 Solution

For $U=y$, Rayleigh's equation reduces to ${\varphi }^{\prime \prime }-k^2\varphi =0$, so $\varphi =A\cosh ky+B\sinh ky$. Applying the free-surface condition at $y=\pm 1$ and setting the determinant to zero givesFor $k>0$, $k-\tanh k>0$, so instability occurs exactly when $c^2<0$, orThus $0<k<k_c$, where the unique positive cutoff satisfies $\boxed{k_c\tanh k_c=1}$.