and Vibration ControlErgodic ProcessesA function may vary in such a manner that there is no narrow time interval that canbe truly representative of the function. However, if the time interval is sufficientlylong and the probability distribution functions are independent of the time intervalduring which they were measured, the function then represents a stationary process.This means that the root-mean-square value measured during one intervalshould be equal to the root-mean-square value measured during a later interval.The autocorrelation function will also be unaffected by a time shift. If root-meansquarevalues and the autocorrelation functions of a number of ensembles of dataare equal to the temporal values, then the process may be considered ergodic aswell as stationary.Spectral DensityThe power spectral density, also known as the root-mean-square spectral density,can be determined from the autocorrelation function as follows:S(ω) =∫ +∞−∞ R(τ)e−iωτ dτ2π(20.51)whereS(ω) = root-mean-square spectral density, in g 2 /(rad/s) or (m/s 2 ) 2 /(rad/s)ω = radial frequency, rad/sEquation (20.51) is used in analytical studies. The inverse relationship isR(τ) =∫ +∞−∞S(ω)e iωτ dωVibration measurements obtained from an FFT analyzer or another spectralinstrument may be expressed in dB re 1gordBre1m/s 2 , within each frequencyband. These values may be converted to root-mean-square spectral density W ( f )(usually expressed in g 2 /Hz or some other engineering units), where W ( f ) andS(ω) are related bywhereW ( f ) = 4π S(ω)W ( f ) = spectral density, units 2 /Hz), defined for positive frequencies onlyf = frequency, HzS(ω) = spectral density, units 2 /(rad/s), defined for both positive andnegative frequencies (a mathematical artifice)

References 615If the vibration measurements are sufficiently representative and the process isstationary, the root-mean-square value is then given by( ∫ 1 T) ∫ ∞xrms 2 = lim x 2 (t) dt = W ( f ) df = R(0)T →∞ T 00wherex 2 rms= root-mean-square value, g2R(0) = the autocorrelation function for τ = 0White NoiseWhite noise is a random signal which has a constant root-mean-square spectraldensity for all frequencies from zero to infinity, i.e.,W white ( f ) = W 0for 0 < f < ∞This idealization cannot be achieved physically, since it would amount to requiringan infinite amount of power. More realistically, band-limited white noise can beachieved and it is a random signal having a constant spectral density over a specifiedrange:W white ( f ) = W 0 f 1 < f < f 2White noise generators are produced to generate signals with random vibrationin amplitude and frequency, with relatively constant spectral density over variousfrequency ranges.ReferencesBeranek, Leo L. (ed.). 1971. Noise and Vibration Control. New York: McGraw-Hill.Broch, J. T. June 1973. Mechanical and Shock Measurements. Nærum, Denmark: Brüel &Kjær Instruments Company.Brooks, P. 1986. Solving vibration problems on a PC. Sound and Vibration 20(11):26–32.Crandall, S. H. and Mark, W. D. 1963. Random Vibration in Mechanical Systems.New York: Academic Press.Crede, C. E. 1965. Shock and Vibration Concepts in Engineering Design. EnglewoodCliffs, NJ: Prentice Hall.Grandin, H. 1986. Fundamentals of the Finite Element Method. New York: Macmillan.Harris, Cyril M. (ed.). 1991. Handbook of Acoustical Measurements and Noise Control,3rd ed. New York: McGraw-Hill, Chapters 6–10.Harris, C. M. (ed.). 1988. Shock and Vibration Handbook, 3rd ed. New York: McGraw-Hill.Huebner, K. H. and Thornton, E. A. 1982. The Finite Element Method for Engineers, 2nded. New York: Wiley-Interscience.Hughes, T. J. R. 1987. The Finite Element Method—Linear Static and Dynamic FiniteElement Analysis. Englewood Cliffs, NJ: Prentice Hall.

616 20. Vibration and Vibration ControlProblems for Chapter 201. In the spring-dashpot system of Figure 20.1, weight W = 50 N, spring constantk = 0.30 N/cm, and damping ratio ξ = 0.35. Find the natural frequency andthe viscous damping of the system.2. Establish the characteristic equation for the spring-dashpot system ofProblem 1 and determine the additional viscous damping needed to yielda critically damped system.3. In the system of Problem 1, the mass is displaced 15 cm and then suddenlyreleased. Set up the equation for the system response, which describes theposition of the mass as a function of time.4. Consider a spring-dashpot system that has the following parameters: W =40 N, C = 0, and k = 0.39 N/cm. If the mass is initially displaced and released,how will the system response?5. In a spring-dashpot system, W = 40 N, C = 0.10 N/cm/s, and a forcing function4.0 sin ω n t drives the system. What kind of system response will occuras a function of time?6. In a spring-damper system of Figure 20.1, W = 250 N, C = 0.13 N/cm/s,and k = 32 N/cm. How much does the spring deflect under the dead weightload of the mass? What is the natural frequency of the system? If the systemundergoes a forced harmonic oscillation described by F = F 0 sin 12t, whichone of the system parameters effectively controls the response of the system?7. A mass is supported by four identical springs, each having a spring constantof 2.6 N/cm. If the spring deflects 1.5 cm, what is the weight of the mass beingsupported?8. Consider the system described in the last problem. It is forced with a sinusoidalsignal with a frequency twice that of its natural frequency. Find the degree ofvibration isolation that will be obtained.9. A mechanical system is being designed for installation in a plant. It bearsthe following parameters: W = 2000 N and k = 3.65 N/cm. If the forcingfrequency = 12 rad/s, how much damping can the system tolerate if the isolationmust exceed 90%?10. In a spacecreaft, delicate electronic sensors have to be isolated from a panelthat vibrates at 50 rad/s. It is required that at least 90% vibration isolationcan be achieved by using springs to protect the equipment. Assume that thedamping ratio ξ = 0. What static deflection is required?

21Nonlinear Acoustics21.1 IntroductionIn most of the preceding chapters we have dealt with acoustics in terms of the linearwave equation. The amplitude of the sound was considered to be virtually infinitesimal,thus paving the path to relatively convenient mathematical analyses. When thewave amplitude becomes sufficiently large, nonlinear effects occur and the linearwave equation no longer meets the situation. Waveform deformation occurs withpossible formation of shock waves, increased absorption, nonlinear interaction (asopposed to superposition) between combined sound waves, amplitude-dependentdirectivity of rays, onset of cavitation, and sonoluminescence.The realm of nonlinear acoustics produced by intense sound levels encompassesa variety of practical cases: mufflers for internal combustion engines, thermoacousticheat engines, shock waves from supersonic aircraft and spacecraft, underwatersonar, formation of bubbles, cavitation, acoustic compression and energy lossesattributable to viscous and thermal boundary layers, sonoluminescence, and sonochemistry.Velocity dispersion also affects the resonance frequencies of rigid wallcavities and the harmonic spectrum of standing waves.Even in cases of relatively small signals, matters are not always so linear asthey appear. Executing a perturbation procedure on the wave equation to considersecond- and third-order effects may bring out rheological characteristics (i.e., thenon-Newtonian behavior of fluid viscosity) of a fluid medium that can be indicativeof its physical and chemical state (Raichel and Kapfer, 1973; Takabayashi andRaichel, 1998).Obviously not all aspects of nonlinear acoustics can be covered in a singlechapter, but in the following sections we will attempt to highlight some of themajor facets of the finite-amplitude category of acoustics.21.2 Wave DistortionsConsider a plane wave propagating in the x-direction in a frictionless fluid.If the amplitude is sufficiently small, the wave phenomenon can be described617

618 21. Nonlinear Acousticsby∇ 2 φ = 1 c 2 0where φ is the velocity potential, t is the time, and c 0 is the small signal soundspeed listed in Tables A, B, and C of Appendix A. In linear acoustics a sound waveordinarily propagates through a medium without changing its shape, because eachpart on the wave travels with the same speed c 0 = dx/dt.In the case of the finite wave, the propagation speed varies from point to point.The variation occurs because a propagating wave engenders a longitudinal velocityfield u in the fluid medium through which it travels. The motion of the fluid addsto the propagation speed with respect to a fixed location:dx= c + u (21.1)dtwhere c represents the speed of sound with respect to the moving fluid (and it isnot the same as c 0 ). To understand this situation better, consider a fluid that is a gasfor which the speed varies as √ T [cf. Equation (2.2)]. The sound speed becomesa bit higher when the acoustic pressure p is positive (i.e., during the compressionphase that increases with the temperature) and a bit lower when p is negative (i.e.,during the expansion phase that lowers the temperature). Thenc = c 0 + γ − 1 u (21.2)2where γ is the ratio of specific heats of the gas. Combining Equations (21.1) and(21.2) results indx= c 0 + βu (21.3)dtwhere β, the coefficient of nonlinearity is given byβ = γ + 12From Equation (21.3) we observe that the propagation speed depends on the particlevelocity, as the result of the convective effect of the moving fluid and thenonlinearity of the traveling wave. Even in nonlinear acoustics, the particle velocityu is normally much smaller than c 0 . The impact of the varying propagationspeed is accumulative and leads to appreciable distortion that becomes even greaterwith increasingly stronger waves.Nonlinearity of the pressure–density relation is considerably more prominentin liquids and solids than for gases. In the case of liquids the coefficient of thefirst-order nonlinear term in the pressure–density relation is B/2A, where B/A istermed the parameter of nonlinearity. In this case, the analog of Equation (21.1)for liquids is given by∂ 2 φ∂t 2β = 1 + B2A

21.3 Progressive Waves in Fluids 619u(x,t)t = 0 t = t 1t = t 2xc 0 t 1 c 0 (t 2 -t 1 )u(x,t)(a)x = 0 x = x 1 x = x 2 x = x 3t(b)Figure 21.1. Cumulative distortion as seen from the viewpoints of (a) initial-value problem,u as a function of x with increasing time t, and (b) signaling problem, u as a functionof x with increasing x, from the signaling location x = 0.Nonlinear acoustics covers a considerable variety of topics. Problems arisingfrom nonlinearity fall into two categories: (1) source problems, where the timevariation of an acoustic field variable (e.g., pressure or particle velocity) is specifiedat a single location and (2) initial value problems where the spatial variationof the field is specified for a specific time. Figure 21.1 illustrates the cumulativedistortion by linear convection as seen from the viewpoints of (a) initialvalue problems, with u as a function of x at increasing times t, and (b) signalingproblems, where u is a function of t at increasing ranges x, from the signalinglocation.21.3 Progressive Waves in FluidsWhen a single wave moves through a fluid, we have a case of a progressive wavefield. The governing second-order wave equation∂ 2 u∂x = 1 ∂ 2 u2 c02 (21.4)∂t 2

