where is the intrinsic wave
energy and is the intrinsic frequency of the slowly modulated waves – intrinsic here implying: as observed in a
frame of reference moving with the
mean velocity of the motion.[4]
where is the wave-action density
flux and is the
divergence of . The description of waves in inhomogeneous and moving media was further elaborated by
Bretherton & Garrett (1968) for the case of small-amplitude waves; they also called the quantity wave action (by which name it has been referred to subsequently). For small-amplitude waves the conservation of wave action becomes:[3][4]
using and
where is the
group velocity and the mean velocity of the inhomogeneous moving medium. While the total energy (the sum of the energies of the mean motion and of the wave motion) is conserved for a non-dissipative system, the energy of the wave motion is not conserved, since in general there can be an exchange of energy with the mean motion. However, wave action is a quantity which is conserved for the wave-part of the motion.
The equation for the conservation of wave action is for instance used extensively in
wind wave models to forecast
sea states as needed by mariners, the offshore industry and for coastal defense. Also in
plasma physics and
acoustics the concept of wave action is used.
The derivation of an exact wave-action equation for more general wave motion – not limited to slowly modulated waves, small-amplitude waves or (non-dissipative)
conservative systems – was provided and analysed by
Andrews & McIntyre (1978) using the framework of the
generalised Lagrangian mean for the separation of wave and mean motion.[4]
Hayes, W.D. (1970), "Conservation of action and modal wave action", Proceedings of the Royal Society of London A: Mathematical and Physical Sciences, 320 (1541): 187–208,
Bibcode:
1970RSPSA.320..187H,
doi:
10.1098/rspa.1970.0205
Sturrock, P.A. (1962), "Energy and momentum in the theory of waves in plasmas", in Bershader, D. (ed.), Plasma Hydromagnetics. Sixth Lockheed Symposium on Magnetohydrodynamics, Stanford University Press, pp. 47–57,
OCLC593979237