2D Boundary Conditions

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 standing-wave products having zero displacement (``nodes'') along all four boundary lines:

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

It remains to be shown that the set of functions is

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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Center for Computer Research in Music and Acoustics (CCRMA), Stanford University