= Square-basin topographic Rossby mode
{title2=$\omega^2=\beta^2/[4(\pi^2(m^2+n^2)/L^2+L_D^{-2})]$}
In a square with impermeable walls and background topographic <potential-vorticity gradient> $\beta$, the linear <shallow-water quasi-geostrophic potential vorticity> equation is $\partial_t(\nabla^2-L_D^{-2})\psi+\beta\psi_x=0$. For a nonzero-frequency <normal mode>, setting $\widehat\psi=\psi e^{i\beta x/(2\omega)}$ gives a <Helmholtz equation> with homogeneous wall data. The <eigenfunctions> are $\widehat\psi=\sin(m\pi x/L)\sin(n\pi y/L)$, with $m,n\geq1$, and the displayed frequencies. The positive and negative frequency representations are complex conjugates for a real wave.
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