620 21. Nonlinear Acousticsmay be integrated at once to yield∂u∂x = 1 ∂u(21.5)c 0 ∂twhich is a first-order differential equation whose solution is u = f (t − x/c). 1Of the two types of problem approaches mentioned in the preceding section,most of the acoustic problems entail radiation, so source problems receive moreemphasis in this chapter.Plane WavesThe model equation for a source-generated finite plane wave in a lossless fluid is∂u∂x = β c02 u ∂u(21.6)∂tConsider a sinusoidal source excitation governed by u = U 0 sin ωt at x = 0. Thesolution to Equation (21.6) is the Fubini solution∑ 2u = U 0nnσ J n(nσ ) sin nωt (21.7)whereσ = βεκx = x¯x or ¯x = 1βεκxHere ¯x is the shock formation distance, ε = U 0 /c 0 , and κ = ω/c 0 is the wavenumber. The size of σ , the so-called dimensionless distortion range, indicates ameasure of the amount of distortion that has occurred. The value of σ = 1 indicatesshock formation.Other One-Dimensional Waves and Ray TheoryIn one-dimensional progressive waves that are not planar, geometric spreadingslows down the rate of distortion and is described mathematically by an extra termin the model equation (21.6). If the wave is spherical or cylindrical, model equation(21.6) becomes∂u∂r + a r = β c02 u ∂u(21.8)r1 We can simplify Equation (21.5) by transforming coordinates x, t to x, τ, where τ = x − c 0 /t, theretarded time. Then Equation (21.5) becomes∂u∂x = 0which provides the building block on which most model equations for more complex progressivewaves are based.

21.3 Progressive Waves in Fluids 621where r represents the radial coordinate. The retarded time is nowτ = t − (r − r 0)c 0where r 0 is a reference distance, which could be the radius of the source, and ahas a value of unity for spherical waves, 1 / 2 for cylindrical waves, and 0 for planewaves (in which case r is replaced by x). Introducing the coordinate stretchingfunctionz = r 0 ln r r 0(spherical waves)and the spreading compensation function= 2( √ rr 0 − r 0 ) (cylindrical waves)w =( rr 0) au (21.9)into Equation (21.8) results in this equation being reduced to the plane wave form∂w∂z = β c02 w ∂w∂τThus, plane wave solutions may be extended to spherical and cylindrical wavesby replacing u and xwith w and z, respectively. The Fubini solution for sphericalwaves has the same form as that of Equation (21.7).One-dimensional propagation is applicable to ducts of slowly varying crosssection, such as horns or ray tubes. In such cases, the spreading compensationfunction is√w =AA 0uDissipative Function: Burgers EquationEquation (21.8) can be useful for a good variety of finite-amplitude problems, butthe distortion nearly always leads to the formation of shocks that are naturallydissipative. The losses must now be taken into account. The first truly successfulmodel was developed by Burgers (1948) originally to model turbulence, but italso turned out to be an excellent approximation of the equation describing finiteamplitudeplane waves of traveling in a thermoviscous fluid. In the form usablefor source problems, the Burgers equation is∂u∂x − β c02 u ∂u∂r =δ2c 3 0∂ 2 u∂t 2 (21.10)

622 21. Nonlinear Acousticswhereδ = υ[ν + (γ − 1) Pr] = diffusivity of soundυ = μ/ρ = kinematic viscosityμ = shear viscosity coefficientρ 0 = static densityν = 4 3 + μ B= viscosity numberμμ B = bulk viscosity numberPr = c pμ= Prandtl number, a property of the fluidk rc p = specific heat at constant pressure of the fluidk r = thermal conductivity of the fluidOn comparing Equations (21.8) and (21.10) we observe that the former equationdoes not have a dissipation term and it is often called a lossless Burgers equation.The Burgers equation is exactly integrable (Cole and Hopf), which renders ita very useful model. Applications of the Burgers equation have been made tosinusoidal source excitations. Although it is exact, the extraction of the solutionis quite complicated (Hopf, 1950; Blackstock, 1964). For distances larger than3 ¯x(σ >3), it reduces to the Fay solution (Fay, 1931)2 ∑u = u 0Ɣsin nωτsinh n(1 + σ )Ɣ(21.11)where Ɣ = βεκ/α = 1/(α ¯x) that characterizes the importance of nonlinear distortionto the absorption process and α = δω 2 /2c0 3 represents the small signalabsorption at source frequency.In addition to one-dimensional sound waves in thermoviscous fluids, the Burgersequation can also be applied to spherical and cylindrical waves, with the modelequation (21.8) now including the right-hand side of Equation (21.10), i.e.,∂u∂r + u r − βu ∂uc02 ∂r = δ ∂ 2 u2c02 (21.12)∂t 2But Equation (21.12) is not exactly integrable (i.e., no exact solution is known)and numerical methods of solution would therefore be required.Shock WavesSounds from impulsive sources strong enough to result in shock waves includeblast waves from explosions, thunder from lightning, aircraft sonic booms, ballisticmissiles, and N-waves from spark sources.The basic equations describing shock-wave propagation incorporate conservationof mass, Newton’s second law, conservation of energy, and an equation of

21.3 Progressive Waves in Fluids 623state relating pressure to density. These equations are∂ρ+∇·(ρu) = 0 (21.13)∂t( )∂uρ∂t + u ·∇u =−∇p (21.14)ρ ∂ ( )1 1∂t 2 u2 + e)+ u ·∇(2 u2 + e = 0 (21.15)andp = p(ρ,T ) (21.16)where ρ is the fluid density, u is the particle velocity, T is the temperature, p isthe pressure, and e is the internal energy. We have noted in Chapter 2 how theseequations were linearized to derive the basic acoustic equation. When Equations(21.13)–(21.16) are solved for a frictionless fluid, the solutions yield multivaluedwaveforms. But with increasing amplitude, attenuation mechanisms, such as viscosityand heat conduction, assume increasingly greater roles, especially in thecase of large pressure gradients which leads to small but finite times rise of thepressure waveform.For sufficiently strong waves, shock waves or flow discontinuities occur, andEquations (21.13)–(21.15) no longer apply. The governing equations across theshocks are the Rankine-Hugoniot equations:[ρ(u − u sh )] + = [ρ(u − u sh )] − (21.17)[ρu(u − u sh ) + p] + = [ρu(u − u sh ) + p] − (21.18)[ ( )] [ ( )]1 1p2 u2 + e (u − u sh ) + pu = p+ 2 u2 + e (u − u sh ) + pu−(21.19)Here e represents the internal energy per unit mass. The subscript “−” denotesthe variable just behind the shock, and the subscript “+” denotes the variableimmediately in front of the shock. Equations (21.17), (21.18), and (21.19) expressthe conservation of mass, Newton’s second law, and work–energy conservation,respectively.In many situations, we can consider the air or gas surrounding an explosion orenergy release to be a polytropic gas with a constant ratio of specific heats. Theshock relations for a polytropic gas are expressed in terms of shock strength zdefined byz = (p − − p + )(21.20)p +and the Mach number M of the shock relative to the flow ahead:M = (u sh − u + )c +

624 21. Nonlinear AcousticsThe shock relations are as follows:M = u − − u +c +=c −c +=ρ −ρ +=√z√ { ( (γ + 1)γ 1 +2γ1 + γ + 12γz1 + γ − 1 z2γ( ) γ − 11 +2γ[( ) ] γ + 11 + z2γ{1 + z}z) }zThe simplest model of an explosion is that of a point blast and the exact solutionof this problem has been developed by von Neumann (1963), Taylor (1950), andSedov (1961). The explosion is depicted as an energy release E concentrated at apoint in space with ambient density ρ 0 . The relations between the position of theshock r(t) and overpressure in terms of E, ρ 0 , and r arer(t) = k( Eρ 0) 1/5t 2/5 (21.21)or equivalently,p = 8 k 2 ( )ρ 0 E 2/5t −6/525 γ + 1 ρ 0p = 8 25k 5γ + 1 Er−3 (21.22)The decay of a strong spherical shock with r −3 constitutes a general behavior ofthe strong shock region. The dimensionless constant k is established by requiringconservation of energy and is a function only of the ratio γ of specific heats. Thisnecessitates that∫ r(t)( )pE =0 γ − 1 + ρu2 4πr 2 dr (21.23)2remains constant.Equation (21.23) has been evaluated numerically by Taylor and analyticallyby Sedov and von Neumann. For γ = 1.4 at standard atmosphere, the pressureequation (21.22) becomesp = 0.155Er 3

21.3 Progressive Waves in Fluids 625Analytical studies have been made of cylindrical explosions and line sourceenergy releases (Lin, 1954; Plooster, 1970, 1971). Lin was working on the problemof shock generation by meteors or missiles, whereas Plooster was studying thedevelopment and decay of cylindrical shocks with more realistic initial energydistributions and atmospheres. A strong shock estimate of the decay of cylindricalwaves from an instantaneous line source in air (γ = 1.4) at standard pressure isgiven byp = 0.216Er 2Thus, a cylindrical decay proportional to r –2 is characteristic of cylindrical strongshocks.Nonlinearity in SolidsThe propagation of ultrasonic waves of finite amplitudes in a crystal of cubicsymmetry can be analyzed with a nonlinear differential equation similar to thatused for fluids.The propagation of a finite-amplitude ultrasonic wave in any direction in anisotropic solid is given by∂ 2 ξρ 0∂t = κ ∂ 2 ξ2 2∂a + (3κ 2 2 + k 3 ) ∂2 ξ ∂ξ(21.24)∂a 2 ∂awhere ξ is the particle displacement and a is the distance measured in the directionof propagation. κ 2 and κ 3 represent elastic constants that depend on the directionof propagation in the solid.The solution of the nonlinear equation (21.24) is made by assuming that the waveis initially sinusoidal at a = 0, with ξ = A 1 sin (ka – ωt). On this assumption wecan obtain a perturbation solution in the formξ = A 1 sin(ka − ωt) + β A2 2 k2 acos 2(ka − ωt) +···. (21.25)4where β is the nonlinear parameter. The negative ratio of the coefficient of thenonlinear term in Equation (21.24)2β =−(3 + κ 3k 2)is the quantity to be determined from measurement. From 2β we can establish κ 3since κ 2 = ρc0 2 is known.Equation (21.25) indicates that an initially sinusoidal ultrasonic wave generatesa second harmonic as it propagates. In order to measure the nonlinear parameter2β, the absolute value of the fundamental displacement amplitude A 1 needs to bemeasured and also that of the second harmonicA 2 = β A2 1 k2 a4

