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For small irrotational surface gravity waves, write the velocity field as $\nabla \varphi$. Incompressibility gives ${\nabla }^2\varphi =0$. The rigid walls have ${\varphi }_x=0$ at $x=0,a$, ${\varphi }_y=0$ at $y=0,b$, and ${\varphi }_z=0$ at $z=-h$. At $z=0$, the linear kinematic boundary condition and pressure condition areSeparate variables for a rectangular standing surface-gravity mode:The horizontal cosine factors and vertical hyperbolic cosine satisfy all rigid-wall conditions. Put $\eta =A(t)\cos (m\pi x/a)\cos (n\pi y/b)$ and $\lambda =k\tanh kh$. The free-surface conditions give $\dot{A}=\lambda B$ and $\dot{B}=-gA$, henceBoth integers are nonnegative and not both zero. The zero pair would be a spatially uniform displacement, excluded by fixed fluid volume; a spatially constant potential is only a gauge mode.

Forthe normal derivative at $x=0,L$ contains $\sin (\pi x/L)$ and the normal derivative at $y=0,L$ contains $\sin (2\pi y/L)$. Both therefore vanish at the side walls.

At $z=0$, ${\varphi }_t$ has exactly the same horizontal dependence as the imposed pressure. The dynamic boundary condition is consequently consistent provided $\eta$ has that same horizontal mode. This is the rectangular standing surface-gravity mode selected by the forcing.