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Incontinuum mechanics,wave action refers to aconservable measure of thewave part of amotion.^[2] For small-amplitude andslowly varying waves, thewave action density is:^[3]

${\displaystyle \mathcal{A}=\frac{E}{{\omega }_i},}$

where${\displaystyle E}$ is the intrinsic waveenergy and${\displaystyle {\omega }_i}$ is the intrinsic frequency of the slowly modulated waves – intrinsic here implying: as observed in aframe of reference moving with themean velocity of the motion.^[4]

Theaction of a wave was introduced bySturrock (1962) in the study of the (pseudo) energy and momentum of waves inplasmas.Whitham (1965) derived the conservation of wave action – identified as anadiabatic invariant – from anaveraged Lagrangian description of slowly varyingnonlinear wave trains ininhomogeneousmedia:

${\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }t}\mathcal{A}+\mathbf{\nabla }\cdot \mathcal{B}=0,}$

where${\displaystyle \mathcal{B}}$ is the wave-action densityflux and${\displaystyle \mathbf{\nabla }\cdot \mathcal{B}}$ is thedivergence of${\displaystyle \mathcal{B}}$. The description of waves in inhomogeneous and moving media was further elaborated byBretherton & Garrett (1968) for the case of small-amplitude waves; they also called the quantitywave action (by which name it has been referred to subsequently). For small-amplitude waves the conservation of wave action becomes:^[3][4]

${\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }t}\left(\frac{E}{{\omega }_i}\right)+\mathbf{\nabla }\cdot \left[\left(\mathit{U}+{\mathit{c}}_g\right)\frac{E}{{\omega }_i}\right]=0,}$  using ${\displaystyle \mathcal{A}=\frac{E}{{\omega }_i}}$  and ${\displaystyle \mathcal{B}=\left(\mathit{U}+{\mathit{c}}_g\right)\mathcal{A},}$

where${\displaystyle {\mathit{c}}_g}$ is thegroup velocity and${\displaystyle \mathit{U}}$ the mean velocity of the inhomogeneous moving medium. While thetotal 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 inwind wave models to forecastsea states as needed by mariners, the offshore industry and for coastal defense. Also inplasma physics andacoustics 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 byAndrews & McIntyre (1978) using the framework of thegeneralised Lagrangian mean for the separation of wave and mean motion.^[4]

• Continuum mechanics
• Waves

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