The
wave equation in 1D, 2D, or 3D may be written as

(B.47) 
where, in 3D,
denotes the amplitude of the wave at time
and position
, and
denotes the
Laplacian operator in Euclidean coordinates. (In
general coordinates, it is often denoted by
.)
To investigate solutions of the
wave equation, as pursued in
§
B.8.3 below, it is useful to first develop some simple
expressions and notations for elementary waves in 2D and 3D.
Figure
B.9 shows a 2D
crosssection of a snapshot (in time)
of the
sinusoidal plane wave
for
and
, with
and
in the range
.
Figure:
Grayscale density plot of the
crosssection of a sinusoidal plane wave
, at with vector wavenumber
.

Figure
B.10 depicts a more mathematical schematic of a
sinusoidal plane wave traveling toward the upperright of the figure.
The dotted lines indicate the
crests (peak amplitude location)
along the wave.
Figure:
Wave crests of the sinusoidal traveling
plane wave
, for some
fixed time and
in the plane.

The direction of travel and
spatial frequency are indicated by the
vector wavenumber
, as discussed in in the following section.
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.
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
.
We often wish to find solutions of the 2D
wave equation that obey
certain known
boundary conditions. An example is
transverse
waves on an ideal elastic membrane, rigidly clamped on its boundary to
form a rectangle with dimensions
meters.
Similar to the derivation of Eq.
(
B.49), we can
subtract
the second
sinusoidal traveling wave from the first to yield
which satisfies the zero
displacement boundary condition along the
axis. If we restrict the wavenumber
to the set
, where
is any positive integer, then we also satisfy the boundary
condition along the line parallel to the
axis at
. Similar
standing waves along
will satisfy both boundary conditions along
and
.
Note that we can also use
products of horizontal and vertical
standing waves
because, when taking the partial derivative with respect to
, the
term
is simply part of the constant coefficient, and vice
versa.
To build solutions to the
wave equation that obey all of the boundary
conditions, we can form
linear combinations of the above standingwave
products having zero displacement (``
nodes'') along all four boundary
lines:

(B.50) 
where
By construction, all linear combinations of the form Eq.
(
B.50)
are solutions of the
wave equation that satisfy the zero boundary
conditions along the rectangle


. Since
sinusoids at
different frequencies are
orthogonal,
the solution buildingblocks
are orthogonal under the
inner product
It remains to be shown that the set of functions
is
complete, that is, that they form a
basis for the set of
all solutions to the
wave equation satisfying the boundary
conditions. Given that, we can solve the problem of
arbitrary
initial conditions. That is, given any initial
over the
membrane (subject to the boundary conditions, of course), we can find
the amplitude of each excited mode by simple projection:
Showing completeness of the basis
in the desired solution
space is a special case (zero boundary conditions) of the problem of
showing that the 2D
Fourier series expansion is complete in the space
of all continuous rectangular surfaces.
The
Wikipedia page (as of 1/31/10) on the
Helmholtz equation
provides a nice ``entry point'' on the above topics and further
information.
3D Sound
The mathematics of 3D sound is quite elementary, as we will see below.
The hard part of the theory of practical systems typically lies in the
mathematical approximation to the ideal case. Examples include
Ambisonics [
158] and
wave field synthesis
[
49].
Consider a point source at position
. Then the
acoustic complex amplitude at position
is given by
where
denotes the complex amplitude one
meter from the
point source in any direction, and
denotes the wavenumber
(spatial radian frequency). Distributed acoustic sources are
handled as a superposition of point sources, so the point source is a
completely general building block for all types of sources in linear
acoustics.
The fundamental approximation problem in 3D sound is to approximate
the complex acoustic field at one or more listening points using a
finite set of
loudspeakers, which are often modeled as a point
source for each speaker.
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