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| {{For|the different concept in [[general relativity]]|gravitational wave}}
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| [[Image:Waves.jpg|thumb|right|180px|Surface gravity wave, breaking on an ocean beach.]]
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| [[Image:wave clouds.jpg|thumb|right|180px|Wave clouds over [[Theresa, Wisconsin]], United States.]]
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| [[Image:GravityWaves ArabianSea.MODIS.2005may23.jpg|right|180px|thumb|Atmospheric gravity waves as seen from space.]]
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| In [[fluid dynamics]], '''gravity waves''' are waves generated in a [[fluid]] medium or at the [[Interface (chemistry)|interface]] between two media which has the restoring [[force]] of [[gravity]] or [[buoyancy]]. An example of such an interface is that between the [[atmosphere]] and the [[ocean]], which gives rise to [[wind waves]].
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| When a fluid element is displaced on an interface or internally to a region with a different [[density]], gravity will try to restore it toward [[mechanical equilibrium|equilibrium]], resulting in an [[oscillation]] about the [[equilibrium state]] or ''wave orbit''.<ref>{{Citation | publisher = Cambridge University Press | isbn = 9780521010450 | last = Lighthill | first = James |author-link = James Lighthill | title = Waves in fluids | year = 2001 | page = 205 }}</ref> Gravity waves on an air–sea interface of the ocean are called '''surface gravity waves''' or [[surface wave]]s, while gravity waves that are ''within'' the body of the water (such as between parts of different densities) are called [[internal wave]]s. [[Wind wave|Wind-generated waves]] on the water surface are examples of gravity waves, and [[tsunami]]s and ocean [[tide]]s are others.
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| Wind-generated gravity waves on the [[free surface]] of the Earth's ponds, lakes, seas and oceans have a period of between 0.3 and 30 seconds (3 Hz to 0.03 Hz). Shorter waves are also affected by [[surface tension]] and are called [[gravity–capillary wave]]s and (if hardly influenced by gravity) [[capillary wave]]s. Alternatively, so-called [[infragravity waves]], which are due to [[subharmonic]] [[nonlinear]] wave interaction with the wind waves, have periods longer than the accompanying wind-generated waves.<ref>{{Citation
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| | last1 = Bromirski
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| | first1 = Peter D.
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| | first2 = Olga V.
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| | last2 = Sergienko
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| | first3 = Douglas R.
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| | last3 = MacAyeal
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| | title = Transoceanic infragravity waves impacting Antarctic ice shelves
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| | journal = Geophysical Research Letters
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| | volume = 37
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| | issue = L02502
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| | publisher = [[American Geophysical Union]]
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| | location =
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| | year = 2010
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| | doi = 10.1029/2009GL041488
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| | postscript = .
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| | bibcode=2010GeoRL..3702502B}}</ref>
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| ==Atmosphere dynamics on Earth==
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| {{See also|Undular bore}}
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| In the Earth's [[Earth's atmosphere|atmosphere]], gravity waves are a mechanism for the transfer of [[momentum]] from the [[troposphere]] to the [[stratosphere]]. Gravity waves are generated in the troposphere by [[Weather front|frontal systems]] or by airflow over [[mountain]]s. At first, waves propagate through the atmosphere without appreciable change in [[arithmetic mean|mean]] [[velocity]]. But as the waves reach more rarefied air at higher [[altitude]]s, their [[amplitude]] increases, and [[nonlinearity|nonlinear effects]] cause the waves to break, transferring their momentum to the mean flow.
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| This process plays a key role in studying the [[Dynamics (mechanics)|dynamics]] of the middle atmosphere.{{Citation needed|date=August 2009}}
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| The clouds in gravity waves can look like [[altostratus undulatus cloud]]s, and are sometimes confused with them, but the formation mechanism is different.{{Citation needed|date=August 2009}}
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| ==Quantitative description==
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| {{See|Airy wave theory|Stokes wave}}
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| The [[phase velocity]] <math>\scriptstyle c</math> of a linear gravity wave with [[wavenumber]] <math>\scriptstyle k</math> is given by the formula
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| <math>c=\sqrt{\frac{g}{k}},</math>
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| where ''g'' is the acceleration due to gravity. When surface tension is important, this is modified to
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| <math>c=\sqrt{\frac{g}{k}+\frac{\sigma k}{\rho}},</math>
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| where ''σ'' is the surface tension coefficient and ''ρ'' is the density.
