Pressure matching at a piecewise-linear shear interface ($[(U-c){\varphi }^{\prime }-U^{\prime }\varphi ]=0$)

For a normal mode of an inviscid parallel flow, the perturbation pressure is $p=-(U-c){\varphi }^{\prime }+U^{\prime }\varphi$. At an interface where $U$ is continuous but $U^{\prime }$ jumps, continuity of normal velocity and pressure gives continuity of $\varphi$ and the displayed flux. Consequently $(U-c)[{\varphi }^{\prime }]=[U^{\prime }]\varphi$. A sinuous mode of a planar jet may therefore have a cusp at a symmetric velocity-profile corner: even parity alone does not force its one-sided derivative to vanish.