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Physical Audio Signal Processing
    Elementary Physics, Mechanics, and Acoustics
       Wave Equation in Higher Dimensions
          Vector Wavenumber

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Vector Wavenumber

Mathematically, a sinusoidal plane wave, as in Fig.B.9 or Fig.B.10, can be written as

$\displaystyle p(t,\underline{x}) \eqsp p_0 + A\cos\left(\omega t - \underline{k}^T\underline{x}+ \phi\right), \quad \underline{x}\in{\bf R}^3 \protect$ (B.48)

where p(t,x) is the pressure at time $ t$ (seconds) and position $ \underline{x}\in{\bf R}^3$ (3D Euclidean space). The amplitude $ A$, phase $ \phi$, and radian frequency $ \omega $ are ordinary sinusoid parameters [451], and $ \underline{k}$ is the vector wavenumber:

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

where
  • $ \underline{u}= $ (unit) vector of direction cosines
  • $ k = 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}$ [451].B.35Thus, $ (\underline{u}^T\underline{x})\underline{u}$ is the component of $ \underline{x}$ lying along the direction of wave propagation indicated by $ \underline{u}$. The norm of this component is $ \vert\vert\,(\underline{u}^T\underline{x})\underline{u}\,\vert\vert =\vert\underline{u}^T\underline{x}\vert$, since $ \underline{u}$ is unit-norm by construction. More generally, $ \underline{u}^T\underline{x}$ is the signed length (in meters) of the component of $ \underline{x}$ along $ \underline{u}$. This length times wavenumber $ k$ gives the spatial phase advance along the wave, or, $ \theta(\underline{x})=k\cdot(\underline{u}^T\underline{x}) \isdeftext \underline{k}^T\underline{x}$.

For another point of view, consider the plane wave $ \cos(\underline{k}^T\underline{x})$, which is the varying portion of the general plane-wave of Eq.$ \,$(B.48) at time $ t=0$, with unit amplitude $ A=1$ and zero phase $ \phi=0$. The spatial phase of this plane wave is given by

$\displaystyle \theta(\underline{x}) \isdefs \underline{k}^T\underline{x}\eqsp k_x x + k_y y + k_z z. $

Recall that the general equation for a plane in 3D space is

$\displaystyle \alpha x + \beta y + \gamma z =$   constant

where $ \alpha$, $ \beta$, and $ \gamma$ are real constants, and $ x$, $ y$, and $ z$ are 3D spatial coordinates. Thus, the set of all points $ \underline{x}^T=(x,y,z)$ yielding the same value $ \theta(\underline{x})=\theta_0$ define a plane of constant phase $ \theta_0$ in $ {\bf R}^3$.

As we know from elementary vector calculus, the direction of maximum phase advance is given by the gradient of the phase $ \theta(\underline{x})$:

$\displaystyle \underline{\nabla }\theta(\underline{x}) \isdefs \left[\begin{ar... ...rray}{c} k_x \\ [2pt] k_y \\ [2pt] k_z\end{array}\right] \isdefs \underline{k} $

This shows that the vector wavenumber $ \underline{k}$ is equal to the gradient of the phase $ \theta(\underline{x})$, so that $ \underline{k}$ points in the direction of maximum spatial-phase advance.

Since the wavenumber $ k$ is the spatial frequency (in radians per meter) along the direction of travel, we should be able to compute it as the directional derivative of $ \theta(\underline{x})$ along $ \underline{k}$, i.e.,

$\displaystyle k \isdefs d_{\underline{\nabla \theta}}\theta(\underline{x}) \isd... ...ta(\underline{x})}{\delta \left\Vert\,\underline{\nabla \theta}\,\right\Vert}. $

An explicit calculation yields

$\displaystyle k \eqsp \left\Vert\,\underline{\nabla \theta}\,\right\Vert \eqsp \sqrt{k_x^2+k_y^2+k_z^2} \isdefs \left\Vert\,\underline{k}\,\right\Vert $

as needed.

Scattering of plane waves is discussed in §C.8.1.

Previous: Plane Waves in Air
Next: Solving the 2D Wave Equation

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About the Author: Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.


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