Past exam of the mathematics course of the University of Cambridge / 2016 / ib / Paper 3 / 18C / b / Solution

Linearize about the flat interface at rest, $\eta =0$, ${\phi }_j=0$. Both potentials still satisfy Laplace equation, now on the fixed strips $0<y<h_1$ and $-h_2<y<0$, with the same rigid-wall conditions. Discarding products of perturbations and evaluating the interface conditions at $y=0$ givesFor a normal mode proportional to $e^{i(kx-\omega t)}$, the vertical profiles satisfy ${\Phi }_j^{\prime \prime }-k^2{\Phi }_j=0$, the interface velocity is $-i\omega \eta$, and the wall-normal derivatives vanish. Smallness of the amplitude relative to wavelength and layer depths justifies this linearization.