- Airy wave theory
In

fluid dynamics ,**Airy wave theory**(often referred to as**linear wave theory**) gives a linearised description of the propagation of gravity waves on the surface of a homogeneousfluid layer. The theory assumes that the fluid layer has a uniform mean depth, and that thefluid flow isinviscid ,incompressible andirrotational . This theory was first published, in correct form, byGeorge Biddell Airy in the 19^{th}century.Craik (2004).]Airy wave theory is often applied in

ocean engineering andcoastal engineering for the modelling ofrandom sea state s — giving a description of the wavekinematics anddynamics of high-enough accuracy for many purposes.Cite book

last=Goda

first=Y.

title=Random Seas and Design of Maritime Structures

series=Advanced Series on Ocean Engineering | volume=15

year=2000

publisher=World Scientific Publishing Company

location=Singapore

isbn=981-02-3256-X

oclc=45200228 ] [*Dean & Dalrymple (1991).*] Further, several second-ordernonlinear properties of surface gravity waves, and their propagation, can be estimated from its results. [*Phillips (1977), §3.2, pp. 37–43 and §3.6, pp. 60–69.*] This linear theory is often used to get a quick and rough estimate of wave characteristics and their effects.**Description**Airy wave theory uses a

potential flow approach to describe the motion of gravity waves on a fluid surface. The use of — inviscid and irrotational — potential flow in water waves is remarkably successful, giving its failure to describe many other fluid flows where it is often essential to takeviscosity ,vorticity ,turbulence and/orflow separation into account. This is due to the fact that for the oscillatory part of the fluid motion, wave-induced vorticity is restricted to some thin oscillatoryStokes boundary layer s at the boundaries of the fluid domain. [*cite journal*]

first =

last = Lighthill

authorlink=M. J. Lighthill

year = 1986

title = Fundamentals concerning wave loading on offshore structures

journal = J. Fluid Mech.

volume = 173

pages = 667–681

doi = 10.1017/S0022112086001313Airy wave theory is often used in

ocean engineering andcoastal engineering . Especially forrandom waves, sometimes calledwave turbulence , the evolution of the wave statistics — including the wavespectrum — is predicted well over not too long distances (in terms of wavelengths) and in not too shallow water.Diffraction is one of the wave effects which can be described with Airy wave theory. Further, by using theWKBJ approximation ,wave shoaling andrefraction can be predicted.Earlier attempts to describe gravity surface waves using potential flow were made by, among others, Laplace, Poisson, Cauchy and Kelland. But Airy was the first to publish the correct derivation and formulation in 1841. Soon after, in 1847, the linear theory of Airy was extended by Stokes for

non-linear wave motion, correct up to third order in the wave steepness.Stokes (1847).] Even before Airy's linear theory, Gerstner derived a nonlineartrochoid al wave theory in 1804, which however is notirrotational .Airy wave theory is a linear theory for the propagation of waves on the surface of a potential flow and above a horizontal bottom. The free surface elevation "η"("x","t") of one wave component is

sinusoidal , as a function of horizontal position "x" and time "t"::$eta(x,t),\; =,\; a,\; cos,\; left(\; kx,\; -,\; omega\; t\; ight)$

where

*"a" is the waveamplitude inmetre ,

*cos is thecosine function,

*"k" is theangular wavenumber inradian per metre, related to thewavelength "λ" as:$k,=,frac\{2pi\}\{lambda\},,$

*"ω" is the

angular frequency in radian persecond , related to the period "T" andfrequency "f" by:$omega,=,frac\{2pi\}\{T\},=,2pi,f.,$

The waves propagate along the the water surface with the

phase speed "c_{p}"::$c\_p,\; =,\; frac\{omega\}\{k\},\; =,\; frac\{lambda\}\{T\}.$

The angular wavenumber "k" and frequency "ω" are not independent parameters (and thus also wavelength "λ" and period "T" are not independent), but are coupled. Gravity surface waves on a fluid are dispersive waves — exhibiting frequency dispersion — meaning that each wavenumber has its own frequency and phase speed.

