A planar flow of an inviscid, incompressible fluid is everywhere in the $x$-direction and has velocity profile

$$u=\left\{\begin{array}{cc} U & y>0, \\ 0 & y<0 . \end{array}\right.$$

By examining linear perturbations to the vortex sheet at $y=0$ that have the form $e^{i k x-i \omega t}$, show that

$$\omega=\frac{1}{2} k U(1 \pm i)$$

and deduce the temporal stability of the sheet to disturbances of wave number $k$.

Use this result to determine also the spatial growth rate and propagation speed of disturbances of frequency $\omega$ introduced at a fixed spatial position.