Wave action flux: a physical interpretation

Abstract

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.