= Jump conditions for stratified inviscid shear flow
At a material interface without <surface tension>, continuity of displacement and of the pressure evaluated on the displaced interface gives
$$
\left[\frac{\widehat w}{U-c}\right]=0,\qquad
\left[(U-c)\widehat w'-U'\widehat w-\frac{g\rho}{\rho_0}\frac{\widehat w}{U-c}\right]=0.
$$
The first follows from the <kinematic boundary condition> $ik(U-c)\widehat\eta=\widehat w$. The second uses $\widehat p=\rho_0[(U-c)\widehat w'-U'\widehat w]/(ik)$ and hydrostatic displacement $[\widehat p-g\rho\widehat\eta]=0$. These conditions apply to density, velocity and vorticity jumps within the <Boussinesq approximation>.
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