Vector Wavenumber

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

$$p(t,\underline{x}) \eqsp p_0 + A\cos\left(\omega t - \underline{k}^T\underline{x}+ \phi\right), \quad \underline{x}\in\mathbb{R}^3 \protect$$ (B.50)

where p(t,x) is the pressure at time $t$ (seconds) and position $\underline{x}\in\mathbb{R}^3$ (3D Euclidean space). The amplitude $A$ , phase $\phi$ , and radian frequency $\omega$ are ordinary sinusoid parameters [454], and $\underline{k}$ is the vector wavenumber:

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

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 $(\underline{u}^T\underline{x})\underline{u}$ [454].^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.50) at time $t=0$ , with unit amplitude $A=1$ and zero phase $\phi=0$ . The spatial phase of this plane wave is given by

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

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

$$\alpha x + \beta y + \gamma z =$$   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 $\mathbb{R}^3$ .

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

$$\underline{\nabla }\theta(\underline{x}) \isdefs \left[\begin{array}{c} \frac{\partial}{\partial x} \\ [2pt] \frac{\partial}{\partial y} \\ [2pt] \frac{\partial}{\partial z}\end{array}\right] \theta(\underline{x}) \eqsp \left[\begin{array}{c} k_x \\ [2pt] k_y \\ [2pt] k_z\end{array}\right] \isdefs \underline{k}$$

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.,

$$k \isdefs d_{\underline{\nabla \theta}}\theta(\underline{x}) \isdefs \lim_{\delta\to 0} \frac{\theta\left(\underline{x}+ \delta\underline{\nabla \theta}\right) - \theta(\underline{x})}{\delta \left\Vert\,\underline{\nabla \theta}\,\right\Vert}.$$

An explicit calculation yields

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

as needed.

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