Rossby-wave isofrequency circle (${\left(k+\beta /(2\omega )\right)}^2+l^2={\beta }^2/(4{\omega }^2)-a^2$)

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.