626 21. Nonlinear AcousticsBut the propagation constant k = 2 π/λ = ω/c0 2 is known and sample length a canbe measured. We can then write2β = 8A 2c 2 0A 2 1 ω2 aand determine 2β directly. A 2 is usually plotted as a function of A 2 1, and the slopeis then evaluated to yield 2β.ReferencesBeyer, Robert T. 1974. Nonlinear Acoustics. Washington, DC: Navy Sea Systems Command.(Reissued in 1997 by the Acoustical Society of America, with an appendix containingreferences to new developments.)Beyer, Robert T. (ed.). 1984. Nonlinear Acoustics in Fluids. New York, NY: Van Nostrand.(A collection of fundamental papers on nonlinear acoustics.)Blackstock, David T. 1964. On plane, spherical, and cylindrical sound waves of finiteamplitude in lossless fluids. Journal of the Acoustical Society of America 36:217–219.Blackstock, David T. 1985. Generalized Burgers equation for plane waves. Journal of theAcoustical Society of America 77:2050–2053.Blackstock, David T. 1972. Nonlinear acoustics (theoretical). In: Gray, D. E. (ed.).American Institute of Physics Handbook. New York, NY: McGraw-Hill, pp. 3-183–3-205.Burgers, J. M. 1948. A mathematical model illustrating the theory of turbulence. In: VonMises, R. and von Kàrmàn, T. (eds.). Advances in Applied Mechanics, Vol. I. New York,NY: Academic, pp. 171–191.Crocker, Malcolm F. (ed.). 1998. Part II: Nonlinear Acoustics and Cavitation. Handbookof Acoustics, pp. 175–253.Fay, R. D. 1931. Plane sound waves of finite amplitude. Journal of the Acoustical Societyof America 3:222–241.Fubini, E. 1935. Anomalies in the propagation of acoustic waves of great amplitude. AltaFrequenza 4:530–581 (in Italian).Hamilton, Mark F. and Blackstock, David T. 1988. On the coefficient of nonlinearity ofnonlinear acoustics. Journal of the Acoustical Society of America 83:1555–1561.Hopf, E. 1950. The partial differential equation u t + uu x = μu xx . Communications in Pureand Applied Mathematics 3:201–230.Landau, Lev D. 1945. On shock waves at large distances from the place of their origin.U.S.S.R. Journal of Physics 9:496–503.Lighthill, James. 1972. The propagation of sound through moving fluids. Journal of Soundand Vibration 24:471–492.Lighthill, James. 1978. Waves in Fluids. New York, NY: Cambridge University Press.(Classic text by a former holder of the Lucasian chair at Cambridge University oncehold by Isaac Newton and now occupied by Stephen Hawking.)Lin, S.-C. 1954. Cylindrical shock waves produced by instantaneous energy release.Journal of Applied Physics 25:54–57.Plooster, M. N. 1970. Shock waves from line sources. Numerical Solutions and experimentalmeasurements. Physics of Fluids 13:2665–2675.

Problems for Chapter 21 627Plooster, M. N. 1971. Numerical simulation of spark discharges in air. Physics of Fluids14:2111–2123.Raichel, Daniel R. and Kapfer, William H. 1973. Sound propagation in non-Newtonianfluids. Journal of Applied Mechanics 40(Series E, 1):1–6.Sedov, L. I. 1961. Similarity and Dimensional Methods of Mechanics. New York, NY:Academic.Takabayashi, Kenneth and Raichel, Daniel R. 1998. Discernment by acoustic means ofnon-Newtonian behavior in liquids. Rheologica Acta 37(6):593–600.Taylor, G. I. 1950. The formation of a blast wave by a very intense explosion: I. Theoreticaldiscussion. Proceedings of the Royal Society 201:159–174.Von Neumann, John. 1963. The point source solution. In: Taub, A. H. (ed.). The CollectedWorks of John von Neumann, Vol VI. New York, NY: Pergamon, pp. 219–237.Whitham, G. B. 1974. Linear and Nonlinear Waves. Wiley Interscience.Problems for Chapter 211. What is the change in timing for a sound traveling at c 0 = 1100 m/s from aspherical source of radius 1mat25cmawayfrom the source center due tononlinearity?2. What is the coordinate stretching function z for (1) a spherical wave and (2) acylindrical wave?3. What is the spreading compensation function for a divergent nozzle in terms ofpercentage of particle velocity u for a throat diameter of 2.5 cm at a locationwhere the diameter is 5.6 cm?4. Why is it necessary to resort to Rankine-Hugoniot relations instead of thecustomary fluid mechanics equations in flows entailing shock waves?

Appendix APhysical Properties of MatterA.1. SolidsYoung’s Shear CharacteristicDensity, ρ Modulus, E Modulus, Poisson’s Speed, c Impedance, ρ 0 cSolid (kg/m 3 ) (GPa) G (GPa) Ratio, ν (m/s) (10 6 Pa·s/m)Aluminum 2700 71 24 0.33 5150 13.9Brass 8500 104 38 0.37 3500 29.8Copper 8900 122 44 0.35 3700 33.0Iron (cast) 7700 105 44 0.28 3700 28.5Lead 11,300 16.5 5.5 0.44 1200 13.6Nickel 8800 210 80 0.31 4900 43.0Silver 11,300 78 28 0.37 2700 28.4Steel 7700 195 83 0.28 5050 39.0Glass (Pyrex) 2300 62 25 0.24 5200 12.0Quartz (X-cut) 2650 79 39 0.33 5450 14.5Lucite 1200 4 1.4 0.4 1800 2.15Concrete 2600 — — — — —Ice 920 — — — — —Cork 240 — — — — —Wood, oak 720 — — — — —Wood, pine 450 — 1 0.4 1450 1.6Hard rubber 1100 2.3 — 0.5 70 0.065Soft rubber 950 0.005 — — — —Rubber, ρ–c 1000 — — — — —629

630 Appendix A. Physical Properties of MatterA.2. LiquidsBulk Ratio of Characteristic Coefficient ofTemperature Density, ρ0 Modulus, B Specific Speed, Impedance, ρ0/c Shear Viscosity, ηLiquid ( ◦ C) (kg/m 3 ) (GPa) Heats, γ (m/s) (10 6 Pa·s/m) (Pa·s)Alcohol (ethyl) 20 790 — — 1150 0.91 0.0012Castor oil 20 950 — — 1540 1.45 0.96Glycerine 20 1260 — — 1980 2.5 1.2Mercury 20 13600 25.3 1.13 1450 19.7 0.0016Seawater 13 1026 2.28 1.01 1500 1.54 0.001Turpentine 20 870 1.07 1.27 1250 1.11 0.0015Water (fresh) 20 998 2.18 1.004 1481 1.48 0.001Fluidlike sea bottomsRed clay 1340 — — 1460 1.96 —Calcareous ooze 1570 — — 1470 2.31 —Coarse silt 1790 — — 1540 2.76 —Quartz sand 2070 — — 1730 3.58 —

Appendix A. Physical Properties of Matter 631A.3. Gases (at 1 atmosphere pressure, 101.3 kPa)Characteristic CoefficientRatio of Impedance, of ShearTemperature Density, ρ 0 Specific Speed ρ 0 /c Viscosity ηGas ( ◦ C) (kg/m 3 ) Heats, γ (m/s) (Pa·s/m) (Pa·s)Air 0 1.293 1.402 331.6 428 0.000017Air 20 1.21 1.402 343 415 0.0000181Hydrogen 0 0.09 1.41 1269.5 114 0.0000088CO 2 (low frequency) 0 1.98 1.304 258 512 0.0000145CO 2 (high frequency) 0 1.98 1.40 268.6 532 0.0000145Oxygen 0 1.43 1.40 317.2 453 0.00002Steam 100 0.6 1.324 404.8 242 0.000013A.4. Conversion FactorsConversely,To Convert Into Multiply by Multiply byatm (atmosphere) mmHg at 0 ◦ C 760 1.316 × 10 −3lb/in. 2 14.70 6.805 × 10 −2N/m 2 (Pa) 1.0132 × 10 5 9.872 × 10 −6kg/m 2 1.033 × 10 4 9.681 × 10 −5◦ C (Celsius) ◦ F (fahrenheit) [( ◦ C × 9)/5] + 32 ( ◦ F − 32) × 5/9cm (centimeter) in. (inch) 0.3937 2.540ft (foot) 3.281 × 10 −2 30.48m (meter) 10 −2 10 2cm 2 (square centimeter) in. 2 0.1550 6.452ft 2 1.0764 × 10 −3 929m 2 10 −4 10 4cm 3 (cubic centimeter) in. 3 0.06102 16.387ft 3 3.531 × 10 −5 2.832 × 10 4m 3 10 −6 10 6dyne lb (force) 2.248 × 10 −6 4.448 × 10 5N (newton) 10 −5 10 5dynes/cm 2 lb/ft 2 (force) 2.090 × 10 −3 478.5N/m 2 (Pa) 10 −1 10ft (foot) in. (inch) 12 0.08333cm (centimeter) 30.48 3.281 × 10 −2m (meter) 0.3048 3.281ft 2 (square foot) in. 2 144 6.945 × 10 −3cm 2 9.290 × 10 2 0.010764m 2 9.290 × 10 −2 10.764ft 3 (cubic foot) in. 2 1728 5.787 × 10 −4cm 3 2.832 × 10 4 3.531 × 10 −5m 3 2.832 × 10 −2 35.31hp (horsepower) W (watt) 745.7 1.341 × 10 −3in. (inch) ft (foot) 0.0833 12cm (centimeter) 2.540 0.3937m (meter) 0.0254 39.37(continued)

