 Solution

For a short channel slice, the fluid volume per unit length is $b(H+\eta )$ and the axial volume flux is $b(H+\eta )u$. The quasi-one-dimensional conservation of mass equation is thereforeNeglecting products of perturbations gives ${\eta }_t+(H/b)(bu{)}_x=0$. This uses the section-averaged approximation stipulated in the problem; small transverse velocities ensure impermeability of varying-width walls but are not leading axial terms.

The Euler equations are ${\partial }_t\mathbf{u}+(\mathbf{u}\cdot \nabla )\mathbf{u}=-{\rho }^{-1}\nabla p-g{\mathbf{e}}_z$. For irrotational flow, $\mathbf{u}=\nabla \varphi$ and $(\mathbf{u}\cdot \nabla )\mathbf{u}=\nabla (|\nabla \varphi {|}^2/2)$, soAt the zero-pressure free surface, linearization about rest gives ${\varphi }_t+g\eta =C(t)$; differentiate in $x$ to get $u_t+g{\eta }_x=0$. Differentiating the mass equation in time and substituting givesPut $c_0=\sqrt{gH}$. For the exponential width, a time-harmonic amplitude satisfiesWriting $N=e^{-\mu x/2}V$ removes the first derivative and gives $V^{\prime \prime }+k^{2}V=0$, where $k^2={\omega }^2/c_0^2-{\mu }^2/4$. A nonzero real wavenumber, and hence a spatially travelling phase, exists precisely whenFor this case an outward-going solution matching the oscillating boundary isSubstitution in both linear equations verifies it, and $\mathrm{Im}(U(0)e^{i\omega t})=u_0\sin \omega t$. Below the cutoff the spatial exponents are real, giving evanescent responses; at equality the repeated-root solution has no travelling spatial phase.

This is the cutoff frequency in an exponentially widening shallow-water channel.