= Solution
Let $\widehat\eta$ be the displacement amplitude of an interface. Its <kinematic boundary condition> is $ik(U-c)\widehat\eta=\widehat w$, on either side, so the first of the <jump conditions for stratified inviscid shear flow> is
$$
\boxed{\left[\frac{\widehat w}{U-c}\right]_-^+=0.}
$$
For zero <surface tension>, pressure is continuous at the displaced interface. Since the background <hydrostatic pressure> has derivative $-g\rho$, linearization gives $[\widehat p-g\rho\widehat\eta]_-^+=0$. Using $\widehat p=\rho_0[(U-c)\widehat w'-U'\widehat w]/(ik)$ and the kinematic condition gives
$$
\boxed{\left[(U-c)\widehat w'-U'\widehat w-\frac{g\rho}{\rho_0}\frac{\widehat w}{U-c}\right]_-^+=0.}
$$
This 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.
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