Past exam of the mathematics course of the University of Cambridge / 2018 / ib / Paper 4 / 18D (Fluid Dynamics)

Let $\varphi$ be the velocity potential. The linearized deep-water equations are ${\nabla }^2\varphi =0$ for $z<0$, ${\eta }_t={\varphi }_z$ and ${\varphi }_t+g\eta =0$ at $z=0$, with zero normal velocity at the four side walls. The normal modes arewhere $m,n\ge 0$ are not both zero. The free-surface conditions give the dispersion relation for a deep-water gravity wave:

The initial displacement is the $(m,n)=(3,4)$ mode, so $k=5\pi /a$. Release from rest selects a cosine in time:Velocity decays as $e^{kz}$, so it falls to $1/e$ of its surface value at depth $a/(5\pi )$.