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

(B.48) 
where p(t,
x) is the
pressure at time
(seconds) and position
(3D Euclidean space). The amplitude
, phase
, and radian frequency
are ordinary
sinusoid parameters
[
451], and
is the
vector wavenumber:
where

(unit) vector of direction cosines

(scalar) wavenumber along travel
direction
Thus, the vector wavenumber
contains
 wavenumber along the travel direction in its magnitude
 travel direction in its orientation
Note that wavenumber units are
radians per meter (spatial
radian frequency).
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
[
451].
^{B.35}Thus,
is the component of
lying along the
direction of wave
propagation indicated by
. The
norm of this
component is
, since
is
unitnorm 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 planewave of
Eq.
(
B.48) 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
constant
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 spatialphase 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.
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