= Solution
{c}
Take the undisturbed surface to be $y=0$ and treat $\eta$, $\phi$, 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 is
$$
\boxed{\phi_{xx}+\phi_{yy}=0\quad(-H<y<0),}
$$
$$
\boxed{\eta_t=\phi_y,\quad \phi_t+g\eta=0\quad(y=0),\qquad
\phi_y=0\quad(y=-H).}
$$
Here $u=\phi_x$, $v=\phi_y$. The linear free-surface conditions can also be combined into $\phi_{tt}+g\phi_y=0$ at zero.
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