View source for Wave action (continuum mechanics)

{{short description|Conservable measure of the wave part of a motion}}
{{Continuum mechanics| cTopic=Fluid mechanics}}
[[File:NOAA Wavewatch III Sample Forecast.gif|thumb|right|Five-day forecast of the [[significant wave height]] for the [[North Atlantic]] on November 22, 2008, by [[NOAA]]'s Wavewatch III model. This [[wind wave model]] generates forecasts of wave conditions through the use of wave-action conservation and the wind-field forecasts (from [[weather forecasting models]]).<ref>{{Citation |url=http://polar.ncep.noaa.gov/waves/wavewatch/wavewatch.shtml |title=WAVEWATCH III Model |publisher=[[National Weather Service]], [[NOAA]] |access-date=2013-11-14}}</ref>]]
In [[continuum mechanics]], '''wave action''' refers to a [[conserved quantity|conservable measure]] of the [[wave]] part of a [[motion (physics)|motion]].<ref name="AM1978">{{harvtxt|Andrews|McIntyre|1978}}</ref> For small-[[amplitude]] and [[Slowly varying envelope approximation|slowly varying]] waves, the '''wave action density''' is:<ref name="BG1968">{{harvtxt|Bretherton|Garrett|1968}}</ref>

:<math>\mathcal{A} = \frac{E}{\omega_i},</math>

where <math>E</math> is the intrinsic wave [[energy]] and <math>\omega_i</math> is the intrinsic frequency of the slowly modulated waves – intrinsic here implying: as observed in a [[frame of reference]] moving with the [[average|mean]] velocity of the motion.<ref name="Craik1988">{{harvtxt|Craik|1988|pp=98–110}}</ref>

The [[action (physics)|action]] of a wave was introduced by {{harvtxt|Sturrock|1962}} in the study of the (pseudo) energy and momentum of waves in [[plasma (physics)|plasma]]s. {{harvtxt|Whitham|1965}} derived the conservation of wave action – identified as an [[adiabatic invariant]] – from an [[averaged Lagrangian]] description of slowly varying [[nonlinear system|nonlinear]] wave trains in [[homogeneity and heterogeneity|inhomogeneous]] [[transmission medium|media]]:

:<math>\frac{\partial}{\partial t}\mathcal{A} + \boldsymbol{\nabla} \cdot \boldsymbol{\mathcal{B}} = 0,</math>

where <math>\boldsymbol{\mathcal{B}}</math> is the wave-action density [[flux]] and <math>\boldsymbol{\nabla}\cdot\boldsymbol{\mathcal{B}}</math> is the [[divergence]] of <math>\boldsymbol{\mathcal{B}}</math>. The description of waves in inhomogeneous and moving media was further elaborated by {{harvtxt|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:<ref name="BG1968"/><ref name="Craik1988"/>

:<math>\frac{\partial}{\partial t}\left( \frac{E}{\omega_i} \right) + \boldsymbol{\nabla} \cdot \left[ \left( \boldsymbol{U} + \boldsymbol{c}_g \right)\, \frac{E}{\omega_i} \right] = 0,</math> {{pad|2em}} using {{pad|2em}} <math>\mathcal{A} = \frac{E}{\omega_i}</math> {{pad|1em}} and {{pad|1em}} <math>\boldsymbol{\mathcal{B}} = \left( \boldsymbol{U} + \boldsymbol{c}_g \right) \mathcal{A},</math>

where <math>\boldsymbol{c}_g</math> is the [[group velocity]] and <math>\boldsymbol{U}</math> 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 model]]s to forecast [[sea state]]s 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 system]]s – was provided and analysed by {{harvtxt|Andrews|McIntyre|1978}} using the framework of the [[generalised Lagrangian mean]] for the separation of wave and mean motion.<ref name="Craik1988"/>

==Notes==
{{reflist}}

==References==
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{{Refend}}

{{physical oceanography}}

[[Category:Continuum mechanics]]
[[Category:Waves]]