= 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 therefore
$$
\partial_t[b(H+\eta)]+\partial_x[b(H+\eta)u]=0.
$$
Neglecting 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>, $\mathbf u=\nabla\phi$ and $(\mathbf u\cdot\nabla)\mathbf u=\nabla(|\nabla\phi|^2/2)$, so
$$
\phi_t+\frac12|\nabla\phi|^2+\frac p\rho+gz=C(t).
$$
At the zero-pressure free surface, linearization about rest gives $\phi_t+g\eta=C(t)$; differentiate in $x$ to get $u_t+g\eta_x=0$. Differentiating the mass equation in time and substituting gives
$$
\boxed{\eta_{tt}=\frac{gH}b(b\eta_x)_x}.
$$
Put $c_0=\sqrt{gH}$. For the exponential width, a time-harmonic amplitude satisfies
$$
N''+\mu N'+\frac{\omega^2}{c_0^2}N=0.
$$
Writing $N=e^{-\mu x/2}V$ removes the first derivative and gives $V''+k^2V=0$, where $k^2=\omega^2/c_0^2-\mu^2/4$. A nonzero real wavenumber, and hence a spatially travelling phase, exists precisely when
$$
\boxed{\omega>\frac\mu2\sqrt{gH}}.
$$
For this case an outward-going solution matching the oscillating boundary is
$$
U(x)=u_0e^{-\mu x/2-ikx},\qquad
N(x)=\frac H\omega(k+i\mu/2)U(x).
$$
Substitution in both linear equations verifies it, and $\operatorname{Im}(U(0)e^{i\omega t})=u_0\sin\omega t$. 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>.
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