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 [[Physics:Significant wave height|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 [[Physics:Continuum mechanics|continuum mechanics]], '''wave action''' refers to a [[Conserved quantity|conservable measure]] of the [[Wave|wave]] part of a [[Physics:Motion|motion]].<ref name="AM1978">{{harvtxt|Andrews|McIntyre|1978}}</ref> For small-[[Physics:Amplitude|amplitude]] and [[Physics: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 [[Physics:Energy|energy]] and <math>\omega_i</math> is the intrinsic frequency of the slowly modulated waves – intrinsic here implying: as observed in a [[Astronomy:Frame of reference|frame of reference]] moving with the [[Average|mean]] velocity of the motion.<ref name="Craik1988">{{harvtxt|Craik|1988|pp=98–110}}</ref>

The [[Physics:Action|action]] of a wave was introduced by {{harvtxt|Sturrock|1962}} in the study of the (pseudo) energy and momentum of waves in [[Physics:Plasma|plasma]]s. {{harvtxt|Whitham|1965}} derived the conservation of wave action – identified as an [[Physics:Adiabatic invariant|adiabatic invariant]] – from an [[Physics:Averaged Lagrangian|averaged Lagrangian]] description of slowly varying nonlinear wave trains in [[Chemistry:Homogeneity and heterogeneity|inhomogeneous]] [[Physics: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 [[Physics:Flux|flux]] and <math>\boldsymbol{\nabla}\cdot\boldsymbol{\mathcal{B}}</math> is the [[Divergence|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 [[Physics:Group velocity|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 [[Physics:Wind wave model|wind wave model]]s to forecast [[Earth:Sea state|sea state]]s as needed by mariners, the offshore industry and for coastal defense. Also in [[Physics:Plasma physics|plasma physics]] and [[Physics:Acoustics|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|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==
{{Ref begin}}
* {{Citation
| doi = 10.1017/S0022112078002785
| volume = 89
| issue = 4
| pages = 647–664
| last1 = Andrews
| first1 = D.G.
| last2 = McIntyre
| first2 = M.E.
| title = On wave-action and its relatives
| journal = Journal of Fluid Mechanics
| year = 1978
|bibcode = 1978JFM....89..647A }}
* {{Citation
| doi = 10.1098/rspa.1968.0034
| volume = 302
| issue = 1471
| pages = 529–554
| last1 = Bretherton
| first1 = F.P.
| first2 = C.J.R.
| last2 = Garrett
| title = Wavetrains in inhomogeneous moving media
| journal = Proceedings of the Royal Society of London A: Mathematical and Physical Sciences
| year = 1968
| bibcode = 1968RSPSA.302..529B }}
* {{Citation
| title = Waves and mean flows
| last = Bühler
| first = O.
| isbn = 978-1-107-66966-6
| series = Cambridge Monographs on Mechanics
| edition = 2nd
| year = 2014
| publisher = Cambridge University Press
}}
* {{Citation
| publisher = Cambridge University Press
| isbn = 9780521368292
| last = Craik
| first = A.D.D.
| title = Wave interactions and fluid flows
| year = 1988
}}
* {{Citation
| doi = 10.1063/1.1692854
| issn = 0031-9171
| volume = 13
| issue = 11
| pages = 2710–2720
| last = Dewar
| first = R.L.
| title = Interaction between hydromagnetic waves and a time‐dependent, inhomogeneous medium
| journal = Physics of Fluids
| year = 1970
|bibcode = 1970PhFl...13.2710D }}
* {{Citation
| doi = 10.1146/annurev.fl.16.010184.000303
| volume = 16
| pages = 11–44
| last = Grimshaw
| first = R.
| title = Wave action and wave–mean flow interaction, with application to stratified shear flows
| journal = Annual Review of Fluid Mechanics
| year = 1984
|bibcode = 1984AnRFM..16...11G }}
* {{Citation
| doi = 10.1098/rspa.1970.0205
| volume = 320
| issue = 1541
| pages = 187–208
| last = Hayes
| first = W.D.
| title = Conservation of action and modal wave action
| journal = Proceedings of the Royal Society of London A: Mathematical and Physical Sciences
| year = 1970
|bibcode = 1970RSPSA.320..187H }}
* {{Citation
| last = Sturrock
| first= P.A.
| contribution = Energy and momentum in the theory of waves in plasmas
| title = Plasma Hydromagnetics. Sixth Lockheed Symposium on Magnetohydrodynamics
| editor-last = Bershader
| editor-first = D.
| pages = 47–57
| publisher = Stanford University Press
| year = 1962
| oclc = 593979237
}}
* {{Citation
| doi = 10.1017/S0022112065000745
| volume = 22
| issue = 2
| pages = 273–283
| last = Whitham
| first = G.B.
| title = A general approach to linear and non-linear dispersive waves using a Lagrangian
| journal = Journal of Fluid Mechanics
| year = 1965
|bibcode = 1965JFM....22..273W }}
* {{Citation 
| first = G.B. 
| last = Whitham 
| year = 1974  
| title = Linear and nonlinear waves 
| publisher = Wiley-Interscience 
| isbn = 0-471-94090-9 
}}
{{Ref end}}

{{physical oceanography}}

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

{{Sourceattribution|Wave action (continuum mechanics)}}