632 Appendix A. Physical Properties of MatterA.4. (Continued)Conversely,To Convert Into Multiply by Multiply byin. 2 (square inch) ft 2 6.945 × 10 −3 144cm 2 6.452 0.1550m 2 6.452 × 10 −4 1550in. 3 (cubic inch) ft 3 5.787 × 10 −4 1.728 × 10 3cm 3 16.387 6.102 × 10 −2m 3 1.639 × 10 −5 6.102 × 10 4kg (kilogram) lb (weight) 2.2046 0.4536slug 0.06852 14.594g (gram) 10 3 10 −3kg/m 2 lb/in. 2 (weight) 1.422 × 10 −3 703.0lb/ft 2 (weight) 0.2048 4.882g/cm 2 10 −1 10m (meter) in. (inch) 39.371 2.540 × 10 −2ft (foot) 3.2808 0.30481cm (centimeter) 10 2 10 −2m 2 (square meter) in. 2 1550 6.452 × 10 −4ft 2 10.764 9.290 × 10 −2cm 2 10 4 10 −4m 3 (cubic meter) in. 3 6.102 × 10 4 1.639 × 10 −5ft 3 35.31 2.832 × 10 −2cm 3 10 6 10 −6microbar (dynes/cm 2 ) lb/in. 2 1.4513 × 10 −5 6.890 × 10 4lb/ft 2 2.090 × 10 −3 478.5N/m 2 (Pa) 10 −1 10Np (neper) dB (decibel) 8.686 0.1151N (newton) lb (force) 0.2248 4.448dynes 10 5 10 −5N/m 2 (pascal, Pa) lb/in. 2 (force) 1.4513 × 10 −2 6.890 × 10 3lb/ft 2 (force) 2.090 × 10 −2 47.85dynes/cm 2 10 10 −1lb (force) (pound) N (newton) 4.448 0.2248lb (weight) (pound) slug 0.03108 32.17kg (kilogram) 0.4536 2.2046lb/in. 2 (weight) lb/ft 2 (weight) 144 6.945 × 10 −3kg/m 2 703 1.422 × 10 −3lb/in. 2 (force) lb/ft 2 (force) 144 6.945 × 10 −3N/m 2 (Pa) 6894 1.4506 × 10 −4lb/ft 2 (weight) lb/in. 2 (weight) 6.945 × 10 −3 144g/cm 2 0.4882 2.0482kg/m 2 4.882 0.2048lb/ft 2 (force) lb/in. 2 (force) 6.945 × 10 −3 144N/m 2 (Pa) 47.85 2.090 × 10 −2Slugs lb (weight) 32.17 3.108 × 10 −2kg (kilogram) 14.594 6.852 × 10 −2W (watt) hp (horsepower) 1.341 × 10 −3 745.7

Appendix BBessel FunctionsB.1 The Bessel Differential EquationThe Bessel differential equation of order n, expressed as[x 2 d2dx + x d]2 dx + (x 2 − n 2 ) f (x) = 0carries the solutions consisting of (1) the Bessel functions of the first kind J n (x) forall x, and (2) the Bessel functions of the second kind Y n (x) (also called Neumanfunctions), and the Bessel functions of the third kind H n (1)(2)(x) and H n (x) for all xgreater than zero.B.2 Relationships Between SolutionsH (1)nH (2)n= J n + iY n= J n − iY nJ −n = (−1) n J nY −n = (−1) n Y nB.3 Series Expansions for J 0 and J 1J 0 = 1 − x 22 2 + x 42 2 · 4 2 − x 62 2 · 4 2 · 6 2 +···J 1 = x 2 − 2x 32 · 4 2 + 3x 52 · 4 2 · 6 2 +··· 633

634 Appendix B. Bessel FunctionsB.4 Approximations for Small Argument x < 1J 0 → 1 − x 24J 1 → x 2 − x 3Y 0 → 2 π ln xY 1 →− 2 πB.5 Approximations for Large Arguments x > 2πH (1)nH (2)nB.6 Recursion Relations1x16√2J n →(xπ x cos − nπ 2 − π )4√2Y n →(xπ x sin − nπ 2 − π )4√2→π x ei(x− nπ 2 − π 4 )√2→π x e−i(x− nπ 2 − π 4 )The relations listed below hold true for C being any of the Bessel functions of thefirst, second, or third kind or for linear combinations of these functions.C n+1 + C n−1 = 2nx C nddxddxdC 0dx =−C 1dC ndx = 1 2 (C n−1 − C n+1 )(x n )C n = x n C n−1( ) 1=− 1 x n x C n n+1

B.8 Tables of Bessel Functions, Zeros, and Inflection Points 635B.7 Modified Bessel FunctionsI n (x) = i −n J n (ix)I 0 (x) = J 0 (ix) = 1 + x 22 + x 42 2 2 · 4 + x 62 2 2 · 4 2 · 6 +···∫2 xI 0 (x)dx = xI 1 (x)∫I 1 (x)dx = I 0 (x)I 0 (x) − I 2 (x) = 2 x I 1(x)B.8 Tables of Bessel Functions, Zeros, and Inflection PointsB.1. Bessel Functions or Order 0, 1, and 2x J 0 (x) J 1 (x) J 2 (x) I 0 (x) I 1 (x) I 2 (x)0.0 1.0000 0.0000 0.0000 1.0000 0.0000 0.00000.2 0.9900 0.0995 0.0050 0.0100 0.1005 0.00500.4 0.9604 0.1960 0.0197 0.0404 0.2040 0.02030.6 0.9120 0.2867 0.0437 1.0921 0.3137 0.04640.8 0.8463 0.3688 0.0758 1.1665 0.4329 0.08431.0 0.7652 0.4401 0.1149 1.2661 0.5652 0.13581.2 0.6711 0.4983 0.1593 1.3937 0.7147 0.20261.4 0.5669 0.5419 0.2074 1.5534 0.8861 0.28761.6 0.4554 0.5699 0.2570 1.7500 1.0848 0.39401.8 0.3400 0.5815 0.3061 1.9895 1.3172 0.52602.0 0.2239 0.5767 0.3528 2.2796 1.5906 0.68902.2 0.1104 0.5560 0.3951 2.6292 1.9141 0.88912.4 +0.0025 0.5202 0.4310 3.0492 2.2981 1.11112.6 −0.0968 0.4708 0.4590 3.5532 2.7554 1.43382.8 −0.1850 0.4097 0.4777 4.1575 3.3011 1.79943.0 −0.2601 0.3391 0.4861 4.8808 3.9534 2.24523.2 −0.3202 0.2613 0.4835 5.7472 4.7343 2.78843.4 −0.3643 0.1792 0.4697 6.7848 5.6701 3.44953.6 −0.3918 0.0955 0.4448 8.0278 6.7926 4.25383.8 −0.4026 +0.0128 0.4093 9.5169 8.1405 5.23234.0 −0.3971 −0.0660 0.3641 11.302 9.7594 6.42244.2 −0.3766 −0.1386 0.3105 13.443 11.705 7.86834.4 −0.3423 −0.2028 0.2501 16.010 14.046 9.62594.6 −0.2961 −0.2566 0.1846 19.097 16.863 11.7614.8 −0.2404 −0.2985 0.1161 22.794 20.253 14.3555.0 −0.1776 −0.3276 +0.0466 27.240 24.335 17.5055.2 −0.1103 −0.3432 −0.0217 32.584 29.254 21.3325.4 −0.0412 −0.3453 −0.0867 39.010 35.181 25.9805.6 +0.0270 −0.3343 −0.1464 46.738 42.327 31.6215.8 0.0917 −0.3110 −0.1989 56.039 50.945 38.472(Continued)

636 Appendix B. Bessel FunctionsB.1. (Continued)x J 0 (x) J 1 (x) J 2 (x) I 0 (x) I 1 (x) I 2 (x)6.0 0.1507 −0.2767 −0.2429 67.235 61.341 46.7886.2 0.2017 −0.2329 −0.2769 80.717 73.888 56.8826.4 0.2433 −0.1816 −0.3001 96.963 89.025 69.1436.6 0.2740 −0.1250 −0.3119 116.54 107.31 84.0216.8 0.2931 −0.0652 −0.3123 140.14 129.38 102.087.0 0.3001 −0.0047 −0.3014 168.59 156.04 124.017.2 0.2951 +0.0543 −0.2800 202.92 188.25 150.637.4 0.2786 0.1096 −0.2487 244.34 227.17 182.947.6 0.2516 0.1592 −0.2097 294.33 274.22 222.177.8 0.2154 0.2014 −0.1638 354.68 331.10 269.798.0 0.1716 0.2346 −0.1130 427.57 399.87 327.60B.2. Zeros of J m : J m ( j mn ) = 0.j mnm 0 1 2 3 4 50 — 2.40 5.52 8.65 11.79 14.931 0 3.83 7.02 10.17 13.32 16.472 0 5.14 8.42 11.62 14.80 17.963 0 6.38 9.76 13.02 16.22 19.414 0 7.59 11.06 14.37 17.62 20.835 0 8.77 12.34 15.70 18.98 22.22nB.3. Inflection Points of J m :(dJ m /dx) j ′ mn= 0.j ′ mnm 1 2 3 4 50 0 3.83 7.02 10.17 13.321 1.84 5.33 8.54 11.71 14.862 3.05 6.71 9.97 13.17 16.353 4.20 8.02 11.35 14.59 17.794 5.32 9.28 12.68 15.96 19.205 6.41 10.52 13.99 17.31 20.58n

Appendix CUsing Laplace Transforms to SolveDifferential EquationsC.1 IntroductionNot only all aspects of the of Laplace transforms will be presented, but also enoughof the characteristics will be presented, discussed here to help the reader understandand appreciate the elegance and powerfulness of the procedures that canbe effectively applied to solve linear differential equations. The essential idea ofLaplace transforms is that they are used to convert a differential equation into analgebraic one in order to solve the differential equation.Let f (t) be a function of t for t > 0. Its Laplace transform is defined asF(s) =∫ ∞0f (t)e −st dt = L[ f (t), s]where s is a complex variable which can be expressed as(C.1)s = α + iβ(C.2)The function f (t) is Laplace transformable for α>0if∫ TlimT →∞ 0| f (t)|e −αt dt < ∞If f (t) = A (a constant), for t > 0, the Laplace transform of the constant A isL[A] =∫ ∞0Ae −st dt =− 1 s e−st∣ ∣ ∞ 0We therefore have the Laplace transform pair= A s = F(s)f (t) = A ⇔ F(s) = A sIf f (t) = e −at for t > 0, then the Laplace transform of this exponential functionisLe −at =∫ ∞0e −at e −st dt =∫ ∞0e −(s+a)t =As + a637

638 Appendix C. Using Laplace Transforms to Solve Differential Equationsand hencef (t) = e −at ⇔ F(s) = 1s + aC.2 Laplace Transforms of DerivativesConsider a function f (t) and its derivative df (t)/dt that are both Laplace transformable.If the function f (t) has the Laplace transform F(s), thenL df(t) = sF(s) − f (0+) (C.3)dtEquation (C.3) is readily proven by using integration by parts, i.e.,SetIt then follows thatF(s) = uv| ∞ 0F(s) =u = f (t),dv = e −st dt,∫ ∞0f (t)e −st dtdu = df(t)dtv =− 1 s e−st∫ ∞− vdu =− 1 ∣ ∣∣∣∞s f (t)e−st + 1 s0= f (0+)s+ 1 s0∫ ∞0dt∫ ∞0df(t)e −st dtdtdf(t)e −st dtdtor∫ ∞df(t)e −st dt = sF(s) − f (0+)0 dtThe term f (0+) represents the value of f (t)ast → 0 from the positive side. In asimilar manner, it can be demonstrated thatL d2 f (t)= s 2 F(s) − sf(0+) − f (0+) (C.4)dt 2C.3 Solving Differential EquationsWe discuss here a differential equation that is of greatest interest in treatment ofvibration problems:mẍ(t) + Cẋ(t) + kx(t) = f (t)(C.5)

