Steady, gravity water waves on a constant-over-depth current, progressing
over a
slowly varying bed, are studied with the purpose of connecting the wave
action flux
concept with conventional energy flux considerations. The analysis is two-dimensional
and dissipation is neglected. A new relation between integral properties
containing the
energy flux referred to a ‘global’ level, the
so-called mean energy level, gives the
surprising result that this flux is simply the product of absolute angular
frequency and
wave action flux. An alternative, less physical, proof of this result is
also presented. A
general equation for the action velocity is set out and for linear waves
shown to equal
a well-known expression. Also presented are new expressions for relative
phase velocity
in terms of kinetic energy and mean momentum for the wave, and the kinetic
energy
in terms of the characteristic velocities for the combined wave and current
motion. In
the Appendix a simple relation between energy and action fluxes for small-amplitude
waves on a linear shear current is found which resembles the irrotational
theory,
finite-height result. A possible extension of this relation to finite-height
waves on a general
shear current is discussed.