Note that in engineering the

wave height "H" — the difference in elevation between crest and trough — is often used::$H,\; =,\; 2,\; a\; qquad\; ext\{and\}\; qquad\; a,\; =,\; frac12,\; H,$

valid in the present case of linear periodic waves.

Underneath the surface, there is a fluid motion associated with the free surface motion. While the surface elevation shows a propagating wave, the fluid particles are in an orbital motion. Within the framework of Airy wave theory, the orbits are in deep water closed

circle s, and in finite depth closedellipsoid s — with the ellipsoids becoming flatter near the bottom of the fluid layer. So while the wave propagates, the fluid particles just orbit (oscillate) around theiraverage position. With the propagating wave motion, the fluid particles transfer energy in the wave propagation direction, without having a mean velocity. The diameter of the orbits reduces with depth below the free surface. In deep water, the orbit's diameter is reduces to 4% of its free-surface value at a depth of half a wavelength.In a similar fashion, there is also a pressure oscillation underneath the free surface, with wave-induced pressure oscillations reducing with depth — in the same way as for the orbital motion of fluid parcels.

**Mathematical formulation of the wave motion****Flow problem formulation**The waves propagate in the horizontal direction, with coordinate "x", and a fluid domain bound above by a free surface at "z" = "η"("x","t"), with "z" the vertical coordinate (positive in the upward direction) and "t" being

time .For the equations, solution and resulting approximations in deep and shallow water, see Dingemans (1997), Part 1, §2.1, pp. 38–45. Or: Phillips (1977), pp. 36–45.] The level "z" = 0 corresponds with the mean surface elevation. The impermeable bed underneath the fluid layer is at "z" = -"h". Further, the flow is assumed to be incompressible and irrotational — a good approximation of the flow in the fluid interior for waves on a liquid surface — andpotential theory can be used to describe the flow. The velocity potential "Φ"("x","z","t") is related to theflow velocity components "u"_{"x"}and "u"_{"z"}in the horizontal ("x") and vertical ("z") directions by::$u\_x,\; =,\; frac\{partialPhi\}\{partial\; x\}\; quad\; ext\{and\}\; quad\; u\_z,\; =,\; frac\{partialPhi\}\{partial\; z\}.$

Then, due to the continuity equation for an incompressible flow, the potential "Φ" has to satisfy the

Laplace equation ::$(1)\; qquad\; frac\{partial^2Phi\}\{partial\; x^2\},\; +,\; frac\{partial^2Phi\}\{partial\; z^2\},\; =,\; 0.$

Boundary condition s are needed at the bed and the free surface in order to close the system of equations. For their formulation within the framework of linear theory, it is necessary to specify what the base state (or zeroth-order solution) of the flow is. Here, we assume the base state is rest, implying the mean flow velocities are zero.The bed being impermeable, leads to the kinematic bed boundary-condition:

:$(2)\; qquad\; frac\{partialPhi\}\{partial\; z\},\; =,\; 0\; quad\; ext\{\; at\; \}\; z,\; =,\; -h.$

In case of deep water — by which is meant

infinite water depth, from a mathematical point of view — the flow velocities have to go to zero in the limit as the vertical coordinate goes to minus infinity: "z" → -∞.At the free surface, for

infinitesimal waves, the vertical motion of the flow has to be equal to the vertical velocity of the free surface. This leads to the kinematic free-surface boundary-condition::$(3)\; qquad\; frac\{partialeta\}\{partial\; t\},\; =,\; frac\{partialPhi\}\{partial\; z\}\; quad\; ext\{\; at\; \}\; z,\; =,\; eta(x,t).$

If the free surface elevation "η"("x","t") was a known function, this would be enough to solve the flow problem. However, the surface elevation is an extra unknown, for which an additional boundary condition is needed. This is provided by Bernoulli's equation for an unsteady potential flow. The pressure above the free surface is assumed to be constant. This constant pressure is taken equal to zero, without loss of generality, since the level of the pressure does not alter the flow. After linearisation, this gives the dynamic free-surface boundary condition:

:$(4)\; qquad\; frac\{partialPhi\}\{partial\; t\},\; +,\; g,\; eta,\; =,\; 0\; quad\; ext\{\; at\; \}\; z,\; =,\; eta(x,t).$

Because this is a linear theory, in both free-surface boundary conditions — the kinematic and the dynamic one, equations (3) and (4) — the value of "Φ" and ∂"Φ"/∂"z" at the fixed mean level "z" = 0 is used.

