= Solution
Let the disturbed sheet be $y=h+\eta$ and take a <normal mode> proportional to $e^{i(kx-\omega t)}$, with $K=|k|$. <Irrotational flow> and the rigid-wall condition at $y=0$ give perturbation <velocity potentials>
$$
\phi_-'=A\cosh(Ky)e^{i(kx-\omega t)},
\qquad
\phi_+'=Be^{-K(y-h)}e^{i(kx-\omega t)}.
$$
The two linearized <kinematic boundary conditions> at $y=h$ are
$$
-i(\omega-kU)\eta=AK\sinh(Kh),
\qquad
-i\omega\eta=-KB.
$$
The <Unsteady Bernoulli equation> gives $p_-'=i\rho(\omega-kU)A\cosh(Kh)$ and $p_+'=i\rho\omega B$. Continuity of pressure therefore yields the <finite-depth vortex-sheet dispersion relation>
$$
\boxed{(\omega-kU)^2\coth(Kh)+\omega^2=0},
$$
or, with $T=\tanh(Kh)$,
$$
(\omega-kU)^2+T\omega^2=0.
$$
For $k>0$ its two roots are
$$
\boxed{\omega=\frac{kU}{1+T}\left(1\pm i\sqrt T\right)}.
$$
One root has positive temporal growth for every $k\ne0$, so every finite <wavelength> is subject to the <Kelvin-Helmholtz instability>.
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