An incompressible, inviscid fluid occupies the region beneath the free surface $y=\eta(x, t)$ and moves with a velocity field given by the velocity potential $\phi(x, y, t)$; gravity acts in the $-y$ direction. Derive the kinematic and dynamic boundary conditions that must be satisfied by $\phi$ on $y=\eta(x, t)$.

[You may assume Bernoulli's integral of the equation of motion:

$$\left.\frac{p}{\rho}+\frac{\partial \phi}{\partial t}+\frac{1}{2}|\nabla \phi|^{2}+g y=F(t) .\right]$$

In the absence of waves, the fluid has uniform velocity $U$ in the $x$ direction. Derive the linearised form of the above boundary conditions for small amplitude waves, and verify that they and Laplace's equation are satisfied by the velocity potential

$$\phi=U x+\operatorname{Re}\left\{b e^{k y} e^{i(k x-\omega t)}\right\}$$

where $|k b| \ll U$, with a corresponding expression for $\eta$, as long as

$$(\omega-k U)^{2}=g k$$

What are the propagation speeds of waves with a given wave-number $k ?$