**Solution for a progressive monochromatic wave**For a propagating wave of a single frequency — a

monochromatic wave — the surface elevation is of the form::$eta,\; =,\; a,\; cos,\; (\; k\; x,\; -,\; omega\; t\; ).$

The associated velocity potential, satisfying the Laplace equation (1) in the fluid interior, as well as the kinematic boundary conditions at the free surface (2), and bed (3), is:

:$Phi,\; =,\; frac\{omega\}\{k\},\; a,\; frac\{cosh,\; igl(\; k,\; (z+h)\; igr)\}\{sinh,\; (k,\; h)\},\; sin,\; (\; k\; x,\; -,\; omega\; t),$

with sinh and cosh the

hyperbolic sine andhyperbolic cosine function, respectively.But "η" and "Φ" also have to satisfy the dynamic boundary condition, which results in non-trivial (non-zero) values for the wave amplitude "a" only if the linear dispersion relation is satisfied::$omega^2,\; =,\; g,\; k,\; anh,\; (\; k\; h\; ),$

with tanh the

hyperbolic tangent . So angular frequency "ω" and wavenumber "k" — or equivalently period "T" and wavelength "λ" — cannot be chosen independently, but are related. This means that wave propagation at a fluid surface is aneigenproblem . When "ω" and "k" satisfy the dispersion relation, the wave amplitude "a" can be chosen freely (but small enough for Airy wave theory to be a valid approximation).**Table of wave quantities**In the table below, several flow quantities and parameters according to Airy wave theory are given. The given quantities are for a bit more general situation as for the solution given above. Firstly, the waves may propagate in an arbitrary horizontal direction in the

**"x**" = ("x","y") plane. Thewavenumber vector is**"k**", and is perpendicular to the cams of the wave crests. Secondly, allowance is made for a mean flow velocity**"U**", in the horizontal direction and uniform over (independent of) depth "z". This introduces aDoppler shift in the dispersion relations. The table only gives the oscillatory parts of flow quantities — velocities, particle excursions and pressure — and not their mean value or drift.The oscillatory particle excursions

**"ξ**"_{"x"}and "ξ"_{"z"}are the timeintegral s of the oscillatory flow velocities**"u**"_{"x"}and "u"_{"z"}respectively.Water depth is classified into three regimes: [

*Dean & Dalrymple (1991) pp. 64–65*]

***deep water**— for a water depth larger than half thewavelength , "h" > ½ "λ", thephase speed of the waves is hardly influenced by depth (this is the case for most wind waves on the sea and ocean surface), [*The error in the phase speed is less than 0.2% if depth "h" is taken to be infinite, for "h" > ½ "λ".*]

***shallow water**— for a water depth smaller than the wavelength divided by 20, "h" < frac|1|20 "λ", the phase speed of the waves is only dependent on water depth, and no longer a function ofperiod or wavelength; [*The error in the phase speed is less than 2% if wavelength effects are neglected for "h" <frac|1|20 "λ".*] and

***intermediate depth**— all other cases, frac|1|20 "λ" < "h" < ½ "λ", where both water depth and period (or wavelength) have a significant influence on the solution of Airy wave theory. In the limiting cases of deep and shallow water, simplifying approximations to the solution can be made. While for intermediate depth, the full formulations have to be used.**Surface tension effects**Due to

surface tension , the dispersion relation changes to:Phillips (1977), p. 37.]:$Omega^2(k),\; =,\; left(\; g,\; +,\; frac\{gamma\}\{\; ho\},\; k^2\; ight),\; k;\; anh,\; (\; k,\; h\; ),$

with "γ" the surface tension, with

SI units in N/m^{2}. All above equations for linear waves remain the same, if the gravitational acceleration "g" is replaced by [*Lighthill (1978), p. 223.*]:$ilde\{g\},\; =,\; g,\; +,\; frac\{gamma\}\{\; ho\},\; k^2.$

