= 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 $y=0$ gives
$$
\boxed{\phi_{xx}+\phi_{yy}=0\quad(-H<y<0)},
$$
$$
\boxed{\eta_t=\phi_y,\qquad\phi_t+g\eta=0\quad(y=0),\qquad\phi_y=0\quad(y=-H)}.
$$
The velocity components are $u=\phi_x$, $v=\phi_y$. 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.
Back to article page
