 Solution

Write the surface displacement and the perturbation velocity potential as small quantities about rest, and drop their quadratic products. Evaluating the remaining surface terms at the unperturbed level $y=0$ givesThe velocity components are $u={\varphi }_x$, $v={\varphi }_y$. The shift of the evaluation point from zero to $\eta$ contributes a product of perturbations, as do $u{\eta }_x$ and the kinetic-energy term, so they disappear at this order.