A layer of water of depth $h$ flows along a wide channel with uniform velocity $(U, 0)$, in Cartesian coordinates $(x, y)$, with $x$ measured downstream. The bottom of the channel is at $y=-h$, and the free surface of the water is at $y=0$. Waves are generated on the free surface so that it has the new position $y=\eta(x, t)=a e^{i(\omega t-k x)}$.

Write down the equation and the full nonlinear boundary conditions for the velocity potential $\phi$ (for the perturbation velocity) and the motion of the free surface.

By linearizing these equations about the state of uniform flow, show that

where $g$ is the acceleration due to gravity.

Hence, determine the dispersion relation for small-amplitude surface waves

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