The solutions of the wave equation for a circular drum with fixed boundary are well known. What do the solutions look like for a spherical drum in three spatial dimensions? What about higher dimensions?

You're asking about eigenfunctions of the Laplacian on a sphere. The keyword "spherical harmonic" will be useful in your search.
– Aaron HoffmanOct 4 '11 at 16:16

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The 3d "drum" is a ball and is fixed on the boundary (sphere). So solutions will be time dependent functions on the 3d ball which are constantly 0 on the spherical boundary.
– antianticamperOct 4 '11 at 16:43

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Sorry. I meant (Dirichlet) eigenfunctions of the Laplacian on the ball. Take e.g. section 10.3 in Strauss.
– Aaron HoffmanOct 4 '11 at 17:50