 Solution

Let $\hat{\eta }$ be the displacement amplitude of an interface. Its kinematic boundary condition is $ik(U-c)\hat{\eta }=\hat{w}$, on either side, so the first of the jump conditions for stratified inviscid shear flow isFor zero surface tension, pressure is continuous at the displaced interface. Since the background hydrostatic pressure has derivative $-g\rho$, linearization gives $[\hat{p}-g\rho \hat{\eta }{]}_{-}^{+}=0$. Using $\hat{p}={\rho }_0[(U-c){\hat{w}}^{\prime }-U^{\prime }\hat{w}]/(ik)$ and the kinematic condition givesThis derivation also works when $U$ jumps: the displacement is shared, while the vertical velocity need not be. It avoids ambiguous products of distributions at a discontinuous velocity profile.