As a result of surface tension, the waves propagate faster. Surface tension only has influence for short waves, with wavelengths less than a few

decimeter s in case of a water–air interface. For very short wavelengths — twomillimeter in case of the interface between air and water – gravity effects are negligible.**Interfacial waves**Gravity surface waves are a special case of interfacial waves, on the interface between two fluids of different

density . Consider two fluids separated by an interface, and without further boundaries. Then their dispersion relation becomes: [*Lamb (1994), §267, page 458–460.*] [*Dingemans (1997), Section 2.1.1, p. 45.*]:$Omega^2(k),\; =,\; |k|,\; left(\; frac\{\; ho-\; ho\text{'}\}\{\; ho+\; ho\text{'}\}\; g,\; +,\; frac\{sigma\}\{\; ho+\; ho\text{'}\},\; k^2\; ight),$

where "ρ" and "ρ‘" are the densities of the two fluids, below ("ρ") and above ("ρ‘") the interface, respectively. For interfacial waves to exits, the lower layer has to be heavier than the upper one, "ρ > ρ‘". Otherwise, the interface is unstable and a

Rayleigh–Taylor instability develops.**Second-order wave properties**Several second-order wave properties, "i.e."

quadratic in the wave amplitude "a", can be derived directly from Airy wave theory. They are of importance in many practical applications, "e.g"forecast s of wave conditions. [*See for example: the [*]*http://www.weather.gov/om/marine/zone/hsmz.htm "High seas forecasts"*] ofNOAA 's National Weather service.**Wave energy density**Wave energy is a quantity of primary interest, since it is a primary quantity that is transported with the wave trains. [

*Phillips (1977), p. 23–25.*] As can be seen above, many wave quantities like surface elevation and orbital velocity are oscillatory in nature with zero mean (within the framework of linear theory). In water waves, the most used energy measure is the mean wave energy density per unit horizontal area. It is the sum of the kinetic andpotential energy density, integrated over the depth of the fluid layer and averaged over the wave phase. Simplest to derive is the mean potential energy density per unit horizontal area "E"_{pot}of the gravity surface waves, which is the deviation of the potential energy due to the presence of the waves:Phillips (1977), p. 39.]:$E\_\; ext\{pot\},\; =,\; overline\{int\_\{-h\}^\{eta\}\; ho,g,z;\; ext\{d\}z\},\; -,\; int\_\{-h\}^0\; ho,g,z;\; ext\{d\}z,\; =,\; overline\{frac12,\; ho,g,eta^2\},\; =,\; frac14,\; ho,g,a^2,$

with an overbar denoting the mean value (which in the present case of periodic waves can be taken either as a time average or an average over one wavelength in space).

The mean kinetic energy density per unit horizontal area "E"

_{kin}of the wave motion is similarly found to be::$E\_\; ext\{kin\},\; =,\; overline\{int\_\{-h\}^0\; frac12,\; ho,\; left\; [,\; left|\; \backslash boldsymbol\{U\},\; +,\; \backslash boldsymbol\{u\}\_x\; ight|^2,\; +,\; u\_z^2,\; ight]\; ;\; ext\{d\}z\},\; -,\; int\_\{-h\}^0\; frac12,\; ho,\; left|\; \backslash boldsymbol\{U\}\; ight|^2;\; ext\{d\}z,\; =,\; frac14,\; ho,\; frac\{sigma^2\}\{k,\; anh,\; (k,\; h)\},a^2,$with "σ" the intrinsic frequency, see the table of wave quantities. Using the dispersion relation, the result for gravity surface waves is:

