Solution
= Solution
For $U=y$, Rayleigh's equation reduces to $\phi''-k^2\phi=0$, so $\phi=A\cosh ky+B\sinh ky$. Applying the free-surface condition at $y=\pm1$ and setting the determinant to zero gives
$$
\boxed{c^2=\frac{(k\tanh k-1)(k-\tanh k)}{k^2\tanh k}}.
$$
For $k>0$, $k-\tanh k>0$, so instability occurs exactly when $c^2<0$, or
$$
k\tanh k<1.
$$
Thus $0<k<k_c$, where the unique positive cutoff satisfies $\boxed{k_c\tanh k_c=1}$.