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| {{hidden begin
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| |toggle = left
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| |title = Details of the phase-speed derivation
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| }}
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| The gravity wave represents a perturbation around a stationary state, in which there is no velocity. Thus, the perturbation introduced to the system is described by a velocity field of infinitesimally small amplitude, <math>\scriptstyle (u'(x,z,t),w'(x,z,t)).</math> Because the fluid is assumed incompressible, this velocity field has the [[streamfunction]] representation
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| :<math>\textbf{u}'=(u'(x,z,t),w'(x,z,t))=(\psi_z,-\psi_x),\,</math>
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| where the subscripts indicate [[partial derivatives]]. In this derivation it suffices to work in two dimensions <math>\scriptstyle \left(x,z\right)</math>, where gravity points in the negative ''z''-direction. Next, in an initially stationary incompressible fluid, there is no vorticity, and the fluid stays [[irrotational]], hence <math>\scriptstyle\nabla\times\textbf{u}'=0.\,</math> In the streamfunction representation, <math>\scriptstyle\nabla^2\psi=0.\,</math> Next, because of the translational invariance of the system in the ''x''-direction, it is possible to make the [[ansatz]]
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| :<math>\psi\left(x,z,t\right)=e^{ik\left(x-ct\right)}\Psi\left(z\right),\,</math>
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| where ''k'' is a spatial wavenumber. Thus, the problem reduces to solving the equation
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| :<math>\left(D^2-k^2\right)\Psi=0,\,\,\,\ D=\frac{d}{dz}.</math>
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| We work in a sea of infinite depth, so the boundary condition is at <math>\scriptstyle z=-\infty.</math> The undisturbed surface is at <math>\scriptstyle z=0</math>, and the disturbed or wavy surface is at <math>\scriptstyle z=\eta,</math> where <math>\scriptstyle\eta</math> is small in magnitude. If no fluid is to leak out of the bottom, we must have the condition
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| :<math>u=D\Psi=0,\,\,\text{on}\,z=-\infty.</math>
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| Hence, <math>\scriptstyle\Psi=Ae^{k z}</math> on <math>\scriptstyle z\in\left(-\infty,\eta\right)</math>, where ''A'' and the wave speed ''c'' are constants to be determined from conditions at the interface.
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| ''The free-surface condition:'' At the free surface <math>\scriptstyle z=\eta\left(x,t\right)\,</math>, the kinematic condition holds:
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| :<math>\frac{\partial\eta}{\partial t}+u'\frac{\partial\eta}{\partial x}=w'\left(\eta\right).\,</math>
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| Linearizing, this is simply
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| :<math>\frac{\partial\eta}{\partial t}=w'\left(0\right),\,</math>
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| where the velocity <math>\scriptstyle w'\left(\eta\right)\,</math> is linearized on to the surface <math>\scriptstyle z=0.\,</math> Using the normal-mode and streamfunction representations, this condition is <math>\scriptstyle c \eta=\Psi\,</math>, the second interfacial condition.
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| ''Pressure relation across the interface'': For the case with [[surface tension]], the pressure difference over the interface at <math>\scriptstyle z=\eta</math> is given by the [[Young–Laplace]] equation:
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| :<math>p\left(z=\eta\right)=-\sigma\kappa,\,</math>
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| where ''σ'' is the surface tension and ''κ'' is the [[curvature]] of the interface, which in a linear approximation is
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| :<math>\kappa=\nabla^2\eta=\eta_{xx}.\,</math>
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| Thus,
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| :<math>p\left(z=\eta\right)=-\sigma\eta_{xx}.\,</math>
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| However, this condition refers to the total pressure (base+perturbed), thus
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| :<math>\left[P\left(\eta\right)+p'\left(0\right)\right]=-\sigma\eta_{xx}.</math>
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| (As usual, The perturbed quantities can be linearized onto the surface ''z=0''.) Using [[hydrostatic balance]], in the form <math>\scriptstyle P=-\rho g z+\text{Const.},</math>
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| this becomes
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| :<math>p=g\eta\rho-\sigma\eta_{xx},\qquad\text{on }z=0.\,</math>
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| The perturbed pressures are evaluated in terms of streamfunctions, using the horizontal momentum equation of the linearised [[Euler equations]] for the perturbations,
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| :<math>\frac{\partial u'}{\partial t} = - \frac{1}{\rho}\frac{\partial p'}{\partial x}\,</math>
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| to yield <math>\scriptstyle p'=\rho c D\Psi.</math>
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| Putting this last equation and the jump condition together,
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| :<math>c\rho D\Psi=g\eta\rho-\sigma\eta_{xx}.\,</math>
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| Substituting the second interfacial condition <math>\scriptstyle c\eta=\Psi\,</math> and using the normal-mode representation, this relation becomes <math>\scriptstyle c^2\rho D\Psi=g\Psi\rho+\sigma k^2\Psi.</math>
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| Using the solution <math>\scriptstyle \Psi=e^{k z}</math>, this gives
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| <math>c=\sqrt{\frac{g}{k}+\frac{\sigma k}{\rho}}.</math>
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| {{hidden end}}
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| Since <math>\scriptstyle c=\omega/k</math> is the phase speed in terms of the angular frequency <math>\scriptstyle\omega</math> and the wavenumber, the gravity wave angular frequency can be expressed as
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| <math>\omega=\sqrt{gk}.</math>
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| The [[group velocity]] of a wave (that is, the speed at which a wave packet travels) is given by
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| <math>c_g=\frac{d\omega}{dk},</math>
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| and thus for a gravity wave,
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| <math>c_g=\frac{1}{2}\sqrt{\frac{g}{k}}=\frac{1}{2}c.</math>
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| The group velocity is one half the phase velocity. A wave in which the group and phase velocities differ is called dispersive.
