 Solution

Set $v=0$. For $F_{+}(x+ct)$ the equations require ${\tilde{\varphi }}_{+}=-c{\tilde{u}}_{+}$ and ${\tilde{u}}_{+}^{\prime }=(f/c){\tilde{u}}_{+}$; for $F_{-}(x-ct)$ they require ${\tilde{\varphi }}_{-}=c{\tilde{u}}_{-}$ and ${\tilde{u}}_{-}^{\prime }=-(f/c){\tilde{u}}_{-}$. Hence the channel supports the Kelvin wavesEach wave propagates along one wall and is exponentially trapped toward that wall on the deformation scale $c/|f|$.