 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 ${\partial }_{t}v+(v\cdot \nabla )v=-\nabla p/\rho -g\hat{z}$. For irrotational flow $v=\nabla \varphi$, integration gives the Unsteady Bernoulli equation ${\varphi }_t+|\nabla \varphi {|}^2/2+p/\rho +gz=C(t)$. On the zero-pressure surface $z=\eta$, linearization gives ${\varphi }_t+g\eta =C(t)$. 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 getFor $b=b_0{e}^{\mu x}$ this is ${\eta }_{tt}=gH({\eta }_{xx}+\mu {\eta }_x)$. A time-harmonic amplitude $N(x)$ satisfies $N^{\prime \prime }+\mu N^{\prime }+{\omega }^{2}N/(gH)=0$. Write $N=e^{-\mu x/2}W$ to obtainSpatially oscillating waves therefore occur exactly when the coefficient is positive: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 $U(x)=u_0{e}^{-\mu x/2}{e}^{-i\kappa x}$ 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 $x>0$.