:$E\_\; ext\{kin\},\; =,\; frac14,\; ho,\; g,\; a^2.$

As can be seen, the mean kinetic and potential energy densities are equal. This is a general property of energy densities of progressive linear waves in a

conservative system . [*cite journal | title=On progressive waves | author=Lord Rayleigh (J. W. Strutt) | authorlink=Lord Rayleigh | year=1877 | journal=Proceedings of the London Mathematical Society | volume=9 | pages=21–26 | doi=10.1112/plms/s1-9.1.21 Reprinted as Appendix in: "Theory of Sound"*] . Adding potential and kinetic contributions, "E"**1**, MacMillan, 2nd revised edition, 1894._{pot}and "E"_{kin}, the mean energy density per unit horizontal area "E" of the wave motion is::$E,\; =,\; E\_\; ext\{pot\},\; +,\; E\_\; ext\{kin\},\; =,\; frac12,\; ho,\; g,\; a^2.$

In case of surface tension effects not being negligible, their contribution also adds to the potential and kinetic energy densities, givingPhillips (1977), p. 38.]

:$E\_\; ext\{pot\},\; =,\; E\_\; ext\{kin\},\; =,\; frac14,\; left(\; ho,\; g,\; +,\; gamma,\; k^2\; ight),\; a^2,\; qquad\; ext\{so\}\; qquad\; E,\; =,\; E\_\; ext\{pot\},\; +,\; E\_\; ext\{kin\},\; =,\; frac12,\; left(\; ho,\; g,\; +,\; gamma,\; k^2\; ight),\; a^2,$

with "γ" the

surface tension .**Wave action, wave energy flux and radiation stress**In general, there can be an energy transfer between the wave motion and the mean fluid motion. This means, that the wave energy density is not in all cases a conserved quantity (neglecting dissipative effects), but the total energy density — the sum of the energy density per unit area of the wave motion and the mean flow motion — is. However, there is for slowly-varying wave trains, propagating in slowly-varying

bathymetry and mean-flow fields, a similar and conserved wave quantity, the wave action $mathcal\{A\}=E/sigma$: Phillips (1977), p. 26.] citation | first=G.B. | last=Whitham | title=Linear and nonlinear waves | publisher = Wiley-Interscience | year=1974 | isbn=0 471 94090 9 | oclc=815118 , p. 559.] [*citation | title = Wavetrains in inhomogeneous moving media | first1=F. P. | last1=Bretherton | first2=C. J. R. | last2=Garrett | year = 1968 | journal = Proceedings of the Royal Society of London, Series A | volume = 302 | issue = 1471 | pages = 529–554 | doi = 10.1098/rspa.1968.0034*]:$frac\{partial\; mathcal\{A\{partial\; t\},\; +,\; ablacdotleft\; [\; left(\backslash boldsymbol\{U\}+\backslash boldsymbol\{c\}\_g\; ight),\; mathcal\{A\}\; ight]\; ,\; =,\; 0,$

with $left(\backslash boldsymbol\{U\}+\backslash boldsymbol\{c\}\_g\; ight),\; mathcal\{A\}$ the action

flux and $\backslash boldsymbol\{c\}\_g=c\_g,\backslash boldsymbol\{e\}\_k$ thegroup velocity vector. Action conservation forms the basis for many wave energy (orwave turbulence ) models. [*Phillips (1977), pp. 179–183.*] It is also the basis ofcoastal engineering models for the computation ofwave shoaling . [*Phillips (1977), pp. 70–74.*] Expanding the above wave action conservation equation leads to the following evolution equation for the wave energy density:Phillips (1977), p. 66.]:$frac\{partial\; E\}\{partial\; t\},\; +,\; ablacdotleft\; [left(\; \backslash boldsymbol\{U\}+\backslash boldsymbol\{c\}\_g\; ight),\; E\; ight]\; ,\; +,\; mathbb\{S\}:left(\; abla\backslash boldsymbol\{U\}\; ight),\; =,\; 0,$

with:

*$left(\; \backslash boldsymbol\{U\}+\backslash boldsymbol\{c\}\_g\; ight),\; E$ is the mean wave energy density flux,

