# 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 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.Craik (2004).]

Airy wave theory is often applied in ocean engineering and coastal engineering for the modelling of random sea states — giving a description of the wave kinematics and dynamics 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-order nonlinear 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 take viscosity, vorticity, turbulence and/or flow 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 oscillatory Stokes boundary layers at the boundaries of the fluid domain. [cite journal
first =
last = Lighthill
year = 1986
journal = J. Fluid Mech.
volume = 173
pages = 667–681
doi = 10.1017/S0022112086001313
]

Airy wave theory is often used in ocean engineering and coastal engineering. Especially for random waves, sometimes called wave turbulence, the evolution of the wave statistics — including the wave spectrum — 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 the WKBJ approximation, wave shoaling and refraction 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 nonlinear trochoidal wave theory in 1804, which however is not irrotational.

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":

:$eta\left(x,t\right), =, a, cos, left\left( kx, -, omega t ight\right)$

where
*"a" is the wave amplitude in metre,
*cos is the cosine function,
*"k" is the angular wavenumber in radian per metre, related to the wavelength "λ" as

:$k,=,frac\left\{2pi\right\}\left\{lambda\right\},,$

*"ω" is the angular frequency in radian per second, related to the period "T" and frequency "f" by

:$omega,=,frac\left\{2pi\right\}\left\{T\right\},=,2pi,f.,$

The waves propagate along the the water surface with the phase speed "cp":

:$c_p, =, frac\left\{omega\right\}\left\{k\right\}, =, frac\left\{lambda\right\}\left\{T\right\}.$

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:

:$H, =, 2, a qquad ext\left\{and\right\} qquad a, =, frac12, H,$

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 circles, and in finite depth closed ellipsoids — with the ellipsoids becoming flatter near the bottom of the fluid layer. So while the wave propagates, the fluid particles just orbit (oscillate) around their average 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 — and potential theory can be used to describe the flow. The velocity potential "Φ"("x","z","t") is related to the flow velocity components "u""x" and "u""z" in the horizontal ("x") and vertical ("z") directions by:

:$u_x, =, frac\left\{partialPhi\right\}\left\{partial x\right\} quad ext\left\{and\right\} quad u_z, =, frac\left\{partialPhi\right\}\left\{partial z\right\}.$

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

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

Boundary conditions 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:

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

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:

:$\left(3\right) qquad frac\left\{partialeta\right\}\left\{partial t\right\}, =, frac\left\{partialPhi\right\}\left\{partial z\right\} quad ext\left\{ at \right\} z, =, eta\left(x,t\right).$

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:

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

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 &part;"Φ"/&part;"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:

:$eta, =, a, cos, \left( k x, -, omega t \right).$

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:

:

with sinh and cosh the hyperbolic sine and hyperbolic 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:

:$omega^2, =, g, k, anh, \left( k h \right),$

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 an eigenproblem. 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. The wavenumber 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 a Doppler 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 ""x" and "ξ""z" are the time integrals 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 the wavelength, "h" > ½ "λ", the phase 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 of period 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.]

:$Omega^2\left(k\right), =, left\left( g, +, frac\left\{gamma\right\}\left\{ ho\right\}, k^2 ight\right), k; anh, \left( k, h \right),$

with "γ" the surface tension, with SI units in N/m2. All above equations for linear waves remain the same, if the gravitational acceleration "g" is replaced by [Lighthill (1978), p. 223.]

:$ilde\left\{g\right\}, =, g, +, frac\left\{gamma\right\}\left\{ ho\right\}, k^2.$

As a result of surface tension, the waves propagate faster. Surface tension only has influence for short waves, with wavelengths less than a few decimeters in case of a water–air interface. For very short wavelengths — two millimeter 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.]

:$Omega^2\left(k\right), =, |k|, left\left( frac\left\{ ho- ho\text{'}\right\}\left\{ ho+ ho\text{'}\right\} g, +, frac\left\{sigma\right\}\left\{ ho+ ho\text{'}\right\}, k^2 ight\right),$

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" forecasts of wave conditions. [See for example: the [http://www.weather.gov/om/marine/zone/hsmz.htm "High seas forecasts"] of NOAA'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 and potential 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.]

