Square-basin topographic Rossby mode (${\omega }^2={\beta }^2/[4({\pi }^2(m^2+n^2)/L^2+L_D^{-2})]$)

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 $\hat{\psi }=\psi e^{i\beta x/(2\omega )}$ gives a Helmholtz equation with homogeneous wall data. The eigenfunctions are $\hat{\psi }=\sin (m\pi x/L)\sin (n\pi y/L)$, with $m,n\ge 1$, and the displayed frequencies. The positive and negative frequency representations are complex conjugates for a real wave.