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

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.