Shallow-water Rossby-wave dispersion relation ($\omega =-\beta k/(k^2+l^2+a^2)$)

For a resting midlatitude layer, the slow Rossby wave equation is $({\nabla }_h^2-a^2){\eta }_t=-\beta {\eta }_x$, where $a=f_0/\sqrt{gH}$. With the plane wave convention $e^{i(kx+ly-\omega t)}$,The zonal phase velocity is westward for $\beta >0$, but the zonal group velocity changes sign at $k^2=l^2+a^2$.

A stationary straight meridional wall preserves frequency and tangential wavenumber $l$. The two normal wavenumbers obeyIncidence and reflection are distinguished by the normal group velocity. In the leading quasi-geostrophic approximation, a nonzero $l$ and the impermeability condition give reflected height amplitude $A_r=-A_i$. The $l=0$ no-normal-flow condition is degenerate in that reduced model; exact ageostrophic transport generally changes the amplitude ratio.

The wavevectors of a nonzero-frequency Rossby wave in a resting shallow layer satisfyThis follows by completing the square in the shallow-water Rossby-wave dispersion relation. A real circle requires $|\omega |\le |\beta |/(2|a|)$. Its two intersections with a given tangential wavenumber provide the incident and reflected normal wavenumbers at a meridional wall.