A stationary inviscid fluid of thickness $h$ and density $\rho$ is located below a free surface at $y=h$ and above a deep layer of inviscid fluid of the same density in $y<0$ flowing with uniform velocity $U>0$ in the $\mathbf{e}_{x}$ direction. The base velocity profile is thus

$$u=U, y<0 ; \quad u=0,0<y<h$$

while the free surface at $y=h$ is maintained flat by gravity.

By considering small perturbations of the vortex sheet at $y=0$ of the form $\eta=\eta_{0} e^{i k x+\sigma t}, k>0$, calculate the dispersion relationship between $k$ and $\sigma$ in the irrotational limit. By explicitly deriving that

$$\operatorname{Re}(\sigma)=\pm \frac{\sqrt{\tanh (h k)}}{1+\tanh (h k)} U k$$

deduce that the vortex sheet is unstable at all wavelengths. Show that the growth rates of the unstable modes are approximately $U k / 2$ when $h k \gg 1$ and $U k \sqrt{h k}$ when $h k \ll 1$.