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Assume $k$ is constant, as for constant resting long-wave speed. With $S_0^{\prime }/S_0=\beta$, put $\eta =e^{-\beta x/2}Y$. The amplitude equation becomesIf $k^2>{\beta }^2/4$, write $\mu =\sqrt{k^2-{\beta }^2/4}$: the solution is an oscillatory sine-cosine combination multiplied by $e^{-\beta x/2}$. If $k^2={\beta }^2/4$, it is $e^{-\beta x/2}(A+Bx)$. If $k^2<{\beta }^2/4$, it is a combination of real exponentials, $e^{-\beta x/2}(A{e}^{\gamma x}+B{e}^{-\gamma x})$, where $\gamma =\sqrt{{\beta }^2/4-k^2}$. These classify the local spatial behavior.

For a nonzero-frequency mode, zero velocity at both dams and $v_t=-g{\eta }_x$ impose ${\eta }^{\prime }(0)={\eta }^{\prime }(L)=0$. In the oscillatory case, the first condition givesIts derivative at $L$ is $-A{k}^2{e}^{-\beta L/2}\sin (\mu L)/\mu$. Hence the standing modes in an exponentially tapered channel haveThe hyperbolic and repeated-root cases provide no additional nonzero-frequency mode satisfying both end conditions. There is a separate zero-frequency uniform-level mode $\eta =\mathrm{constant}$, corresponding to a change of the basin's conserved water volume.