5.11 Wave Action

Ray theory is applicable in the fully dispersed region, where waves of different wavelengths and amplitudes are sufficiently spread out that wavelengths and amplitudes vary by negligible amounts over a wavelength or wave period. It is then convenient to speak of a local wave-energy density E, which is the wave energy per unit surface area (or per unit volume in more general systems), and is defined by averaging over a wavelength or wave period. We have relied on Rayleigh s proof that E in the fully dispersed region propagates at the group velocity to justify the calculation of wave amplitudes in ray theory. In Example 2 of 5.10, however, it was remarked that conservation of wave energy could not be ensured when waves propagate through a region that is itself in slowly varying mean motion ( nonuniform on a scale of many wavelengths). The energy of the whole system might be conserved, but there is no reason to suppose that there is no transfer of energy between the wave and the background flow.

This also raises the question of how wave energy should be defined in a medium in nonuniform motion. The simplest procedure is to use the same definition of E in both moving and stationary media for a moving medium we must then adopt a local reference frame fixed relative to the local flow and define wave properties relative to this frame. The size of the region covered by the local...