= Cutoff frequency in an exponentially widening shallow-water channel
{title2=$\omega_c=\mu\sqrt{gH}/2$}
In a quasi-one-dimensional inviscid channel with depth $H$ and width $b(x)=b_0e^{\mu x}$, the linear <shallow water equations> are $\eta_t+(H/b)(bu)_x=0$ and $u_t+g\eta_x=0$. Combining them gives $\eta_{tt}=gH(\eta_{xx}+\mu\eta_x)$.
For a time dependence $e^{i\omega t}$, write the elevation amplitude as $N(x)=e^{-\mu x/2}V(x)$. The spatial equation becomes
$$
V''+\left(\frac{\omega^2}{gH}-\frac{\mu^2}4\right)V=0.
$$
There is a travelling spatial phase only when the bracket is positive, so the <cutoff frequency> is $\omega_c=\mu\sqrt{gH}/2$. Above it, an outgoing wave has a phase proportional to $\omega t-kx$ and amplitude proportional to $e^{-\mu x/2}$; below it the spatial response is evanescent. Geometric widening introduces the cutoff even though a constant-width channel has the nondispersive speed $\sqrt{gH}$.
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