:$E_ ext\left\{pot\right\}, =, overline\left\{int_\left\{-h\right\}^\left\{eta\right\} ho,g,z; ext\left\{d\right\}z\right\}, -, int_\left\{-h\right\}^0 ho,g,z; ext\left\{d\right\}z, =, overline\left\{frac12, ho,g,eta^2\right\}, =, frac14, ho,g,a^2,$

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:

:with "σ" the intrinsic frequency, see the table of wave quantities. Using the dispersion relation, the result for gravity surface waves is:

:$E_ ext\left\{kin\right\}, =, frac14, ho, g, a^2.$

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" 1, MacMillan, 2nd revised edition, 1894.] . Adding potential and kinetic contributions, "E"pot and "E"kin, the mean energy density per unit horizontal area "E" of the wave motion is:

:$E, =, E_ ext\left\{pot\right\}, +, E_ ext\left\{kin\right\}, =, frac12, ho, g, a^2.$

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

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

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 $mathcal\left\{A\right\}=E/sigma$: 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 ]

:

with the action flux and the group velocity vector. Action conservation forms the basis for many wave energy (or wave turbulence) models. [Phillips (1977), pp. 179–183.] It is also the basis of coastal engineering models for the computation of wave 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.]

:

with:
* is the mean wave energy density flux,
*$mathbb\left\{S\right\}$ is the radiation stress tensor and
* is the mean-velocity shear-rate tensor.In this equation in non-conservation form, the Frobenius inner product 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, the mean wave energy density $E$ is conserved. The two tensors $mathbb\left\{S\right\}$ and are in a Cartesian coordinate system of the form: [Phillips (1977), p. 68.]

:

with $k_x$ and $k_y$ the components of the wavenumber vector and similarly $U_x$ and $U_y$ the components in of the mean velocity vector .

Wave mass flux and wave momentum

The mean horizontal momentum per unit area induced by the wave motion — and also the wave-induced mass flux or mass transport — is:Phillips (1977), pp. 39–40 & 61.]

:

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 of phase velocity for periodic nonlinear waves:
*"Stokes first definition of wave celerity" — with the mean Eulerian flow velocity equal to zero for all elevations "z" below the wave troughs, 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 an undertow.

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 defined as:Phillips (1977), pp. 61–63.]

:

Note that for deep water, when the mean depth "h" goes to infinity, the mean Eulerian velocity and mean transport velocity become equal.

The equation for mass conservation is:

:

where "h"("x","t") is the mean water-depth, slowly varying in space and time. Similarly, the mean horizontal momentum evolves as:

:with "d" the still-water depth (the sea bed is at "z"=–"d"), $mathbb\left\{S\right\}$ is the wave radiation-stress tensor, $mathbb\left\{I\right\}$ is the identity matrix and $otimes$ is the dyadic product:

: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 with Noether'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.]

:

The wave kinematics are described through the wave-crest conservation equation: [Phillips (1977), p. 23.]

:

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 the curl of the wave-crest conservation, it can be seen that an initially irrotational wavenumber field stays irrotational.

Stokes drift

When following a single particle in pure wave motion 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 , which is the Stokes drift after one wave cycle divided by the period, can be estimated using the results of linear theory:Phillips (1977), p. 44.]

:

so it varies as a function of elevaton. The given formula is for Stokes first definition of wave celerity. When is integrated over depth, the expression for the mean wave momentum is recovered.

* Boussinesq approximation (water waves)nonlinear theory for waves in shallow water.
* Capillary wave — surface waves under the action of surface 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

* 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
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
volume=13
publisher=World Scientific
location=Singapore
isbn=981 02 0427 2
oclc=36126836
Two parts, 967 pages.
* cite book
first=H.
last=Lamb
year=1994
title=Hydrodynamics
publisher=Cambridge University Press
edition=6th edition
isbn=978 0 521 45868 9
oclc=30070401
Originally published in 1879, the 6th extended edition appeared first in 1932.
* cite book
title=Fluid mechanics
first=L. D.
last=Landau
coauthors=Lifshitz, E. M.
year=1986
publisher=Pergamon Press
series=Course of Theoretical Physics
volume=6
edition=2nd revised edition
isbn=0 08 033932 8
oclc=15017127

* cite book
first = M. J.
last = 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=2nd edition
isbn=0 521 29801 6
oclc=7319931

Notes

* [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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