A layer of water of depth flows along a wide channel with uniform velocity , in Cartesian coordinates , with measured downstream. The bottom of the channel is at , and the free surface of the water is at . Waves are generated on the free surface so that it has the new position .
Write down the equation and the full nonlinear boundary conditions for the velocity potential (for the perturbation velocity) and the motion of the free surface.
By linearizing these equations about the state of uniform flow, show that
where is the acceleration due to gravity.
Hence, determine the dispersion relation for small-amplitude surface waves