C.3 Solving Differential Equations 639which also can be expressed asẍ + 2ξω n ẋ(t) + ω 2 n x(t) = f α(t)We now apply Equations (C.3) and (C.4)–(C.6) to obtains 2 X(s) − sx(0+) − ẋ(0+) + 2ξω n [sX(s) − x(0+)] + ω 2 n X(s) = F α(s)Solving for X(s):(C.6)X(s) = F α(x) + x(0+)(s + 2ξω n ) + ẋ(0+)(C.7)s 2 + 2ξω n s + ωn2Equation (C.7) can be written in the formX(s) = A(s)(C.8)B(s)Equation (C.7) and its variation (C.6) have been derived from a second-orderequation, and the techniques we describe below are valid for any order function.For equations with simple roots, we can write Equation (C.8) in the formX(s) = A(s)B(s) =A(s)(C.9)(s + a 1 )(s + a 2 ) ···(s + a n )It is assumed that B(s) is of higher order than A(s) throughout the discussion below.We can expand Equation (C.9) through the use of partial fraction expansions, whichwill result inwhereX(s) = C 1s + a 1+ C 2s + a 2+···+C k =lim (s + a k ) A(s)s→−a k B(s)C ns + a n(C.10)(C.11)Example Problem 1Find the inverse Laplace transform of the functionSolutionX(s) =13(s + 3)s(s + 1)(s + 2)The roots of the numerator (for example, s =−3) are called zeros of the function.The roots of the denominator (viz., 0, −1, −2) are called poles. A partial fractionexpansion of the poles is written as follows:(13(s + 3)s(s + 1)(s + 2) = 13 C1s + C 2s + 1 + C )3s + 2

640 Appendix C. Using Laplace Transforms to Solve Differential EquationsThe residues C 1 , C 2 , and C 3 are obtained from Equation (C.11) as follows:s + 3C 1 = lims→0 (s + 1)(s + 2) = 3(1)(2) = 3 2s + 3C 2 = lims→−1 s(s + 2) = 2(−1)(1) =−2s + 3C 3 = lims→−2 s(s + 1) = 1(−2)(−1) = 1 2Hence( 3/2X(s) = 13 + −2s s + 1 + 1/2 )s + 2Applying the transform pairs we derived in Section C.2,( 3x(t) = 132 − 2e−t + 1 )2 e−2tWhen we know the form of the inverse transform above we can derive the residuegraphically by inspection. The method for doing this is simply divide the productof all vectors from the zeros to the pole whose residue is being determined by theproduct of the vectors from all the other poles to the pole under consideration. Thistechnique is shown in Figure C.1. In Figure C.1(a), C 1 is derived as3̸0 ◦C 1 =(2̸ 0 ◦ )(1̸ 0 ◦ ) = 3/2In Figure C.1(b), we get2̸ 0 ◦C 2 =(1̸ 0 ◦ )(1̸ 180 ◦ ) = 2(1)(−1) =−2and from Figure C.1(c):1̸ 0 ◦C 3 =(1̸ 180 ◦ )(2̸ 180 ◦ ) = 1(−1)(−2) = 1/2Note that the numbers obtained from the use of the graphical residue techniqueare the same as those derived from Equation (C.11). The sign of the residue can bereadily established by simply counting the number of poles and zeros to the rightin the s plane of the pole whose residue is being determined. If the number is even,the sign of the residue is positive; and if the number is odd, the sign is negative.Example Problem 2Consider the function8(s + 2)(s + 4)X(s) =s(s + 1)(s + 3)Find the inverse Laplace transform.

C.4 Equations with Multiple-Order Roots 641(a)(b)Solution(c)Figure C.1. Example of the graphical residue technique.Use the pole-zero pattern of Figure C.2 to apply the graphical technique. Theinverse transform can be written down by inspection as[ (2)(4)x(t) = 8(1)(3) − (1)(3)(1)(2) e−t − (1)(1) ] [ 8(2)(3) e−3t = 83 − 3 2 e−t − 1 ]6 e−3tC.4 Equations with Multiple-Order RootsConsider a function X(s) that has multiple-order roots of the formX(s) = A(s)B(s) =A(s)(s + a 1 ) m (s + a 2 ) ···(s + a n )

642 Appendix C. Using Laplace Transforms to Solve Differential EquationsFigure C.2. Pole-zero pattern for Example Problem 2.The partial fraction expansion of the function must be of the formatA(s)B(s) = C 11(s + a 1 ) m + C 12(s + a 1 ) m−1 +··· C 1ms + a 1+ C 2s + a 2+···+C ns + a nThe residues C 2 , C 3 ,...,C n are determined in the same manner as was describedin the preceding section but the coefficients C 11 , C 12 ,..., C 1m require specialtreatment. They are evaluated from the following expression:1C 1 j =( j − 1)!d j−1ds j−1 (s + a 1) m A(s)B(s)(C.12)which is evaluated at s =−a 1 . Then the inverse Laplace is obtained from thetransform relation:[ ]L −1 1= t j+1(s + a) j ( j − 1)! e−at (C.13)The following example problem will demonstrate the procedure.Example Problem 3Find the inverse Laplace transform forX(s) =s + 1(s + 2) 2 (s + 3) = C 11(s + 2) 2 + C 12s + 2 + C 2s + 3

SolutionC.5 Equations with Complex Roots 643Use Equation (C.11) to obtainC 11 = s + 1s + 3∣ =−1,s=−2C 2 = s + 1 ∣∣∣∣s=−3=−2(s + 2) 2Equation (C.12) is now applied to obtain the coefficient C 12 :1 d 2−1 ( ) s + 1C 12 == d ( ) ( s + 1 1=(2 − 1)! ds 2−1 s + 3 ds s + 3 s + 3 − s + 1 )∣∣∣∣s=−2= 2(s + 3) 2which now results inX(s) =−1(s + 2) 2 + 2s + 2 − 2s + 3The inverse Laplace transform now becomesx(t) =−te −2t + 2e −2t − 2e −3tC.5 Equations with Complex RootsLet us consider the case of finding the inverse Laplace transform of a function ofthe formX(s) =(s + b 1 )(s + b 2 )[(s + α) 2 + β 2 ](s + a 1 )(s + a 2 )(s + a 3 )We could evaluate the inverse transform using a partial fraction expansion, but thiscan be quite a complicated procedure especially where complex roots are entailed.In order to deal with this problem, it is preferable to use the relation between thetransform pairsas + b(s + α) 2 + β ⇔ 2 2Me−αt cos(βt + φ)where M and φ are unknowns that depend on a and b.In order to demonstrate that M and φ can be evaluated by the graphical approach,suppose that the above function has the pole-zero pattern of Figure C.3(a). Only theresidues of the complex poles at –α ± iβ will be determined. Using the graphicalapproach for simple roots, we get from using Figure C.3(b):X(s) =(z 1 e iθ 2)(z 2 e iθ 6)(z 1 e −iθ 2)(z 2 e −iθ 6)(P 1 e iθ 1 )(P2 e iθ 3 )(P3 e iθ 4 )(P4 e iθ 5 ) (P+ 1 e −iθ 1 )(P2 e −iθ 3 )(P3 e −iθ 4 )(P4 e −iθ 5 )s + α − iβs + α + iβ+ additional terms (C.14)

644 Appendix C. Using Laplace Transforms to Solve Differential Equations(a)(b)Figure C.3. Graphical residue approach for complex roots.We observe that the coefficient of the second term, which represents the residueat s =−α – iβ is the complex conjugate of the coefficient of the first term that isthe residue at s =–α + iβ. Equation (C.14) can be expressed asX(s) =Me iφs + α − iβ +Me−iφs + α + iβ+ additonal termswhereM =z 1 z 2P 1 P 2 P 3 P 4, φ = θ 2 + θ 6 − (θ 1 + θ 3 + θ 4 + θ 5 )

C.5 Equations with Complex Roots 645Thenx(t) = M(e −αt+iβt+iφ + e −iαt−iβt−iφ ) + addtional terms= Me −αt (e i(βt+iφ) + e −i(βt+φ) ) + additonal terms= 2Me −αt (ei(βt+iφ) + e −i(βt+φ) )+ additonal terms2= 2Me −αt cos(βt + φ) + additional termsHere M and φ can be determined graphically using Figure C.3. The use of thegraphical residue technique for equations with complex roots is extremely powerfulsince the inverse transform can be determined by inspection.Example Problem 4GivenX(s) =find the inverse Laplace transform.Solutions + 2s(s 2 + 4s + 5)Figure C.4 shows the function. The residue at the pole at the origin is found byusing Figure C.4(b), and the complex poles are derived by using Figure B.4(c).The ensuing function isx(t) = √ 2 √ + 2 × 15 5 2 √ 5 e−2t cos(t + φ)whereinφ = (90 ◦ ) − (90 ◦ + θ)θ = ( π − tan −1 )12Example Problem 5From vibration theory X(s) [cf. Equation (20.12)] is of the forms + 2ξω nX(s) = x 0s 2 + 2ξω n s + ωn2 x 0 (s + 2ξω n )=√ √(s + ξω n + iω n 1 − ξ 2)(s + ξω n − iω n 1 − ξ 2)Determine the inverse transform.

(a)(b)(c)Figure C.4. Pole-zero diagrams for Example Problem 3.Figure C.5. Pole-zero diagram for Example Problem 4.