*$mathbb\{S\}$ is the radiation stresstensor and

*$abla\backslash boldsymbol\{U\}$ is the mean-velocity shear-rate tensor.In this equation in non-conservation form, theFrobenius inner product $mathbb\{S\}:(\; abla\backslash boldsymbol\{U\})$ is the source term describing the energy exchange of the wave motion with the mean flow. Only in case the mean shear-rate is zero, $abla\backslash boldsymbol\{U\}=mathsf\{0\},$ the mean wave energy density $E$ is conserved. The two tensors $mathbb\{S\}$ and $abla\backslash boldsymbol\{U\}$ are in aCartesian coordinate system of the form: [*Phillips (1977), p. 68.*]:$egin\{align\}\; mathbb\{S\},\; =,\; egin\{pmatrix\}\; S\_\{xx\}\; S\_\{xy\}\; \backslash \; S\_\{yx\}\; S\_\{yy\}\; end\{pmatrix\},\; =,\; mathbb\{I\},\; left(\; frac\{c\_g\}\{c\_p\}\; -\; frac12\; ight),\; E,\; +,\; frac\{1\}\{k^2\},\; egin\{pmatrix\}\; k\_x,\; k\_x\; k\_x,\; k\_y\; \backslash \; [2ex]\; k\_y,\; k\_x\; k\_y,\; k\_y\; end\{pmatrix\},\; frac\{c\_g\}\{c\_p\},\; E,\; \backslash \; mathbb\{I\},\; =,\; egin\{pmatrix\}\; 1\; 0\; \backslash \; 0\; 1\; end\{pmatrix\}\; quad\; ext\{and\}\; \backslash \; abla\; \backslash boldsymbol\{U\},\; =,\; egin\{pmatrix\}\; displaystyle\; frac\{partial\; U\_x\}\{partial\; x\}\; displaystyle\; frac\{partial\; U\_y\}\{partial\; x\}\; \backslash \; [2ex]\; displaystyle\; frac\{partial\; U\_x\}\{partial\; y\}\; displaystyle\; frac\{partial\; U\_y\}\{partial\; y\}\; end\{pmatrix\},\; end\{align\}$

with $k\_x$ and $k\_y$ the components of the wavenumber vector $\backslash boldsymbol\{k\}$ and similarly $U\_x$ and $U\_y$ the components in of the mean velocity vector $\backslash boldsymbol\{U\}$.

**Wave mass flux and wave momentum**The mean horizontal

momentum per unit area $\backslash boldsymbol\{M\}$ induced by the wave motion — and also the wave-inducedmass flux or mass transport — is:Phillips (1977), pp. 39–40 & 61.]:$\backslash boldsymbol\{M\},\; =,\; overline\{int\_\{-h\}^eta\; ho,\; left(\; \backslash boldsymbol\{U\}+\backslash boldsymbol\{u\}\_x\; ight);\; ext\{d\}z\},\; -,\; int\_\{-h\}^0\; ho,\; \backslash boldsymbol\{U\};\; ext\{d\}z,\; =,\; frac\{E\}\{c\_p\},\; \backslash boldsymbol\{e\}\_k,$

which is an exact result for periodic progressive water waves, also valid for

nonlinear waves. [*Phillips (1977), p. 40.*] However, its validity strongly depends on what is called the wave momentum and mass flux. Stokes already identified two possible definitions ofphase velocity for periodic nonlinear waves:

*"Stokes first definition of wavecelerity " — with the mean Eulerian flow velocity equal to zero for all elevations "z" below the wavetrough s, and

*"Stokes second definition of wave celerity" — with the mean mass transport equal to zero.The above used definition of wave momentum corresponds with Stokes' first definition. However, for waves perpendicular to a coast line or in closed laboratory wave channel, the second definition is more appropriate, with the waves having zero mass flux and momentum. [*Phillips (1977), p. 70.*] Then the above mass flux is compensated by anundertow .So in general, there are quite some subtleties involved. Therefore also the term pseudo-momentum of the waves is used instead of wave momentum. [

