= Solution
Set $v=0$. For $F_+(x+ct)$ the equations require $\widetilde\phi_+=-c\widetilde u_+$ and $\widetilde u_+'=(f/c)\widetilde u_+$; for $F_-(x-ct)$ they require $\widetilde\phi_-=c\widetilde u_-$ and $\widetilde u_-'=-(f/c)\widetilde u_-$. Hence the channel supports the <Kelvin waves>
$$
\boxed{
\widetilde u_\pm=C_\pm e^{\pm fy/c},
\qquad
\widetilde\phi_\pm=\mp c\widetilde u_\pm.}
$$
Each wave propagates along one wall and is exponentially trapped toward that wall on the deformation scale $c/|f|$.
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