Pinned-square surface displacement mode ($k_{nm}=\pi \sqrt{n^2+m^2}/L$)

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 $n,m\ge 1$. The fundamental therefore has wavenumber $\sqrt{2}\pi /L$. 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.