Wave action (continuum mechanics)

In continuum mechanics, wave action refers to a conservable measure of the wave part of a motion. For small-amplitude and slowly varying waves, the wave action density is:

where ${\displaystyle E}$ is the intrinsic wave energy and ${\displaystyle \omega _{i}}$ 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.

The action of a wave was introduced by Sturrock (1962) in the study of the (pseudo) energy and momentum of waves in plasmas. Whitham (1965) derived the conservation of wave action – identified as an adiabatic invariant – from an averaged Lagrangian description of slowly varying nonlinear wave trains in inhomogeneous media:

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