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.51), 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:
By construction, all linear combinations of the form Eq.(B.52) are solutions of the wave equation that satisfy the zero boundary conditions along the rectangle - - . Since sinusoids at different frequencies are orthogonal, the solution building-blocks 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 orthogonal 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.