An inviscid incompressible fluid occupies a rectangular tank with vertical sides at $x=0, a$ and $y=0, b$ and a horizontal bottom at $z=-h$. The undisturbed free surface is at $z=0$.

(i) Write down the equations and boundary conditions governing small amplitude free oscillations of the fluid, neglecting surface tension, and show that the frequencies $\omega$ of such oscillations are given by

$$\frac{\omega^{2}}{g}=k \tanh k h, \text { where } k^{2}=\pi^{2}\left(\frac{m^{2}}{a^{2}}+\frac{n^{2}}{b^{2}}\right)$$

for non-negative integers $m, n$, which cannot both be zero.

(ii) The free surface is now acted on by a small external pressure

$$p=\epsilon \rho g h \sin \Omega t \cos \frac{m \pi x}{a} \cos \frac{n \pi y}{b},$$

where $\epsilon \ll 1$. Calculate the corresponding oscillation of the free surface when $\Omega$ is not equal to the quantity $\omega$ given by (1).

Why does your solution break down as $\Omega \rightarrow \omega ?$