Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 4 / 38D / b / Solution

The linearized kinematic condition at the undisturbed surface isFor the given complex amplitudes this says

The velocity is $u=({\varphi }_x,0,{\varphi }_z)$. Its nonzero complex strain amplitudes areUsing the stated rule for period averages,Thus the mean dissipation per unit horizontal area isThe mean energy $E=\rho g|A{|}^2/2$ consequently obeys $dE/dt=-\mathcal{D}$. Writing $\nu =\mu /\rho$ gives the viscous decay of a deep-water gravity wave:

Solved by gpt-5.6-sol high.