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| {{Continuum mechanics| cTopic=Fluid mechanics}}
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| [[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]] |accessdate=2013-11-14}}</ref>]]
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| 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>
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| :<math>\mathcal{A} = \frac{E}{\omega_i},</math>
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| 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>
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| 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 mechanics|Lagrangian description]] of slowly varying [[nonlinear]] wave trains in [[homogeneity and heterogeneity|inhomogeneous]] [[transmission medium|media]]:
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| :<math>\frac{\partial}{\partial t}\mathcal{A} + \boldsymbol{\nabla} \cdot \boldsymbol{\mathcal{B}} = 0,</math>
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| 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 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"/>
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| :<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>
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| 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.
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| 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.
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| 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"/>
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| ==Notes==
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| {{reflist}}
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| ==References==
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| {{Ref begin}}
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| * {{Citation
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| | doi = 10.1017/S0022112078002785
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| | volume = 89
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| | issue = 4
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| | pages = 647–664
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| | last1 = Andrews
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| | first1 = D.G.
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| | last2 = McIntyre
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| | first2 = M.E.
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| | title = On wave-action and its relatives
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| | journal = Journal of Fluid Mechanics
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| | year = 1978
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| }}
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| * {{Citation
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| | doi = 10.1098/rspa.1968.0034
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| | volume = 302
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| | issue = 1471
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| | pages = 529–554
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| | last1 = Bretherton
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| | first1 = F.P.
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| | first2 = C.J.R.
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| | last2 = Garrett
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| | title = Wavetrains in inhomogeneous moving media
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| | journal = Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences
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| | year = 1968
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| }}
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| * {{Citation
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| | publisher = Cambridge University Press
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| | isbn = 9780521368292
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| | last = Craik
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| | first = A.D.D.
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| | title = Wave interactions and fluid flows
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| | year = 1988
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| }}
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| * {{Citation
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| | doi = 10.1063/1.1692854
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| | issn = 00319171
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| | volume = 13
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| | issue = 11
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| | pages = 2710–2720
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| | last = Dewar
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| | first = R.L.
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| | title = Interaction between hydromagnetic waves and a time‐dependent, inhomogeneous medium
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| | journal = Physics of Fluids
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| | year = 1970
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| }}
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| * {{Citation
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| | doi = 10.1146/annurev.fl.16.010184.000303
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| | volume = 16
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| | pages = 11–44
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| | last = Grimshaw
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| | first = R.
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| | title = Wave action and wave–mean flow interaction, with application to stratified shear flows
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| | journal = Annual Review of Fluid Mechanics
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| | year = 1984
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| }}
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| * {{Citation
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| | doi = 10.1098/rspa.1970.0205
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| | volume = 320
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| | issue = 1541
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| | pages = 187–208
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| | last = Hayes
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| | first = W.D.
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| | title = Conservation of action and modal wave action
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| | journal = Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences
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| | year = 1970
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| }}
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| * {{Citation
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| | last = Sturrock
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| | first= P.A.
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| | author-link = Peter A. Sturrock
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| | contribution = Energy and momentum in the theory of waves in plasmas
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| | title = Plasma Hydromagnetics. Sixth Lockheed Symposium on Magnetohydrodynamics
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| | editor-last = Bershader
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| | editor-first = D.
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| | pages = 47–57
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| | publisher = Stanford University Press
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| | year = 1962
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| | oclc = 593979237
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| }}
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| * {{Citation
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| | doi = 10.1017/S0022112065000745
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| | volume = 22
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| | issue = 2
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| | pages = 273–283
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| | last = Whitham
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| | first = G.B.
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| | author-link = Gerald B. Whitham
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| | title = A general approach to linear and non-linear dispersive waves using a Lagrangian
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| | journal = Journal of Fluid Mechanics
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| | year = 1965
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| }}
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| * {{Citation
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| | first = G.B.
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| | last = Whitham
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| | author-link = Gerald B. Whitham
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| | year = 1974
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| | title = Linear and nonlinear waves
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| | publisher = Wiley-Interscience
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| | isbn = 0-471-94090-9
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| }}
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| {{Ref end}}
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| {{physical oceanography}}
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| [[Category:Continuum mechanics]]
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| [[Category:Waves]]
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Hello!
I'm Norwegian female :).
I really like The Simpsons!
My web page ... sacramento homes for sale