SolutionC.5 Equations with Complex Roots 647From Figure C.5 the inverse transform can be evaluated to yield[ω(n√ ) ]x(t) = x 0 √ω n 1 − ξ2 e−ζω nt cos ω n 1 − ξ 2t + φwhereφ = θ − π/2θ = cos −1 ξand we now havex(0√ )x(t) = √1 − ξ2 e−2ω nt cos ω n 1 − ξ 2t − π/2 + cos −1 ξ

IndexA-scan, 473A-weighting, 52–54Absorbent effects, growth of sound with,251–252Absorptionin seawater, parametric variation of, 419–421spherical spreading combined with, 421Absorption coefficients, sound, 249, 251Absorption losses, 416Academy of Music, Philadelphia, 266Accelerometers, 611–612Accordion, 541, 542–543Acoustic analogues, 151–168Acoustic barriers, 310–314Acoustic element, lumped, 145Acoustic energy, 248–249Acoustic energy density, 258Acoustic equations, derivation of, 25–29Acoustic filters, 158–165Acoustic hemostasis, 504Acoustic impedance, 151distributed, 154–155lumped, 152–154Acoustic lenses, 494–495Acoustic measurements, 173–209Acoustic microscope, 42–43Acoustic ohm, 152Acoustic propagation constant, 499Acoustic reflex, 219Acoustical instruments, characteristics of,173–174Acoustical shadow, 44–45Acoustical surgery, 504Acoustics, 2–3, 13architectural, 243–278of enclosed spaces, 243–278fundamentals of, 13–28future of, 10–11history of, 1–11importance of, 15musical, 509nonlinear, 617–626term, 4underwater, 409–439Active isolation and absorption systems, 606Active noise control, 404–405Addition method for measuring sound powerlevel, 207–208Adiabatic process, 120Adiabatic relaxation time, 425Aeolian tones, 386Afternoon effect, 423Air compressor noise, 370–371Air-reed instruments, 537Air shroud silencer nozzles, 394–395Aircraft noise, rating of, 332–335Aliasing, 195Alpha factor, 341Alternation method for measuring sound powerlevel, 207Alto clef, 512Amplitude modulation (AM), 208Analogues, acoustic, 151–154Angular frequency, 13Annoyance, 319ANSYS finite element program , 610Antinodes, 77Antiresonance, 134Arau, Higini, 266Architectural acoustics, 243–278Aristotle, 2–3Array gain, 426Articulation index, 227–230Associated liquids, 451Audio range of frequencies, 45–47Audiometer, 222649

650 IndexAuditoriums, design of, 261–278outdoor, 271–276Augmented reality system, 505Autocorrelation, mean-square value and, 613Autocorrelation function (ACF), 276Avogadro’s number, 19Axial flow fans, 360–361Azimuthal nodal circles, 117B-scan, 474B-weighting, 52–54Bacon, Sir Francis, viiBagpipe, 545Balance noise criterion (NCB) curves, 327–328Ball bearing noise, 379–382Band, 562–563, 564Band pass filters, 162–164noise measurement and, 189–193Band shells, design of, 271–276Bandwidth, 46Banjo, 525, 527Barrier insertion loss, approximations for,315–316Barriers, 310–316in free fields, 314–315walls and enclosures and, 281–316Bars, vibrating, see Vibrating barsBass drum, 555–556Bass trombone, 551Bassoon, 546–547Bats, 238, 443–444B.C.S. theory, 457, 458Beat frequency, 35–36, 60Bell lyre, 553Belt drive noise, 385Bekésy, Georg von, 9Bell Telephone Laboratories, 8, 9, 245Benaroya Hall, 270–271Beranek, Leo L., 9Bernoulli’s principle, 541nBessel functions, 116, 633–636modified, 126n, 635Beyer, Robert T., 9Bible, 1Blade frequency, 361Blade rate component, 370–371Blower noise, 359–366Blu-ray, 574, 575Boethius, Sverinus, 2Bolt, Richard H, 9.Boltzmann constant, 19Boom cars, 344, 348–349Boston Symphony Hall, 263–264Botta, Mario, 268Boundaries, reflection of waves at, 74–75Bowed-string instruments, 529–533Boyle, Robert, 4Brasses, 550–551Bugle, 550Bulk viscosity, 419Burgers equation, 621–622C-scan, 476C-weighting, 52–54Calliope, 538Carillon, 554Carnegie Hall, 266Castanets, 556Catgut Acoustical Society, 533–534Cavitation, 451–453Cavitation noise, 452Cavitation threshold intensity, 452–453Celesta, 553Cembalo, 527Central Artery Tunnel Project (CA/T),351–352Central hearing loss, 221Centrifugal fans, 359–361Ceramic microphones, 174–175Cetaceans, 444Chain drive noise, 382–395Characteristic mechanical impedance, 133Charge sensitivity, 612Chilowsky, Constantin, 7Chimes, 514Chladni, Ernst F. F., 4–5Chorusing, 559Chrysippus, 2Clamped-free bar, 106, 107Clarinet, 544Clavichord, 527Closed-ended pipes, resonances in, 132–134Cochlea, 215–219Cochlear microphonic effect, 219Coincidence effect, 286–287Collision number, 446Color schlieren photography, 496Community noise, evaluation of, 344–345Community response, 319Composite noise rating (CNR), 333Compressive forces, 91Computers, integration of measurementfunctions in, 208–209Concert halls, design of, 261–271Condenser microphones, 175–176Conductive hearing loss, 221

Index 651Conservationof energy, 25of mass, 20–22of momentum, 22–25Construction noise, general, 350–352Contact ratio (CR) for gears, 376–377Continuity equation, 21expansion of, 26Contra bassoon, 547Contrast agents, 501–502Cooley-Tukey algorithm, 195Cornet, 550, 551Correlated sound waves, 58–60Corti, organ of, 217–218Council of European Communities, 346Coupled quantum particles, 456–458Creep, 261Crescendo, 520Crest factor capability, 174Critical frequency, 286–288Crossover networks, 578Crum, Lawrence A., 10Cylindrical spreading, 417Curie, Pierre, 7, 459Curie, Paul-Jacques, 7, 459Cymbal, 556Damping, 585–586, 606–610internal, 608–609Damping factor, 123–124Damping materials, 609Damping mechanisms, 609–610Damping ratio, 587Dashpot, 585Data acquisition systems, 208Data windows, 196–198Day-night average sound levels, yearly, 338Day-night equivalent sound pressure level, 57,329–331da Vinci, Leonardo, 3Dead rooms, decay of sound in, 254–256Dead spots, 262Deafness, sensorineural, 221Decibels, 47–49averaging, 49–52Decoupling, 603Decrescendo, 580Deep sound channel, 424Deep sound-channel axis, 414Delay lines, ultrasonic, 484Detection threshold, 430Detuning, 603Diagnostic uses of ultrasound, 498–503Diatomic molecules, 445Diffraction, 44–45Diffuse fields, 178, 244Digital recording, 573–575Direct drive hearing system (DDHS), 236–237Directional characteristic, 200Directivity factor, 258Discord, harmony and, 520–521Discrete Fourier transform (DFT), 194Displacement amplitudes, 16Dissipative function, 621–622Dissipative mufflers and silencers, 398, 403–404Distortion products, 219Distributed acoustic impedance, 154–155Doppler effect, 39–40Dosimeters, 187–189Dorothy Chandler Pavilion, 264Double bass, 528–533Double mechanical-reed instruments, 545–550Double-panel partitions, 289–292Drug therapy, 237Ducted source systems, 165–168Ducts, gaseous flows in, 396–398Dulcimer, 536Duple meter, 517DVD formats, 574–575Dynamic microphones, 174Dynamic range, 174Ear, human, 213–217Ear sensitivity, 222–225Eardrum, 214–215Early decay time (EDT10), 258Eastman Theater, 264Echo, 40, 267, 409Echolocation, 238Echo-ranging (active) sonar, 410Echocardiography, 499Echoencephalography, ultrasonic, 500Effective perceived noise level (EPNL), 324–325Eigenstates, 454Einstein, Albert, 454Ekos device, 505Elastomeric mounts, 604Electret microphones, 174–175Electric acoustic guitar, 556Electric guitar, 556Electric motor noise, 366Electrical and electronic instruments, 521, 522,556–561Electromechanical coupling factor, 465Electron-acoustic image converters, 492–494Electronic organ, 556–559

652 IndexElectrostatic speakers, 580Electrostrictive effect, 461–462Electrostrictive materials, 425Elevation dimension, 470Empirical dosage response relationship, 345Emulsification and flow enhancement, 491Enclosed spaces, acoustics of, 243–271Enclosures, 305–309small, 309–310walls and barriers and, 281–316Energy, conservation of, 25Energy density, 65–67instantaneous, 66Energy flux density, 411of waves, 434Environmental noise, 319performance indices for, 55–57Environmental Protection Agency (EPA), 320Equal energy hypothesis, 345Equal noisiness contours, 323Equivalent simple piston, 120Equivalent sound levels, 56–57, 329–330, 335Ergodic processes, 614Euler, Leonhard, 6, 521European Union (EU), 350–351Eustachian tube, 214–215Evanescent mode, 141Evoked otoacoustic emission, 219Excessive loudness, 261Exchange rate, 188nExcitationharmonic, 593–596by impulse, 597–598motion, 600–603Extracorporeal shock wave lithotripsy, 503–504Eyring equation, 255Fan laws, 365–366Fan noise, 359–366characteristics of, 361–362Far field effect, 62Fast Fourier transform (FFT) technique,194–196Federal Aviation Administration (FAA), 320Federal Highway Administration (FHWA),335–339Federal Interagency Committee on Noise(FICON), 346Fermat, Pierre de, 5Ferroelectrics, 461Field incidence mass law, 284–285Filter networks, 162–165Filter response, 189Filters, acoustic, 158–165Finite element analysis (FEA), 610–611Finite roadway segment adjustment, 342Finite strings, 83–84Fire sensing, ultrasonic, 487–488First harmonic, 76Fixed-fixed bar, 92, 93Flageolet, 539Flanking, 294Flared pipes, 135Flat notes, 512Flaw detection, ultrasonic methods for, 481–483Fletcher, Harvey, 8–9Flowmeter, ultrasonic, 486–487Fluid flow equations, 19–20Fluids, thermodynamic states of, 18–19Flute, 539–540Flutter echo, 261Fohi, 2Forced vibrations, 593–598in finite strings, 83–84in infinite strings, 81–83in membranes, 122–125in plates, 128Forward propagating plane waves, 31–33Forward scanning, 482Fourier series, 34Fourier’s theorem, 79Free-field microphones, 178Free fields, 244barriers in, 314–315Free-fixed bar, 95Free-free bar, 93–94, 105–108Freely vibrating circular membranes, 115–120French horn, 550, 551Frequency, 13–15angular, 13beat, 60Frequency modulation (FM), 208Frequency ranges of musical instruments, 565Frequency response, 173Frequency-weighting, 52–55Fresnel, Augustin Jean, 6Full anechoic chambers, 65, 202–203, 245Functional hearing loss, 221Fundamental mode, 76Gabor, Dennis, 496Galileo Galilei, 3Gas-jet noise, 385–398control of, 393–398Gaseous flows in pipes or ducts, 396–398Gassendi, Pierre, 3

