A source of sound induces a travelling wave of pressure above the free surface of a fluid located in the $z<0$ domain as

$$p=p_{a t m}+p_{0} \cos (k x-\omega t),$$

with $p_{0} \ll p_{a t m}$. Here $k$ and $\omega$ are fixed real numbers. We assume that the flow induced in the fluid is irrotational.

(i) State the linearized equation of motion for the fluid and the free surface, $z=h(x, t)$, with all boundary conditions.

(ii) Solve for the velocity potential, $\phi(x, z, t)$, and the height of the free surface, $h(x, t)$. Verify that your solutions are dimensionally correct.

(iii) Interpret physically the behaviour of the solution when $\omega^{2}=g k$.