= Solution
Set $c=\sqrt{gH_0}$ and seek
$$
\eta=G(x)F(y+\gamma t),\qquad u=0.
$$
The two remaining prognostic equations imply
$$
v=-\frac g\gamma\eta,\qquad
\gamma^2=gH_0=c^2.
$$
The $x$ momentum equation is now geostrophic:
$$
fv=g\eta_x.
$$
Therefore
$$
\frac{G'}G=-\frac f\gamma.
$$
Boundedness for $x>0$ selects $\gamma=\operatorname{sgn}(f)c$ and
$$
\boxed{
G(x)=G(0)e^{-x/L_R}},
\qquad
\boxed{\gamma=\operatorname{sgn}(f)\sqrt{gH_0}}.
$$
Because $F$ depends on $y+\gamma t$, the wave travels with meridional phase speed $-\gamma$. Along this western boundary it travels southward in the Northern Hemisphere and northward in the Southern Hemisphere, keeping the coast on the dynamically required side. This is a coastal <Kelvin wave>.
Moreover,
$$
v=-\frac{\gamma}{H_0}\eta,\qquad
v_x=\frac f{H_0}\eta.
$$
Since $u=0$, its linear potential-vorticity disturbance is
$$
\boxed{q=v_x-u_y-\frac f{H_0}\eta=0}.
$$
Solved by gpt-5.6-sol high.
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