Index 653Gear enclosures, 378Gear noise, 371–378Gear-tooth error, 373Gear train noise, 374–376Gears, helical, 377Gehry, Frank, 273Gene therapy, 236Glissando, 525Glockenspiel, 553Gong, 556Gradient operator, 21–22Grazing incidence, 178Grimaldi, Franciscus Mario, 4Grösser Musikvereinsaal, 263Group speed, 143Guitar, 526Hair cells, 217Half-power point, 191Hanning window, 197Harmonic excitation, 593–596Harmonica, 541, 543Harmonics, 76Harmonium, 542Harmony, discord and, 520–521Harp, 524–525Harpsichord, 527Harris, Cyril M., 8, 270Hausksbee, Francis, 4HD-DVD, 574–575Headphones, 580Hearing aids, 231–233analog, 232digital, 233implantable, 233programmable, 233Hearingcharacteristics of, 222–227human, 213–217in animals, 237–239mechanism of, 217–220physiology of, 213–239Hearing loss, 220–221Helical gears, 377Helmholtz equation, 113Helmholtz, F. L., 6Helmholtz resonator, 145–148Hemianechoic chamber, 65Hemispherical wave, 65Hemostasis, acoustic, 504Hertz (unit), 15High-fidelity reproduction, 10, 570High-pass filters, 161–162Highway construction noise, 350–352Highway traffic noise, 335–344Hollywood Bowl, 271–272Holography, ultrasonic, 496–497Hoods, 306–309Hooke, Robert, 4, 5Hooke’s law, 91Human hearing, 213–217Human voice, 551–552Humidity, reverberation time and, 256–258Hunt, Fredrick V., viiHutchins, Carleen, 9, 531Huygens, Christian, 5Huygens’ principle, 38–39Hydrophone arrays, 425–426Hydrophones, 424Imaging processes, ultrasonic, 492–497Impedance, 82acoustic, see Acoustic impedanceImpedance tube, 137Impingement noise, 395Impulse, excitation by, 597–598In-line silencers, 397Indoor noise criteria, 325–328Industrial applications of ultrasound, 449–497Industrial noise sources, 357–405Inertia blocks, 604Infinite cylindrical pipes, 131–132Infinite strings, forced vibrations in, 81–83Infrasound, 15Ingard, Karl Uno, 9Inner ear, 215Insertion loss, 167Instantaneous energy density, 66Intensity, sound, 60–62Interference patterns, standing wave, 40Interferometers, 483–484Internal damping, 608–609Inverse square law, 64iPod, 582–583Isolators, 603–604Isono system, 580–581Jacobs device, 493–494Joule, James P., 7Journal bearing noise, 378–379Kármán vortex street, 386Kennedy Center for the Performing Arts,2568–270Kettledrum, 554membrane theory and, 120–122

654 IndexKey notation, 518Kinetic energy density, 65–66Kircher, Athenasius, 4Kleinhaus Music Hall, 264Kneser liquids, 451Knudsen, Vern O., 8Kryter, Karl D., 9La Scala Opera House, 264–266Ladder-type acoustic filters, 164–165Lagging pipes, 398Laminar flow, 367Langevin, Paul, 7Latching overload indicator, 187Leakage, 196Leger lines, 510, 511Leibniz, Gottfried Wilhelm, 5Lighthill, Sir James, 9Lighthill’s parameter, 388Linear-array transducer specification, 472Lin-hun, 2Lindsay, R/ Bruce, 9Lip-reed instruments, 550–552Liquid crystal imaging, 496Listening spaces, sound fields of,subjective preferences in, 276–278Lithotripsy, extracorporeal shock wave, 503–504Live rooms, sound intensity growth in, 247–249Longitudinal wave equationderivation of, 89–92solutions of, 92–93Longitudinal waves, 31, 89Loops, 77Lord Rayleigh (see Strutt, John William)Loudness, 18, 52, 225, 323Loudness notation, 518, 519Loudspeakers, 578–580Low-pass filters, 159–160Lumped acoustic element, 145Lumped acoustic impedance, 152–153Lute, 523, 524Lyre, 523, 524Machinery noise control, 357–405Machining, ultrasonic, 488Magnetic recorders, 208Magnetic recording, 571–573Magnetic resonance imaging (MRI), 505Magnetostriction, 466Magnetostrictive materials, 425Magnetostrictive strain coefficient, 468Magnetostrictive stress constant, 468Magnetostrictive transducers, 466–468Magnification factor, 595Major keys, 518, 519Mandolin, 526Marimba, 553Masking, 226–227Mass, conservation of, 20–22Mass concentrated vibrating bars, 95–97Mass control case, 283–284Mass flux, 20–21Mass law, field incidence, 284–245Mean-square value, autocorrelation and, 613Measure, in music, 513Measurement error, noise, 199–200Mechanical drive element noise, 382–385Mechanical impedance, 82characteristic, 133Mechanical reed instruments, 541–542Mechanical stress measurements, 485–486Median noise level, 332Medical uses of ultrasound, 497–506Membrane theory, kettledrum and, 120–122Membranes, 111–125forced vibrations of, 122–125freely vibrating circular, 115–120rectangular, 113–115wave equation for, 111–113Mersenne, Marin, 3Meshing frequencies of gears, 372–373Metals, ultrasonic working of, 488–490Metronome, 516Microphone sensitivity, 170–171Microphones, 171–181selection and position of, 177–181Millikan, Robert, 8–9Millington-Sette theory, 256Minimum audible field (MAF), 222–225Minimum audible pressure (MAP), 223–225Minnesota Orchestra Hall, 267–268Minor keys, 518, 519Mixed hearing loss, 221Mixed layer, 423Momentum, conservation of, 22–25Monopoles, 62–64Morse, Philip M., 9Motion excitation, 600–604Motion sensing, ultrasonic, 484–488Mouth organ, 541, 543MP3, 581MSC-NASTRAN finite element program, 610Muffler system descriptors, 167–168Mufflers and silencers, 398–404dissipative, 398–399, 403–404reactive, 399–404

Index 655Multichannel sound systems, 576–577Multijet diffuser, 394Music, 509musical instruments and, 509–565pitch for, 2Musical acoustics, 509Musical Instrument Digital Interface(MIDI), 561–562Musical instruments, 1–2, 521–565electrical and electronic instruments, 521,522, 556–561frequency ranges of, 565music and, 509–565percussion instruments, 521, 522, 553–556strings, 521, 522, 522–536wind instruments, 521, 522, 536–552Musical notation, 510–513Musical notes, duration of, 513–516Musical staff, 510, 511Narrow band analyzers, 192NASTRAN finite element program, 610National Environmental Policy Act of 1969, 319Near field effect, 62Nematic crystals, 496Neutral axis, 100Newton, Isaac, 5Nobel Prize, 6, 9, 10, 213, 496Nodes, 77true, 106Noise, 15environmental, performance indices for,55–57perception of, 323–325vibration and, 585white, 228, 615Noise and number index (NNI), 334–335Noise bandwidth factor (NBF), 197Noise barriers, 310–316Noise controlactive, 404–405criteria and regulations for, 319–352Noise Control Act of 1972, 320–321Noise criteria, indoor, 325–329Noise criterion curves, 325–326Noise dosimeter, 187–189Noise exposure forecast (NEF), 334Noise insulation ratings, 296–300Noise level, effective perceived (EPNL),324–325Noise-limited range, 433Noise measurement, band pass filters and,189–193Noise measurement error, 199–200Noise rating curves, 323Noise reduction in duct systems, 167Noise reduction of walls, 300–304Noise source sound power, estimation of,358–359Noise sources in workplace, 357–358Noisiness index, 323Nonlinear acoustics, 617–626Nonlinearity in solids, 625–626Normal force, 23Normalized impedance, 133Note values, 515Noy N (unit), 323Nyquist frequency, 195Oboe, 546Ocarina, 539Occupational Safety and Health Act of 1970,321–323Occupational Safety and Health Administration(OSHA), 321–322Octave, 2, 510Octave band analyzer, 191Octave bands, 45–47one-third, 46–47Odontocetes, 237–238Office of Noise Abatement and Control(ONAC), 320Ohm, Simon, 5Olson, Harry F., 9Omnidirectional microphone, 178One-third octave bands, 46–47Open-circuit response, 426Open-ended pipes, 134–136Opera houses, design of, 262–271Optical interference methods, 483OPTIMA computer program, 344Orchestra, 562–563, 564Orchestra Hall, Chicago, 264Organ, 522, 547–550Organ console, 547Organ of Corti, 216–218Organ pipes, 548–549Otoacoustic emission, 219Outdoor auditoriums, design of, 271–278Oval window, 218Panels, sound transmission through,295–286Particle displacement, 58Particle velocity, 16, 58, 151Partitions, double-panel, 289–292

656 IndexPascal (unit), 48Percent isolation, 602Percentile sound levels, 332Percussion instruments, 521, 522, 553–556Performance indices for environmental noise,55–57Period, 13Personal sound exposure meter, 187Petrillo Music Shell, 275Phase angle, 116Phase speed, 103Phased transducer arrays, 469–472Phon curves, 225Phonons, 451–458Physical modeling, electronic music, 562Physiology of hearing, 213–239Piano, 533, 535–536Piccolo, 540–541Picket fence effect, 196Piezoelectric crystals, 459–460Piezoelectric effect, 7Piezoelectric materials, 425Piezoelectric relationships, fundamental,462–464Piezoelectric strain constant, 462Piezoelectric stress constant, 463Piezoelectric transducers, 464–466Piezomagnetic constant, 468Pipes, 131closed-ended, resonances in, 132–134flared, 135gaseous flows in, 396–398infinite cylindrical, 131–132lagging, 398open-ended, 134–135standing waves in, 136–137unflanged, 135waves in, 155–158Piston, equivalent simple, 120Pitch, 2, 225–226, 510Pitch range, 510Plastics, ultrasonic working of, 488–490Plates, 125–128forced vibrations in, 128vibrating thin, 125–128Playback audio equipment, 576–582Plumbing system noise, 366–370Poisson effect, 126Poisson ratio, 125Portable playback equipment, 581–583Power, radiation of, from open-ended pipes,135–136Power spectral density, 614–615Power transmission coefficient, 135–136, 159,160–161, 162, 163Preamplifier, 577Pressure microphone, 178Pritzker Pavilion, 273–276Probability density, 612–613Progressive waves in fluids, 619–626Propagating mode, 141Psychoacoustics, 227Pulse technique in detectinf flaws, 481Pump noise, 366–367Q factor, 465Quadruple meter, 517Quantum particles, coupled, 456–458Quartz crystals, 459–460Quintillianus, 2Radian frequency, 13Radial nodal lines, 117Ragas, 527Railroad noise regulations, 349–350Random-incidence microphone, 178Random vibrations, 612–615Rayl (unit), 58Ray theory, 620–621Rays, 40Reactive mufflers and silencers, 398–403Real strings, 85Real-time analysis, 193Recorder, 539Recording equipment, 569–571, 571–576Recreational noise, 319Rectangular cavity, 138–140Rectangular membranes, 113–115Reed instruments, 536–552Reed organ pipe, 542Reflection, 40–42Reflection coefficient, 249Refraction, 42–44basic laws of, 43–44underwater, 421–423Relaxation, 445Relaxation frequency, 449Relaxation processes, 445–451Relaxation timeadiabatic, 449translational, 445Resolution, 198–199Resonance method of measuring soundpropagation speed, 487Resonances in closed-ended pipes, 132–134Response time, 174

