Linearized free-surface boundary conditions

For a small-amplitude surface gravity wave described by velocity potential $\varphi$ and surface elevation $\eta$, linearizing the kinematic boundary condition for a free-surface graph and the Bernoulli equation at the mean surface $z=0$ givesHere $p$ is the applied excess pressure above constant atmospheric pressure. With $p=0$, differentiate the second equation in time and use the first to obtain ${\varphi }_{tt}+g{\varphi }_z=0$. 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.

In a rectangular box with Neumann side walls, a horizontal modehas horizontal wavenumber $k=[(m\pi /L_x{)}^2+(n\pi /L_y{)}^2{]}^{1/2}$, depth dependence $e^{kz}$ in infinitely deep fluid, and natural frequency $\Omega =\sqrt{gk}$.

Pressure forcing of one rectangular free-surface mode reduces its amplitude toAt $\omega =\sqrt{gk}$, the inviscid undamped response contains $t\sin (\omega t)$ and grows without bound in the linear model.