Past exam of the mathematics course of the University of Cambridge / 2009 / ib / Paper 4 / 18D / i / Solution

For small irrotational surface gravity waves, write the velocity field as $\nabla \varphi$. Incompressibility gives ${\nabla }^2\varphi =0$. The rigid walls have ${\varphi }_x=0$ at $x=0,a$, ${\varphi }_y=0$ at $y=0,b$, and ${\varphi }_z=0$ at $z=-h$. At $z=0$, the linear kinematic boundary condition and pressure condition areSeparate variables for a rectangular standing surface-gravity mode:The horizontal cosine factors and vertical hyperbolic cosine satisfy all rigid-wall conditions. Put $\eta =A(t)\cos (m\pi x/a)\cos (n\pi y/b)$ and $\lambda =k\tanh kh$. The free-surface conditions give $\dot{A}=\lambda B$ and $\dot{B}=-gA$, henceBoth integers are nonnegative and not both zero. The zero pair would be a spatially uniform displacement, excluded by fixed fluid volume; a spatially constant potential is only a gauge mode.