Past exam of the mathematics course of the University of Cambridge / 2023 / ib / Paper 4 / 16C / b / iii / Solution

Substitution into ${\nabla }^2\varphi =0$ givesSetThen $Z=A{e}^{kz}+B{e}^{-kz}$. Since the fluid occupies $z<0$, decay as $z\to -\infty$ forces $B=0$. Absorbing $A$ into $F$ and choosing $Z(0)=1$ givesThis is the usual exponential depth dependence of a deep-water gravity wave.