= Solution
Assume a small <Rossby number>, a <beta plane>, and small fractional changes
$$
\frac{|\beta y|}{f_0},\quad
\frac{|\zeta|}{f_0},\quad
\frac{|h_b|}{H_0},\quad
\frac{|\eta|}{H_0}\ll1.
$$
At leading order, <geostrophic balance> gives
$$
\mathbf u_g=(-\psi_y,\psi_x),
\qquad
\zeta=\nabla^2\psi,
\qquad
\eta=\frac{f_0}{g}\psi.
$$
Expanding the reciprocal depth in the <shallow-water potential vorticity> gives
$$
q=\frac{f_0+\beta y+\zeta}{H_0-h_b+\eta}
\simeq\frac{f_0}{H_0}
\left(1+\frac{\beta y}{f_0}
+\frac{\nabla^2\psi}{f_0}
+\frac{h_b}{H_0}
-\frac{f_0\psi}{gH_0}
\right).
$$
This is the stated <shallow-water quasi-geostrophic potential vorticity> $P_g$. The term $\nabla^2\psi$ is the parcel's relative vorticity, while $-f_0^2\psi/(gH_0)=-\psi/L_D^2$ is vortex stretching caused by free-surface displacement, where
$$
L_D=\frac{\sqrt{gH_0}}{f_0}
$$
is the <barotropic deformation radius>.
For $h_b=\alpha y$, omit the irrelevant constant factor $f_0/H_0$. The active PV anomaly is
$$
Q=\nabla^2\psi-\frac{\psi}{L_D^2}
+\left(\beta+\frac{f_0\alpha}{H_0}\right)y.
$$
Linearizing $Q_t+J(\psi,Q)=0$ about rest and using $\psi\propto e^{i(kx+ly-\omega t)}$ gives the <Topographic Rossby-wave dispersion relation>
$$
\boxed{
\omega=-\frac{(\beta+f_0\alpha/H_0)k}
{k^2+l^2+L_D^{-2}}}.
$$
A 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 $Dq/Dt=0$ gives
$$
\boxed{
\partial_t\nabla^2\psi
+\left(\beta-f\frac{h_y}{h}\right)\psi_x=0}.
$$
For $h=H_0e^{\gamma y}$ and over a region where $f\simeq f_0$, plane waves obey
$$
\boxed{
\omega=-\frac{(\beta-f_0\gamma)k}{k^2+l^2}}.
$$
The 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 $\beta-f\gamma$.
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