A fluid of density $\rho_{1}$ occupies the region $z>0$ and a second fluid of density $\rho_{2}$ occupies the region $z<0$. State the equations and boundary conditions that are satisfied by the corresponding velocity potentials $\phi_{1}$ and $\phi_{2}$ and pressures $p_{1}$ and $p_{2}$ when the system is perturbed so that the interface is at $z=\zeta(x, t)$ and the motion is irrotational.

Seek a set of linearised equations and boundary conditions when the disturbances are proportional to $e^{i(k x-\omega t)}$, and derive the dispersion relation

$$\omega^{2}=\frac{\rho_{2}-\rho_{1}}{\rho_{2}+\rho_{1}} g k,$$

where $g$ is the gravitational acceleration.