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| ==The generation of waves by wind==
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| Wind waves, as their name suggests, are generated by wind transferring energy from the atmosphere to the ocean's surface, and capillary-gravity waves play an essential role in this effect. There are two distinct mechanisms involved, called after their proponents, Phillips and Miles.
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| In the work of Phillips,<ref>{{citation | first=O. M. | last=Phillips | year=1957 | title=On the generation of waves by turbulent wind | journal=J. Fluid Mech. | volume=2 | issue=5 | pages=417–445 | doi=10.1017/S0022112057000233 |bibcode = 1957JFM.....2..417P }}</ref> the ocean surface is imagined to be initially flat (''glassy''), and a [[turbulent]] wind blows over the surface. When a flow is turbulent, one observes a randomly fluctuating velocity field superimposed on a mean flow (contrast with a laminar flow, in which the fluid motion is ordered and smooth). The fluctuating velocity field gives rise to fluctuating [[stress (mechanics)|stress]]es (both tangential and normal) that act on the air-water interface. The normal stress, or fluctuating pressure acts as a forcing term (much like pushing a swing introduces a forcing term). If the frequency and wavenumber <math>\scriptstyle\left(\omega,k\right)</math> of this forcing term match a mode of vibration of the capillary-gravity wave (as derived above), then there is a [[resonance]], and the wave grows in amplitude. As with other resonance effects, the amplitude of this wave grows linearly with time.
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| The air-water interface is now endowed with a surface roughness due to the capillary-gravity waves, and a second phase of wave growth takes place. A wave established on the surface either spontaneously as described above, or in laboratory conditions, interacts with the turbulent mean flow in a manner described by Miles.<ref>{{citation | first=J. W. | last=Miles | authorlink=John W. Miles | year=1957 | title=On the generation of surface waves by shear flows | journal=J. Fluid Mech. | volume=3 | issue=2 | pages=185–204 | doi=10.1017/S0022112057000567 |bibcode = 1957JFM.....3..185M }}</ref> This is the so-called critical-layer mechanism. A critical layer forms at a height where the wave speed ''c'' equals the mean turbulent flow ''U''. As the flow is turbulent, its mean profile is logarithmic, and its second derivative is thus negative. This is precisely the condition for the mean flow to impart its energy to the interface through the critical layer. This supply of energy to the interface is destabilizing and causes the amplitude of the wave on the interface to grow in time. As in other examples of linear instability, the growth rate of the disturbance in this phase is exponential in time.
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| This Miles–Phillips Mechanism process can continue until an equilibrium is reached, or until the wind stops transferring energy to the waves (i.e., blowing them along) or when they run out of ocean distance, also known as [[fetch (geography)|fetch]] length.
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| ==See also==
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| * [[Cloud street]]
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| * [[Lunitidal interval]]
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| * [[Lee waves]]
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| * [[Morning Glory cloud]]
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| * [[Rayleigh–Taylor instability]]
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| * [[Rogue wave]]
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| * [[Orr–Sommerfeld equation]]
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| ==Notes==
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| <references/>
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| ==References==
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| * Gill, A. E., "''[http://amsglossary.allenpress.com/glossary/search?id=gravity-wave1 Gravity wave]''". Atmosphere Ocean Dynamics, Academic Press, 1982.
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| ==Further reading==
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| *{{cite book|last=Nappo|first=Carmen J.|title=An Introduction to Atmospheric Gravity Waves, Second Ed.|year=2012|publisher=Elsevier Academic Press (International Geophysics Volume 102)|location=Waltham, Massachusetts|isbn=978-0-12-385223-6}}
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| ==External links==
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| {{Commons category|Gravity waves}}
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| * {{citation | first1=Steven | last1=Koch | first2=Hugh D. | last2=Cobb, III | first3=Neil A. | last3=Stuart | url=http://www.erh.noaa.gov/er/akq/GWave.htm | title=Notes on Gravity Waves – Operational Forecasting and Detection of Gravity Waves Weather and Forecasting | publisher=[[NOAA]] | accessdate=2010-11-11 }}
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| * {{citation | url=http://www.kcrg.com/news/local/3195031.html | title=Gallery of cloud gravity waves over Iowa| accessdate=2010-11-11 }}
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| * {{citation | url=http://www.youtube.com/watch?v=yXnkzeCU3bE | title=Time-lapse video of gravity waves over Iowa | accessdate=2010-11-11 }}
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| * {{citation | url=http://www.wikiwaves.org/index.php/Main_Page | title=Water Waves Wiki | accessdate=2010-11-11 }}
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| {{physical oceanography}}
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| {{DEFAULTSORT:Gravity Wave}}
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| [[Category:Fluid dynamics]]
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| [[Category:Atmospheric dynamics]]
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| [[Category:Waves]]
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| [[Category:Water waves]]
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