Index 657Rest symbols, 513, 515Restrictive flow silencer nozzle, 394Reverberant effects, 246–247Reverberant fields, 244–245sound absorption in, 256–258sound levels due to, 259–261Reverberation, 409Reverberation chambers, 203–206, 254Reverberation-limited range, 432Reverberation time, 205, 243, 253predicting, 253sound absorption and humidity and, 256–258Reynolds number, 367Rhythm, 517Ribbon-type speaker system, 580Rigid-walled circular waveguide, 144Ring frequency, 397Roller bearing noise, 379–382Room constant, 259Room criterion curves, 327Roomsdead, decay of sound in, 254–256live, sound intensity growth in, 247–249Root-mean-square sound pressure, 47–49Rotational waves, 31Rudnick, Isadore, 9Sabine equation, 243Sabine, Wallace Clement, 6, 243Sarrusophone, 545, 547Sauveur, Joseph, 4Saxophone, 544Scanning plane, 470Schlieren imaging, 495–496Seasonal thermocline, 414Seawaterabsorption in, parametric variation of,419–421speed of sound in, 411–413velocity profiles in, 413–415Secondary emission ratio, 492Semianechoic chamber, 201–202Semitones, 512Sensitivity, 176–179ear, 222–225Sensorineural deafness, 221Sequencer, 561Serial analysis in filters, 191Sextuple meter, 517Shadow zones, 44–45Sharp notes, 512Shear stress, 23Shear viscosity, 419Shear waves, 31Shell isolation rating (SIR), 296, 298–300, 301Shock waves, 622–625Side-lobe ratio (SLR), 197Signal-to-noise ratio (S/N ratio), 181Silencers, see Mufflers and silencersSimple harmonic motion (SHM), 75Simple harmonic solutions of wave equation,75–76Sine function, 13, 14Sine waves, generating, 13, 14Single-reed instruments, 541Sitar, 527Skudrzyk, Eugen, 9Small enclosures, 309–310SNAP 1.1 computer program, 344Snell, Willbrod (Snellius), 5Snell’s law, 44, 422Social surveys of noise, 344–345Sonar, 409Sonar equations, 427–431shortcomings of, 437summarization of, 437transient form of, 434–437Sonar parameters, 429–431Sonar system, 428Sonar transducers, 424–427Sonochemistry, 454Sonoluminescence, 453–454Sonophoresis, 505Sonoporation, 506Soprano clef, 512SORAP acronym, 433–434Sound, 13, 16decay of, 252–256in dead rooms, 254–256growth of, with absorbent effects, 251–252speed of, see Speed of soundunderwater, concepts in, 410–411unwanted, 15wave nature of, 13–15Sound absorptionin reverberant field, 259reverberation time and, 256–258Sound absorption coefficients, 249–251Sound channels, 44Sound decay, 8Sound fields, 244–246of listening spaces, subjective preferences in,276–278Sound focusing, 262Sound generation, 16–17Sound intensity, 60–62

658 IndexSound intensity growth in live rooms, 247–249Sound intensity level, 64Sound intensity probes, vector, 181–182Sound-level meter (SLM), 152–186integrating, 186–187using, 183, 184–185Sound levels, 52–55due to reverberant fields, 259–291equivalent, 56, 329, 335Sound-measuring instrumentation, 173–209Sound power, 200noise source, estimation of, 358–359Sound power level, 64addition method for measuring, 207–208alternation method for measuring, 207specific, 362–365substitution method for measuring, 206–207Sound pressure, 28root-mean-square, 47–48Sound pressure level (SPL), 18, 64at distance from walls, 304–305Sound propagation, 16–18nature of, 31in water, 409–410Sound propagation speed, resonance method ofmeasuring, 487Sound transmission, through panels, 285Sound transmission class (STC), 296–298Sound transmission coefficient, 282combined, 294–296Sound, unwanted, 15Sound velocity, 17–18Sound waves, see WavesSoundboards, 522–523Sousaphone, 551Specific acoustic impedance, 151, 410Specific acoustic resistance, 410Specific sound power level, 362, 363–364Spectral density, 614–615Speech intelligibility, 227–230Speech interference level (SIL), 230–231Speed of sound, 16–18in seawater, 411–413Sphere, theoretical target strength of, 438–439Spherical spreading, 416–418combined with absorption, 421Spherical waves, 64STAMINA 2.0 computer program, 344Standing wave interference patterns, 40Standing-wave ratio, 137Standing waves, 36–38, 76–78in pipes, 136–138Static deflection, 598Steel drum, 554–555Stiffness, 85Strain, 90Stress, 91Strings, musical instruments, 521, 522, 522–536Strings,finite, 83–84infinite, forced vibrations in, 81–83real, 85vibrating, see Vibrating stringsStruck-string instrument, 533, 534–536Strutt, John William (Lord Rayleigh), 5, 6Subjective preferences in sound fields oflistening spaces, 276–278Substitution method for measuring sound powerlevel, 206–207Surface waves, 483Surgery, acoustical, 504–505Synthesizers, 558–561System loss factor, 606Tambourine, 556Tanglewood Music Shed, 272–273Target strengths, 409theoretical, of sphere, 438–439Tenor banjo, 527Tenor clef, 512Tensile forces, 91Therapeutic uses of ultrasound. 503–506Thermocline, 414Thermodynamic states of fluids, 18–19Threshold sound level, 188nTime signature notation, 516–517Timpani, 554Tooth error, 373Torsional waves, 31Traffic conditions, adjustments for, 340–343Traffic noise, evaluation of, 335–344Transducer array-element configurations,470–472Transducer arrays, 469–472phased, 469–470Transducer response, 426–427Transducers, 458–468electrostrictuve effect, 461–462magnetostrictive, 466–470piezoelectric, 462–466sonar, 424–427Transfer function, 189Transformer noise, 366Transistor, 8Translational relaxation time, 445Transmissibility, 598–603

Index 659Transmission coefficientcombined sound, 294–296sound, 282Transmission loss, 282measuring, 292–294underwater, 416–418Transmission loss in duct systems, 167Transmitting-current response, 427Transverse sensitivity, 612Transverse vibrations of vibrating bars, 100–104Transverse wave equation, derivation of, 71–73Transverse waves, 31, 71Triangle, 553Triple meter, 517Trombone, 551True nodes, 106Trumpet, 550Tuba, 551Tubular bells, 553Tuning fork, 105, 107–108, 553Turbulent flow, 367Ukulele, 525–526Ultrasonic cleaning, 479–480Ultrasonic delay lines, 484Ultrasonic diagnosis, safety of, 502Ultrasonic echoencephalography, 500Ultrasonic fire sensing, 487–488Ultrasonic flowmeter, 486–487Ultrasonic holography, 496–497Ultrasonic imaging processes, 492–497Ultrasonic machining, 488Ultrasonic methods for flaw detection, 481–483Ultrasonic motion sensing, 487–488Ultrasonic viscometer, 491Ultrasonic welding, 488–489Ultrasonic working of metals and plastics,488–490Ultrasonics, 443–476Ultrasound, 15Agricultural, 490diagnostic uses of, 498–501industrial applications of, 479–497medical uses of, 497–506safety, 502–503therapeutic uses of, 503–506Uncorrelated sound waves, 58Undamped natural frequency, 587Underwater acoustics, 409–439Underwater refraction, 421–423Underwater sound, concepts in, 410–411Underwater transmission loss, 416–418Unflanged pipes, 135Universal gas constant, 19Universal joint noise, 385Urick, Robert Joseph, 10Vacuum tubes, 8Vector sound intensity probes, 181–182Vehicle noise, 339–344Vehicle noise regulations, 347–349Velocity-depth function, 413Velocity profiles, 413–415in sea, 413–415Vibrating bars, 89–108boundary conditions for, 93–95general boundary conditions for, 98–100mass concentrated, 95–98transverse vibrations of, 100–104Vibrating strings, 71–85assumptions, 71energy of, 80–81Vibrating thin plates, 125–128Vibration(s), 585–615forced, see Forced vibrationsmodifying source of, 603noise and, 585random, 612–615Vibration absorbers, 604–606Vibration control, 598–603techniques for, 603–610Vibration measurements, 611–612Vibration systems, modeling, 585–590Viola, 528–533Violin, 528–533Violin octet, 533, 534Violoncello, 528–533Viscometer, ultrasonic, 491Vitruvius (Marcus Vitruvious Pollo), 3Voice, human, 551–552Voice recognition, 575–576Voicing, 544Voltage-controlled amplifier (VCA), 558–559Voltage-controlled filter (VCF), 558, 559Voltage-controlled oscillator (VCO), 558, 559Voltage sensitivity, 612Volume velocity, 151Volume viscosity, 419Wallsenclosures and barriers and, 281–316noise reduction of, 300–304sound pressure level at distance from,304–305Water, sound propagation in, 409–410Water hammer, 368–370

660 IndexWater-hammer arresters, 370Wave distortion, 617–619Wave equationeffect of initial conditions on, 78–80general solution of, 73–74longitudinal, see Longitudinal wave equationfor membranes, 111–112simple harmonic solutions of, 75–76transverse, see Transverse wave equationWave motion, 16Wave nature of sound, 13–15Waveguideboundary condition at driving end of, 144with constant cross-section, 140–143rigid-walled circular, 144Wavelength, 18Waves, 31–39complex, 33–36energy flux density of, 434forward propagating plane, 31–33hemispherical, 65one-dimensional, 620in pipes, 155–158plane, 620reflection of, at boundaries, 74–75spherical, 64standing, see Standing wavesWebster, Arthur Gordon, 5Weighting curves, 52–55Weighting functions, 196–199Welding, ultrasonic, 489Wever-Bray effect, 219White noise, 228, 615Wind instruments, 521, 522, 536–553Window duration, 196Window error, 196Workplace, noise sources in, 357–358Workplace noise exposure, 346–347Worst noise hour, 336Xylophone, 553Young’s modulus, 18, 91, 629Young, Thomas, 6Zither, 529

THE SCIENCE AND APPLICATIONS OF ACOUSTICS - H. H. Arnold ...

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?