 Cutoff frequency in an exponentially widening shallow-water channel

In a quasi-one-dimensional inviscid channel with depth $H$ and width $b(x)=b_0{e}^{\mu x}$, 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 becomesThere 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 $\sqrt{gH}$.