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In continuum mechanics, **wave action** refers to a conservable measure of the wave part of a motion.^{ [2] } For small-amplitude and slowly varying waves, the **wave action density** is:^{ [3] }

**Continuum mechanics** is a branch of mechanics that deals with the mechanical behavior of materials modeled as a continuous mass rather than as discrete particles. The French mathematician Augustin-Louis Cauchy was the first to formulate such models in the 19th century.

In mathematics, a **conserved quantity** of a dynamical system is a function of the dependent variables whose value remains constant along each trajectory of the system.

In physics, mathematics, and related fields, a **wave** is a disturbance of a field in which a physical attribute oscillates repeatedly at each point or propagates from each point to neighboring points, or seems to move through space.

where is the intrinsic wave energy and is the intrinsic frequency of the slowly modulated waves – intrinsic here implying: as observed in a frame of reference moving with the mean velocity of the motion.^{ [4] }

In physics, **energy** is the quantitative property that must be transferred to an object in order to perform work on, or to heat, the object. Energy is a conserved quantity; the law of conservation of energy states that energy can be converted in form, but not created or destroyed. The SI unit of energy is the joule, which is the energy transferred to an object by the work of moving it a distance of 1 metre against a force of 1 newton.

In physics, a **frame of reference** consists of an abstract coordinate system and the set of physical reference points that uniquely fix the coordinate system and standardize measurements.

In colloquial language, an **average** is a single number taken as representative of a list of numbers. Different concepts of average are used in different contexts. Often "average" refers to the arithmetic mean, the sum of the numbers divided by how many numbers are being averaged. In statistics, mean, median, and mode are all known as measures of central tendency, and in colloquial usage any of these might be called an **average value**.

The action of a wave was introduced by Sturrock (1962) in the study of the (pseudo) energy and momentum of waves in plasmas. Whitham (1965) derived the conservation of wave action – identified as an adiabatic invariant – from an averaged Lagrangian description of slowly varying nonlinear wave trains in inhomogeneous media:

In physics, **action** is an attribute of the dynamics of a physical system from which the equations of motion of the system can be derived. It is a mathematical functional which takes the trajectory, also called *path* or *history*, of the system as its argument and has a real number as its result. Generally, the action takes different values for different paths. Action has the dimensions of [energy]⋅[time] or [momentum]⋅[length], and its SI unit is joule-second.

**Plasma** is one of the four fundamental states of matter, and was first described by chemist Irving Langmuir in the 1920s. Plasma can be artificially generated by heating or subjecting a neutral gas to a strong electromagnetic field to the point where an ionized gaseous substance becomes increasingly electrically conductive, and long-range electromagnetic fields dominate the behaviour of the matter.

A property of a physical system, such as the entropy of a gas, that stays approximately constant when changes occur slowly is called an **adiabatic invariant**. By this it is meant that if a system is varied between two end points, as the time for the variation between the end points is increased to infinity, the variation of an adiabatic invariant between the two end points goes to zero.

where is the wave-action density flux and is the divergence of . The description of waves in inhomogeneous and moving media was further elaborated by 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:^{ [3] }^{ [4] }

**Flux** describes any effect that appears to pass or travel through a surface or substance. A flux is either a concept based in physics or used with applied mathematics. Both concepts have mathematical rigor, enabling comparison of the underlying mathematics when the terminology is unclear. For transport phenomena, flux is a vector quantity, describing the magnitude and direction of the flow of a substance or property. In electromagnetism, flux is a scalar quantity, defined as the surface integral of the component of a vector field perpendicular to the surface at each point.

In vector calculus, **divergence** is a vector operator that produces a scalar field, giving the quantity of a vector field's source at each point. More technically, the divergence represents the volume density of the outward flux of a vector field from an infinitesimal volume around a given point.

- using and

where is the group velocity and 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 **group velocity** of a wave is the velocity with which the overall shape of the wave's amplitudes—known as the *modulation* or *envelope* of the wave—propagates through space.

The equation for the conservation of wave action is for instance used extensively in wind wave models to forecast sea states as needed by mariners, the offshore industry and for coastal defense. Also in plasma physics and acoustics the concept of wave action is used.

In fluid dynamics, **wind wave modeling** describes the effort to depict the sea state and predict the evolution of the energy of wind waves using numerical techniques. These simulations consider atmospheric wind forcing, nonlinear wave interactions, and frictional dissipation, and they output statistics describing wave heights, periods, and propagation directions for regional seas or global oceans. Such **wave hindcasts** and **wave forecasts** are extremely important for commercial interests on the high seas. For example, the shipping industry requires guidance for operational planning and tactical seakeeping purposes.

