4.II.18A

A rectangular tank has a horizontal base and vertical sides. Viewed from above, the cross-section of the tank is a square of side $a$. At rest, the depth of water in the tank is $h$. Suppose that the free-surface is disturbed in such a way that the flow in the water is irrotational. Take the pressure at the free surface as atmospheric. Starting from the appropriate non-linear expressions, obtain free-surface boundary conditions for the velocity potential appropriate for small-amplitude disturbances of the surface.

Show that the governing equations and boundary conditions admit small-amplitude normal mode solutions for which the free-surface elevation above its equilibrium level is everywhere proportional to $e^{i \omega t}$, and find the frequencies, $\omega$, of such modes.

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