Pressure equation for a shear-flow normal mode ($p\prime \prime -{\alpha }^{2}p=-2i\alpha U\prime v$)

For an incompressible parallel flow with base velocity $U(y)$ and perturbation proportional to $e^{i\alpha (x-ct)}$, divergence of the linearized momentum equations gives $p\prime \prime -{\alpha }^{2}p=-2i\alpha U\prime v$. At a no-slip boundary condition, streamwise momentum gives $p(0)=u\prime \prime (0)/(i\alpha R)$. The equation encodes how base shear generates perturbation fluid pressure.