### 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

Note that we can also use *products* of horizontal and vertical
standing waves

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

*orthogonal*, the solution building-blocks are orthogonal under the inner product

*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:

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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3D Sound

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Solving the 2D Wave Equation