  The interaction of surface waves with Couette-type current with uniform
vorticity is a well suited problem for students approaching the theory of
surface waves. The problem, although mathematically simple, contains rich
physics, and is moreover important in several situations from oceanography and
marine technology to microfluidics. We here lay out a simple two-dimensional
theory of waves propagating upon a basic flow of uniform vorticity of constant
depth. The dispersion relation is found, showing how the shearing current
introduces different phase velocities for upstream and downstream propagating
waves. The role of surface tension is discussed and applied to the case of a
wave pattern created by a moving source, stationary as seen by the source. We
conclude by discussing how the average potential and kinetic energies are no
longer equal in the presence of shear.
