Vector Wavenumber

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Vector Wavenumber

Mathematically, a sinusoidal plane wave, as in Fig.F.3 or Fig.F.4, can be written as

$\displaystyle p(t,\underline{x}) = p_0 + A\cos\left(\omega t - \underline{k}^T\underline{x}+ \phi\right), \quad \underline{x}\in{\bf R}^3 \protect$

(F.9)

where the amplitude

, phase $\phi$ , and radian frequency $\omega$ are ordinary sinusoid parameters [462], $\underline{x}$ is position in 3D space, and $\underline{k}$ is the vector wavenumber:

$\displaystyle \underline{k}= \left[\begin{array}{c} k_x \\ [2pt] k_y \\ [2pt] k... ...] \cos{\beta} \\ [2pt] \cos{\gamma}\end{array}\right] \isdef k\,\underline{u},$

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

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

To see that the vector wavenumber $\underline{k}= k\,\underline{u}$ has the claimed properties, consider that the orthogonal projection of any vector $\underline{x}$ onto a vector collinear with $\underline{u}$ is given by $(\underline{u}^T\underline{x})\underline{u}$ [462].

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Physical Audio Signal Processing
    Elementary Physics, Mechanics, and Acoustics
       Properties of Gases
          Vector Wavenumber