Inviscid fluid is contained in a square vessel with sides of length $\pi L$ lying between $x=0, \pi L, y=0, \pi L$. The base of the container is at $z=-H$ where $H \gg L$ and the horizontal surface is at $z=0$ when the fluid is at rest. The variation of pressure of the air above the fluid may be neglected.

Small amplitude surface waves are excited in the vessel.

(i) Now let $H \rightarrow \infty$. Explain why on dimensional grounds the frequencies $\omega$ of such waves are of the form

$$\omega=\left(\frac{\gamma g}{L}\right)^{\frac{1}{2}}$$

for some positive dimensionless constants $\gamma$, where $g$ is the gravitational acceleration.

It is given that the velocity potential $\phi$ is of the form

$$\phi(x, y, z) \approx C \cos (m x / L) \cos (n y / L) \mathrm{e}^{\gamma z / L}$$

where $m$ and $n$ are integers and $C$ is a constant.

(ii) Why do cosines, rather than sines, appear in this expression?

(iii) Give an expression for $\gamma$ in terms of $m$ and $n$.

(iv) Give all possible values that $\gamma^{2}$ can take between 1 and 10 inclusive. How many different solutions for $\phi$ correspond to each of these values of $\gamma^{2} ?$