*citation | title = On the 'wave-momentum' myth | first1= M. E. | last1=McIntyre | year = 1978 | journal = Journal of Fluid Mechanics | volume = 106 | pages = 331–347 | doi = 10.1017/S0022112081001626*]**Mass and momentum evolution equations**For slowly-varying

bathymetry , wave and mean-flow fields, the evolution of the mean flow can de described in terms of the mean mass-transport velocity $ilde\{\backslash boldsymbol\{U$ defined as:Phillips (1977), pp. 61–63.]:$ilde\{\backslash boldsymbol\{U,\; =,\; \backslash boldsymbol\{U\},\; +,\; frac\{\backslash boldsymbol\{M\{\; ho,h\}.$

Note that for deep water, when the mean depth "h" goes to infinity, the mean Eulerian velocity $\backslash boldsymbol\{U\}$ and mean transport velocity $ilde\{\backslash boldsymbol\{U$ become equal.

The equation for mass conservation is:

:$frac\{partial\}\{partial\; t\}left(\; ho,\; h,\; ight),\; +,\; abla\; cdot\; left(\; ho,\; h,\; ilde\{\backslash boldsymbol\{U\; ight),\; =,\; 0,$

where "h"(

**"x**","t") is the mean water-depth, slowly varying in space and time. Similarly, the mean horizontal momentum evolves as::$frac\{partial\}\{partial\; t\}left(\; ho,\; h,\; ilde\{\backslash boldsymbol\{U\; ight),\; +,\; abla\; cdot\; left(\; ho,\; h,\; ilde\{\backslash boldsymbol\{U\; otimes\; ilde\{\backslash boldsymbol\{U,\; +,\; frac12,\; ho,g,h^2,mathbb\{I\},\; +,\; mathbb\{S\}\; ight),\; =,\; ho,\; g,\; h,\; abla\; d,$with "d" the still-water depth (the sea bed is at "z"=–"d"), $mathbb\{S\}$ is the wave radiation-stress

tensor , $mathbb\{I\}$ is theidentity matrix and $otimes$ is thedyadic product ::$ilde\{\backslash boldsymbol\{U\; otimes\; ilde\{\backslash boldsymbol\{U,\; =,\; egin\{pmatrix\}\; ilde\{U\}\_x,\; ilde\{U\}\_x\; ilde\{U\}\_x,\; ilde\{U\}\_y\; \backslash \; [2ex]\; ilde\{U\}\_y,\; ilde\{U\}\_x\; ilde\{U\}\_y,\; ilde\{U\}\_y\; end\{pmatrix\}.$Note that mean horizontal

momentum is only conserved if the sea bed is horizontal ("i.e" the still-water depth "d" is a constant), in agreement withNoether's theorem .The system of equations is closed through the description of the waves. Wave energy propagation is described through the wave-action conservation equation (without dissipation and nonlinear wave interactions):Phillips (1977), p. 66.]

:$frac\{partial\}\{partial\; t\}\; left(\; frac\{E\}\{sigma\},\; ight)\; +,\; abla\; cdot\; left\; [\; left(\; \backslash boldsymbol\{U\}\; +\backslash boldsymbol\{c\}\_g\; ight),\; frac\{E\}\{sigma\}\; ight]\; ,\; =,\; 0.$

The wave kinematics are described through the wave-crest conservation equation: [

*Phillips (1977), p. 23.*]:$frac\{partial\; \backslash boldsymbol\{k\{partial\; t\},\; +,\; abla\; omega,\; =,\; \backslash boldsymbol\{0\},$

with the angular frequency "ω" a function of the (angular)

wavenumber **"k**", related through the dispersion relation. For this to be possible, the wave field must be coherent. By taking thecurl of the wave-crest conservation, it can be seen that an initiallyirrotational wavenumber field stays irrotational.**Stokes drift**When following a single particle in pure wave motion $(\backslash boldsymbol\{U\}=\backslash boldsymbol\{0\}),$ according to linear Airy wave theory the particles are in closed elliptical orbit. However, in nonlinear waves this is no longer the case and the particles exhibit a

