Standing modes in an exponentially tapered channel ($k_n^2=(n\pi /L{)}^2+{\beta }^2/4$)

For constant long-wave speed and $S_0\propto e^{\beta x}$, a standing free-surface amplitude obeys ${\eta }^{\prime \prime }+\beta {\eta }^{\prime }+k^2\eta =0$. The transformation $\eta =e^{-\beta x/2}Y$ gives $Y^{\prime \prime }+(k^2-{\beta }^2/4)Y=0$. Impermeable ends at $0,L$ impose ${\eta }^{\prime }=0$, yielding the displayed nonzero-frequency eigenvalues and ${\eta }_n\propto e^{-\beta x/2}[\cos (n\pi x/L)+\beta L\sin (n\pi x/L)/(2n\pi )]$. A separate zero-frequency mode changes the uniform water level.