= Solution
Let $\widehat\eta$ be the displacement of a material interface and use $[f]_-^+=f_+-f_-$. On each side, the <kinematic boundary condition> is
$$
\widehat w_\pm=ik(\overline U_\pm-c)\widehat\eta.
$$
There is one displaced material interface, so its displacement is continuous. Thus \b[the first jump condition is]
$$
\boxed{\left[\frac{\widehat w}{\overline U-c}\right]_-^+=0.}
$$
In particular, <vertical velocity> need not be continuous when the background <velocity> jumps.
With no <surface tension>, <pressure continuity> applies on the displaced interface, rather than at its undisplaced height. Linearizing gives $[\widehat p+\widehat\eta\,\overline p']_-^+=0$. Since $\overline p'=-g\overline\rho$, this becomes $[\widehat p-g\overline\rho\widehat\eta]_-^+=0$. The horizontal momentum calculation in part (a) gives
$$
\widehat p=\frac{\rho_0}{ik}\bigl[(\overline U-c)\widehat w'-\overline U'\widehat w\bigr].
$$
Substitute this and $\widehat\eta=\widehat w/[ik(\overline U-c)]$ to obtain \b[the second jump condition]:
$$
\boxed{\left[(\overline U-c)\widehat w'-\overline U'\widehat w-\frac{g\overline\rho}{\rho_0}\frac{\widehat w}{\overline U-c}\right]_-^+=0.}
$$
These <jump conditions for stratified inviscid shear flow> use one-sided values of $\overline U'$ and apply to jumps of <mass density>, <vorticity>, or <velocity>. If $\overline U$ is continuous, the first condition reduces to continuity of $\widehat w$; a jump of <vorticity> generally still prevents continuity of $\widehat w'$. This derivation avoids multiplying singular distributional derivatives of a discontinuous <velocity> in the <Taylor–Goldstein equation>.
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