Stokes drift . The Stokes drift velocity $ar\{\backslash boldsymbol\{u\_S$, which is the Stokes drift after one wave cycle divided by theperiod , can be estimated using the results of linear theory:Phillips (1977), p. 44.]:$ar\{\backslash boldsymbol\{u\_S,\; =,\; frac12,\; sigma,\; k,\; a^2,\; frac\{cosh,\; 2,k,(z+h)\}\{sinh^2,\; (k,h)\},\; \backslash boldsymbol\{e\}\_k,$

so it varies as a function of elevaton. The given formula is for Stokes first definition of wave celerity. When $ho,ar\{\backslash boldsymbol\{u\_S$ is integrated over depth, the expression for the mean wave momentum $\backslash boldsymbol\{M\}$ is recovered.

**See also***

Boussinesq approximation (water waves) —nonlinear theory for waves in shallow water.

*Capillary wave — surface waves under the action ofsurface tension

*Ocean surface wave — real water waves as seen in the ocean and sea

*Wave power — using ocean and sea waves for power generation.**References****Historical***citation | first=G. B. | last=Airy | author-link=George Biddell Airy | year=1841 | contribution=Tides and waves | title=Encyclopaedia Metropolitana | publication-date=1817–1845 | series=Mixed Sciences | volume=3 | editors=H.J. Rose, et al. . Also: "Trigonometry, On the Figure of the Earth, Tides and Waves", 396 pp.

*cite journal | first=G. G. | last=Stokes | authorlink=George Gabriel Stokes | year= 1847 | title= On the theory of oscillatory waves | journal= Transactions of the Cambridge Philosophical Society | volume= 8 | pages= 441–455

Reprinted in: cite book | first= G. G. | last= Stokes | year= 1880 | title= Mathematical and Physical Papers, Volume I | publisher= Cambridge University Press | pages= 197–229 | url=http://www.archive.org/details/mathphyspapers01stokrich**Further reading*** cite journal

first=A. D. D.

last=Craik

year=2004

title=The origins of water wave theory

journal=Annual Review of Fluid Mechanics

volume=36

pages=1–28

doi=10.1146/annurev.fluid.36.050802.122118

* cite book

title=Water wave mechanics for engineers and scientists

first=R. G.

last=Dean

coauthors=Dalrymple, R. A.

year=1991

series=Advanced Series on Ocean Engineering

volume=2

publisher=World Scientific

location=Singapore

isbn=978 981 02 0420 4

oclc=22907242

* cite book

title=Water wave propagation over uneven bottoms

first=M. W.

last=Dingemans

year=1997

series=Advanced Series on Ocean Engineering

volume=13

publisher=World Scientific

location=Singapore

isbn=981 02 0427 2

oclc=36126836 Two parts, 967 pages.

* cite book

first=H.

last=Lamb

authorlink=Horace Lamb

year=1994

title=Hydrodynamics

publisher=Cambridge University Press

edition=6^{th}edition

isbn=978 0 521 45868 9

oclc=30070401 Originally published in 1879, the 6^{th}extended edition appeared first in 1932.

* cite book

title=Fluid mechanics

first=L. D.

last=Landau

authorlink=Lev Landau

coauthors=Lifshitz, E. M.

year=1986

publisher=Pergamon Press

series=Course of Theoretical Physics

volume=6

edition=2^{nd}revised edition

isbn=0 08 033932 8

oclc=15017127

* cite book

first = M. J.

last = Lighthill

authorlink = M. J. Lighthill

title = Waves in fluids

publisher = Cambridge University Press

year = 1978

isbn = 0521292336

oclc = 2966533 504 pp.

* cite book

first=O. M.

last=Phillips

title=The dynamics of the upper ocean

publisher=Cambridge University Press

year=1977

edition=2^{nd}edition

isbn=0 521 29801 6

oclc=7319931**Notes****External links*** [

*http://www.wikiwaves.org/index.php/Linear_Theory_of_Ocean_Surface_Waves Linear theory of ocean surface waves*] on WikiWaves.

* [*http://web.mit.edu/fluids-modules/www/potential_flows/LecturesHTML/lec19bu/node1.html Water waves*] at MIT.

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