The wave equation on a plate in two dimensions
f_{tt}(t,x,y) = f_{xx}(t,x,y) + f_{yy}(t,x,y)

can be solved similar as the one dimensional case. Take the basis
f_{n,m} = sin(n x) sin(m y) for functions on the square
[0,pi] x [0,pi] which vanish at the boundary. The wave equation
can be written as f_{tt} = A f. Because
A f_{n,m}=(n^{2}m^{2}) f_{n,m}

we have
f_{n,m}(t)=cos(k t)f_{n,m}(0)+sin(k t)f'_{n,m}(0)/k

with k=(n,m). The general solution is a superposition of such waves.
The wave is periodic in time only if all pairs (n,m) with nonzero Fourier coefficients
are Pythagorean pairs: n^{2} + m^{2} = k^{2}.

