The wave equation on a plate in two dimensions
ftt(t,x,y) = fxx(t,x,y) + fyy(t,x,y)
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can be solved similar as the one dimensional case. Take the basis
fn,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 ftt = A f. Because
we have
fn,m(t)=cos(k t)fn,m(0)+sin(k t)f'n,m(0)/k
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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: n2 + m2 = k2.
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