Plane Wave at an Angle

Planar pressure-wave traveling in an arbitrary direction:

$$p(t,\underline{x}) = \cos\left(\omega t - \underline{k}^T\underline{x}\right), \quad \underline{x}\in{\bf R}^3$$

where $\underline{k}=$ vector wavenumber:

$$\underline{k}= \left[\begin{array}{c} k_x \\ [2pt] k_y \\ [2pt] k... ...ma}\end{array}\right] \mathrel{\stackrel{\mathrm{\Delta}}{=}}k\,\underline{u},$$

where

• $\underline{u}=$ (unit) vector of direction cosines
• $k = \frac{2\pi}{\lambda} =$ (scalar) wavenumber along travel direction

Thus, the vector wavenumber $\underline{k}= k\,\underline{u}$ contains

• wavenumber in its magnitude $k=\left\Vert\,\underline{k}\,\right\Vert$
• travel direction in its orientation $\underline{u}= \underline{k}/k$

Note: wavenumber units are radians per meter
(spatial radian frequency)