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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/iii/paper-331/2/i/a/solution