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]
$$
\boxed{(U-c)(\phi''-k^2\phi)-U''\phi=0}.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 331 / 2 / i / a / Solution

An inviscid parallel shear flow has base velocity $\mathbf u=U(y)\mathbf e_x$. Two-dimensional normal-mode perturbations satisfy the <Rayleigh equation for inviscid shear flow>.

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