Solution
= Solution
The linearized vorticity equation for a <inviscid parallel shear flow> is
$$
(\partial_t+U\partial_x)\nabla^2\psi-U''\partial_x\psi=0.
$$
Substituting $\psi=\phi(y)e^{ik(x-ct)}$ gives <Rayleigh equation for inviscid shear flow>[Rayleigh's equation]
$$
\boxed{(U-c)(\phi''-k^2\phi)-U''\phi=0}.
$$