Past exam of the probability, statistics and options I course of the University of Oxford / 2008 / K / 2

Condition (1b) is the kinematic boundary condition: surface particles remain on the moving free-surface graph. Condition (1c) is the dynamic boundary condition for an inviscid interface from unsteady Bernoulli's equation, with the surface pressure fixed at the atmospheric value and absorbed into the potential's time gauge. Condition (1d) is impermeability of the fixed horizontal bed, so its normal velocity vanishes.

Take the undisturbed surface to be $y=0$ and treat $\eta$, $\varphi$, and their induced velocities as first order in wave amplitude. Products of first-order quantities are second order and are discarded. Evaluating surface conditions at $y=\eta$ differs from evaluation at zero only by second-order terms. Thus the linearized surface gravity wave problem isHere $u={\varphi }_x$, $v={\varphi }_y$. The linear free-surface conditions can also be combined into ${\varphi }_{tt}+g{\varphi }_y=0$ at zero.

For $k>0$, phase velocity is the speed of a fixed phase, $c_p=\omega /k$. The group velocity $c_g=d\omega /dk$ transports a narrow-band wave envelope. With $q=kH$,As $H\to 0$ with $k$ fixed, $\tanh (kH)/(kH)\to 1$. ThereforeBoth velocities tend to the shallow-water speed $\sqrt{gH}$ at leading order.