= Linearized free-surface boundary conditions
For a small-amplitude <surface gravity wave> described by <velocity potential> $\phi$ and <surface elevation> $\eta$, linearizing the <kinematic boundary condition for a free-surface graph> and the <Bernoulli equation> at the mean surface $z=0$ gives
$$
\eta_t=\phi_z,
\qquad
\phi_t+g\eta=-\frac{p}{\rho}
\qquad(z=0).
$$
Here $p$ is the applied excess <pressure> above constant atmospheric pressure. With $p=0$, differentiate the second equation in time and use the first to obtain $\phi_{tt}+g\phi_z=0$. These surface conditions apply at finite depth as well as in the <deep-water gravity wave> limit; depth enters through the bottom condition. The quadratic <kinetic energy> term is omitted at this order, and <surface tension> is neglected.
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