Mathematically, a sinusoidal plane wave, as in Fig.B.9 or Fig.B.10, can be written as
where
To see that the vector wavenumber has the claimed properties, consider that the orthogonal projection of any vector onto a vector collinear with is given by [454].^{B.35}Thus, is the component of lying along the direction of wave propagation indicated by . The norm of this component is , since is unit-norm by construction. More generally, is the signed length (in meters) of the component of along . This length times wavenumber gives the spatial phase advance along the wave, or, .
For another point of view, consider the plane wave , which is the varying portion of the general plane-wave of Eq. (B.50) at time , with unit amplitude and zero phase . The spatial phase of this plane wave is given by
Recall that the general equation for a plane in 3D space is
where , , and are real constants, and , , and are 3D spatial coordinates. Thus, the set of all points yielding the same value define a plane of constant phase in .
As we know from elementary vector calculus, the direction of maximum phase advance is given by the gradient of the phase :
This shows that the vector wavenumber is equal to the gradient of the phase , so that points in the direction of maximum spatial-phase advance.
Since the wavenumber 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 along , i.e.,
An explicit calculation yields
as needed.
Scattering of plane waves is discussed in §C.8.1.