Since solving the
wave equation in 2D has all the essential features
of the 3D case, we will look at the 2D case in this section.
Specializing Eq.
(
B.47) to 2D, the
2D wave equation may
be written as
where
The 2D
wave equation is obeyed by
traveling sinusoidal plane
waves having any amplitude
, radian frequency
, phase
, and direction
:
where
denotes the vectorwavenumber,
denotes
the wavenumber (spatial radian frequency) of the wave along its
direction of travel, and
is a unit vector of direction
cosines. This is the
analyticsignal form of a sinusoidal
traveling plane wave, and we may define the real (physical)
signal as
the real part of the analytic signal, as usual [
451].
We see that the only constraint imposed by the wave equation on this
general
travelingwave is the socalled
dispersion relation:
In particular, the wave can travel in any direction, with any
amplitude, frequency, and phase. The only constraint is that its
spatial frequency is tied to its temporal frequency
by
the dispersion relation.
^{B.36}
The sum of two such waves traveling in opposite directions with the
same amplitude and frequency produces a
standing wave. For example,
if the waves are traveling parallel to the
axis, we have

(B.49) 
which is a
standing wave along
.
Next Section: 2D
Boundary ConditionsPrevious Section: Vector Wavenumber