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On an f-plane, linear shallow-water plane waves have one zero-frequency geostrophic mode and two inertia-gravity modes with ${\omega }^2=f^2+c^2(k^2+l^2)$. The crossover scale $c/|f|$ is the barotropic deformation radius.

Substitution of $e^{i(kx+ly-\omega t)}$ gives a homogeneous three-by-three system. Its determinant issoThe zero-frequency component is in geostrophic balance and carries all conserved $P$. The oscillatory components are inertia-gravity waves with zero $P$. For $K\ll |f|/c$, $\Omega \sim |f|$ and rotation produces an inertial oscillation; for $K\gg |f|/c$, $\Omega \sim cK$ and surface gravity dominates. The crossover length $c/|f|$ is the barotropic deformation radius.

Assume a small Rossby number, a beta plane, and small fractional changesAt leading order, geostrophic balance givesExpanding the reciprocal depth in the shallow-water potential vorticity givesThis is the stated shallow-water quasi-geostrophic potential vorticity $P_g$. The term ${\nabla }^2\psi$ is the parcel's relative vorticity, while $-f_0^2\psi /(g{H}_0)=-\psi /L_D^2$ is vortex stretching caused by free-surface displacement, whereis the barotropic deformation radius.

For $h_b=\alpha y$, omit the irrelevant constant factor $f_0/H_0$. The active PV anomaly isLinearizing $Q_t+J(\psi ,Q)=0$ about rest and using $\psi \propto e^{i(kx+ly-\omega t)}$ gives the Topographic Rossby-wave dispersion relationA northward displacement raises planetary PV when $\beta >0$ and raises topographic PV when the bottom rises northward, $\alpha >0$. PV conservation then requires anticyclonic relative vorticity, producing westward phase propagation. A negative slope opposes $\beta$ and reverses propagation when $f_0\alpha /H_0<-\beta$.

Under a rigid lid, $h=H_0-h_b$ is fixed but need not be close to $H_0$. With $q=(f+{\nabla }^2\psi )/h(y)$, linearization of $Dq/Dt=0$ givesFor $h=H_0{e}^{\gamma y}$ and over a region where $f\simeq f_0$, plane waves obeyThe planetary-vorticity gradient dominates when $|\beta |\gg |f_0\gamma |$, while exponential depth variation dominates when $|f_0\gamma |\gg |\beta |$. If the variation of $f=f_0+\beta y$ across the region is retained, this is the corresponding local WKB approximation with effective gradient $\beta -f\gamma$.