Consider a two-dimensional horizontal vortex sheet of strength $U$ at height $h$ above a horizontal rigid boundary at $y=0$, so that the inviscid fluid velocity is

$$\mathbf{u}= \begin{cases}(U, 0) & 0<y<h \\ (0,0) & y>h\end{cases}$$

Examine the temporal linear instabililty of the sheet and determine the relevant dispersion relationship.

For what wavelengths is the sheet unstable?

Evaluate the temporal growth rate and the wave propagation speed in the limit of both short and long waves. Comment briefly on the significance of your results.