= Solution
Assume $k$ is constant, as for constant resting long-wave speed. With $S_0'/S_0=\beta$, put $\eta=e^{-\beta x/2}Y$. The amplitude equation becomes
$$
Y''+\left(k^2-\frac{\beta^2}{4}\right)Y=0.
$$
If $k^2>\beta^2/4$, 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, $e^{-\beta x/2}(Ae^{\gamma x}+Be^{-\gamma x})$, 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 $\eta'(0)=\eta'(L)=0$. In the oscillatory case, the first condition gives
$$
\eta=Ae^{-\beta x/2}\left[\cos\mu x+\frac{\beta}{2\mu}\sin\mu x\right].
$$
Its derivative at $L$ is $-Ak^2e^{-\beta L/2}\sin(\mu L)/\mu$. Hence the <standing modes in an exponentially tapered channel> have
$$
\boxed{\mu_n=\frac{n\pi}{L},\qquad
k_n^2=\left(\frac{n\pi}{L}\right)^2+\frac{\beta^2}{4},\qquad
\omega_n=c_0k_n,\quad n=1,2,\ldots.}
$$
The 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.
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