= Pinned-square surface displacement mode
{title2=$k_{nm}=\pi\sqrt{n^2+m^2}/L$}
A surface displacement constrained to vanish on all four sides of an ideal square has sine factors $\sin(n\pi x/L)\sin(m\pi y/L)$, with $n,m\geq1$. The fundamental therefore has <wavenumber> $\sqrt2\pi/L$. This differs from the impermeable-wall condition in a closed tank, which permits cosine modes with a zero index. For an elastic plate a simply supported edge also supplies its bending <boundary condition>; clamping imposes a different problem.
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