= Potential-vorticity step edge wave
{title2=$c=U_0-\Delta Q/(2\sqrt{k^2+L_R^{-2}})$}
A positive jump $\Delta Q$ of shallow-water <quasi-geostrophic potential vorticity> supports a current $U=U_0e^{-|y|/L_R}$ with $U_0=L_R\Delta Q/2$. A disturbance has structure $\phi=Ae^{-\sqrt{k^2+L_R^{-2}}|y|}$. Integrating the linear <potential-vorticity conservation> equation across the interface gives $(U_0-c)[\phi_y]+\Delta Q\phi(0)=0$, hence the displayed <dispersion relation>. The wave propagates intrinsically against the mean current because the displaced PV interface induces the <velocity> that advects it; no planetary beta is required.
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