In oceanography, a **sea state** is the general condition of the free surface on a large body of water—with respect to wind waves and swell—at a certain location and moment. A sea state is characterized by statistics, including the wave height, period, and power spectrum. The sea state varies with time, as the wind conditions or swell conditions change. The sea state can either be assessed by an experienced observer, like a trained mariner, or through instruments like weather buoys, wave radar or remote sensing satellites.

**Acoustics** is the branch of physics that deals with the study of all mechanical waves in gases, liquids, and solids including topics such as vibration, sound, ultrasound and infrasound. A scientist who works in the field of acoustics is an **acoustician** while someone working in the field of acoustics technology may be called an acoustical engineer. The application of acoustics is present in almost all aspects of modern society with the most obvious being the audio and noise control industries.

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 by Andrews & McIntyre (1978) using the framework of the generalised Lagrangian mean for the separation of wave and mean motion.^{ [4] }

- ↑
*WAVEWATCH III Model*, National Weather Service, NOAA , retrieved 2013-11-14 - ↑ Andrews & McIntyre (1978)
- 1 2 Bretherton & Garrett (1968)
- 1 2 3 Craik (1988 , pp. 98–110)

The **vorticity equation** of fluid dynamics describes evolution of the vorticity **ω** of a particle of a fluid as it moves with its flow, that is, the local rotation of the fluid . The equation is:

In physics, **Landau damping**, named after its discoverer, the eminent Soviet physicist Lev Davidovich Landau (1908–68), is the effect of damping of longitudinal space charge waves in plasma or a similar environment. This phenomenon prevents an instability from developing, and creates a region of stability in the parameter space. It was later argued by Donald Lynden-Bell that a similar phenomenon was occurring in galactic dynamics, where the gas of electrons interacting by electrostatic forces is replaced by a "gas of stars" interacting by gravitation forces. Landau damping can be manipulated exactly in numerical simulations such as particle-in-cell simulation. It was proved to exist experimentally by Malmberg and Wharton in 1964, almost two decades after its prediction by Landau in 1946.

In mathematics, the **Hamilton–Jacobi equation** (**HJE**) is a necessary condition describing extremal geometry in generalizations of problems from the calculus of variations, and is a special case of the Hamilton–Jacobi–Bellman equation. It is named for William Rowan Hamilton and Carl Gustav Jacob Jacobi.

**Smoothed-particle hydrodynamics** (**SPH**) is a computational method used for simulating the mechanics of continuum media, such as solid mechanics and fluid flows. It was developed by Gingold and Monaghan and Lucy in 1977, initially for astrophysical problems. It has been used in many fields of research, including astrophysics, ballistics, volcanology, and oceanography. It is a meshfree Lagrangian method, and the resolution of the method can easily be adjusted with respect to variables such as density.

In physics, a **ponderomotive force** is a nonlinear force that a charged particle experiences in an inhomogeneous oscillating electromagnetic field.

For a pure wave motion in fluid dynamics, the **Stokes drift velocity** is the average velocity when following a specific fluid parcel as it travels with the fluid flow. For instance, a particle floating at the free surface of water waves, experiences a net Stokes drift velocity in the direction of wave propagation.

In fluid dynamics, a **Stokes wave** is a non-linear and periodic surface wave on an inviscid fluid layer of constant mean depth. This type of modelling has its origins in the mid 19th century when Sir George Stokes – using a perturbation series approach, now known as the **Stokes expansion** – obtained approximate solutions for non-linear wave motion.

In fluid dynamics, **Luke's variational principle** is a Lagrangian variational description of the motion of surface waves on a fluid with a free surface, under the action of gravity. This principle is named after J.C. Luke, who published it in 1967. This variational principle is for incompressible and inviscid potential flows, and is used to derive approximate wave models like the so-called mild-slope equation, or using the averaged Lagrangian approach for wave propagation in inhomogeneous media.

In fluid dynamics, **Airy wave theory** gives a linearised description of the propagation of gravity waves on the surface of a homogeneous fluid layer. The theory assumes that the fluid layer has a uniform mean depth, and that the fluid flow is inviscid, incompressible and irrotational. This theory was first published, in correct form, by George Biddell Airy in the 19th century.

In fluid dynamics, the **mild-slope equation** describes the combined effects of diffraction and refraction for water waves propagating over bathymetry and due to lateral boundaries—like breakwaters and coastlines. It is an approximate model, deriving its name from being originally developed for wave propagation over mild slopes of the sea floor. The mild-slope equation is often used in coastal engineering to compute the wave-field changes near harbours and coasts.

In fluid dynamics, the **radiation stress** is the depth-integrated – and thereafter phase-averaged – excess momentum flux caused by the presence of the surface gravity waves, which is exerted on the mean flow. The radiation stresses behave as a second-order tensor.

