 Solution

Put $q=|k|$. Decay at infinity and the symmetry of the disturbance suggest the mode potentialsThe middle potential is an even function of $y$ because its vertical derivative must take opposite values at the two interfaces. The upper kinematic conditions giveThe lower conditions give the same two relations. The upper dynamic condition isEliminating $A$, $B$ and ${\eta }_0$ yields the dispersion relationWriting $C=\coth (qh)$, its two roots areFor every real $k\ne 0$, $C>0$ and one root has the positive growth rateThus every nonzero wavenumber is linearly unstable in this inviscid vortex sheet model; $k=0$ is neutral.