Wave action (continuum mechanics)

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weather forecasting models).[1]

In

motion.[2] For small-amplitude and slowly varying waves, the wave action density is:[3]

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]

The

nonlinear wave trains in inhomogeneous media
:

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

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)

generalised Lagrangian mean for the separation of wave and mean motion.[4]

Notes

References