In mathematical physics, the **Belinfante–Rosenfeld tensor** is a modification of the energy–momentum tensor that is constructed from the canonical energy–momentum tensor and the spin current so as to be symmetric yet still conserved.

In mathematics, the **Bretherton equation** is a nonlinear partial differential equation introduced by Francis Bretherton in 1964:

The **pressuron** is a hypothetical scalar particle which couples to both gravity and matter theorised in 2013. Although originally postulated without self-interaction potential, the pressuron is also a dark energy candidate when it has such a potential. The pressuron takes its name from the fact that it decouples from matter in pressure-less regimes, allowing the scalar-tensor theory of gravity involving it to pass solar system tests, as well as tests on the equivalence principle, even though it is fundamentally coupled to matter. Such a decoupling mechanism could explain why gravitation seems to be well described by general relativity at present epoch, while it could actually be more complex than that. Because of the way it couples to matter, the pressuron is a special case of the hypothetical string dilaton. Therefore, it is one of the possible solutions to the present non-observation of various signals coming from massless or light scalar fields that are generically predicted in string theory.

In fluid dynamics, a **trochoidal wave** or **Gerstner wave** is an exact solution of the Euler equations for periodic surface gravity waves. It describes a progressive wave of permanent form on the surface of an incompressible fluid of infinite depth. The free surface of this wave solution is an inverted (upside-down) trochoid – with sharper crests and flat troughs. This wave solution was discovered by Gerstner in 1802, and rediscovered independently by Rankine in 1863.

In continuum mechanics, Whitham's **averaged Lagrangian** method – or in short **Whitham's method** – is used to study the Lagrangian dynamics of slowly-varying wave trains in an inhomogeneous (moving) medium. The method is applicable to both linear and non-linear systems. As a direct consequence of the averaging used in the method, wave action is a conserved property of the wave motion. In contrast, the wave energy is not necessarily conserved, due to the exchange of energy with the mean motion. However the total energy, the sum of the energies in the wave motion and the mean motion, will be conserved for a time-invariant Lagrangian. Further, the averaged Lagrangian has a strong relation to the dispersion relation of the system.

In fluid dynamics, the **Craik–Leibovich (CL) vortex force** describes a forcing of the mean flow through wave–current interaction, specifically between the Stokes drift velocity and the mean-flow vorticity. The CL vortex force is used to explain the generation of Langmuir circulations by an instability mechanism. The CL vortex-force mechanism was derived and studied by Sidney Leibovich and Alex D.D. Craik in the 1970s and 80s, in their studies of Langmuir circulations.

In physics and mathematics, the **Clebsch representation** of an arbitrary three-dimensional vector field is:

- Andrews, D.G.; McIntyre, M.E. (1978), "On wave-action and its relatives",
*Journal of Fluid Mechanics*,**89**(4): 647–664, Bibcode:1978JFM....89..647A, doi:10.1017/S0022112078002785 - Bretherton, F.P.; Garrett, C.J.R. (1968), "Wavetrains in inhomogeneous moving media",
*Proceedings of the Royal Society of London A: Mathematical and Physical Sciences*,**302**(1471): 529–554, Bibcode:1968RSPSA.302..529B, doi:10.1098/rspa.1968.0034 - Craik, A.D.D. (1988),
*Wave interactions and fluid flows*, Cambridge University Press, ISBN 9780521368292 - Dewar, R.L. (1970), "Interaction between hydromagnetic waves and a time‐dependent, inhomogeneous medium",
*Physics of Fluids*,**13**(11): 2710–2720, Bibcode:1970PhFl...13.2710D, doi:10.1063/1.1692854, ISSN 0031-9171 - Grimshaw, R. (1984), "Wave action and wave–mean flow interaction, with application to stratified shear flows",
*Annual Review of Fluid Mechanics*,**16**: 11–44, Bibcode:1984AnRFM..16...11G, doi:10.1146/annurev.fl.16.010184.000303 - Hayes, W.D. (1970), "Conservation of action and modal wave action",
*Proceedings of the Royal Society of London A: Mathematical and Physical Sciences*,**320**(1541): 187–208, Bibcode:1970RSPSA.320..187H, doi:10.1098/rspa.1970.0205 - Sturrock, P.A. (1962), "Energy and momentum in the theory of waves in plasmas", in Bershader, D.,
*Plasma Hydromagnetics. Sixth Lockheed Symposium on Magnetohydrodynamics*, Stanford University Press, pp. 47–57, OCLC 593979237 - Whitham, G.B. (1965), "A general approach to linear and non-linear dispersive waves using a Lagrangian",
*Journal of Fluid Mechanics*,**22**(2): 273–283, Bibcode:1965JFM....22..273W, doi:10.1017/S0022112065000745 - Whitham, G.B. (1974),
*Linear and nonlinear waves*, Wiley-Interscience, ISBN 0-471-94090-9

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