= Solution
Conservation of volume in a short channel slice gives $\partial_t[b(H+\eta)]+\partial_x[b(H+\eta)u]=0$. Keeping first-order terms in $u,\eta$ gives $\boxed{\eta_t+(H/b)(bu)_x=0}$.
Euler's momentum equation is $\partial_tv+(v\cdot\nabla)v=-\nabla p/\rho-g\hat z$. For irrotational flow $v=\nabla\phi$, integration gives the <Unsteady Bernoulli equation> $\phi_t+|\nabla\phi|^2/2+p/\rho+gz=C(t)$. On the zero-pressure surface $z=\eta$, linearization gives $\phi_t+g\eta=C(t)$. Differentiate with respect to $x$ to remove the spatially constant term, giving $\boxed{u_t+g\eta_x=0}$. These are the cross-sectionally averaged shallow-water approximations encoded in the stated assumptions.
Differentiate the <mass> equation in time and use the momentum equation to get
$$
\boxed{\eta_{tt}=\frac{gH}b(b\eta_x)_x.}
$$
For $b=b_0e^{\mu x}$ this is $\eta_{tt}=gH(\eta_{xx}+\mu\eta_x)$. A time-harmonic amplitude $N(x)$ satisfies $N''+\mu N'+\omega^2N/(gH)=0$. Write $N=e^{-\mu x/2}W$ to obtain
$$
W''+\left(\frac{\omega^2}{gH}-\frac{\mu^2}4\right)W=0.
$$
Spatially oscillating waves therefore occur exactly when the coefficient is positive:
$$
\boxed{\omega>\frac\mu2\sqrt{gH}.}
$$
Below threshold the solutions have real exponential spatial factors and are evanescent rather than propagating; equality gives the zero-<wavenumber> threshold. For a concrete outgoing wave let $\kappa=\sqrt{\omega^2/(gH)-\mu^2/4}>0$. Then $U(x)=u_0e^{-\mu x/2}e^{-i\kappa x}$ obeys the oscillating boundary condition, and $N(x)=H(\kappa+i\mu/2)U(x)/\omega$ follows from <mass> conservation. The phase $\omega t-\kappa x$ moves into $x>0$.
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