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.
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