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

The linearized vorticity equation for a inviscid parallel shear flow isSubstituting $\psi =\varphi (y)e^{ik(x-ct)}$ gives Rayleigh's equation

No normal flow at a rigid wall gives $\boxed{\varphi (\pm 1)=0}$. At a free surface $y=1+\eta$, the linearized kinematic boundary condition isso $(U-c)\eta =-\varphi$. The linearized tangential momentum equation gives the pressure amplitude $p=(U-c){\varphi }^{\prime }-U^{\prime }\varphi$. Constant surface pressure therefore requires

For $U=y$, Rayleigh's equation reduces to ${\varphi }^{\prime \prime }-k^2\varphi =0$, so $\varphi =A\cosh ky+B\sinh ky$. Applying the free-surface condition at $y=\pm 1$ and setting the determinant to zero givesFor $k>0$, $k-\tanh k>0$, so instability occurs exactly when $c^2<0$, orThus $0<k<k_c$, where the unique positive cutoff satisfies $\boxed{k_c\tanh k_c=1}$.