Negative energy waves in a shear flow with a linear profile

We present a derivation of the time averaged potential and kinetic energies for small-amplitude surface waves on a shear flow with constant vorticity. The effect of surface tension is also taken into consideration. It is demonstrated that the virial theorem of the energy equipartition between the potential and kinetic components is not valid in general for waves on a shear flow. We also show that waves with a negative energy may exist in a shear flow, and we find the condition for existence of such waves.