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| In [[fluid dynamics]], '''Airy wave theory''' (often referred to as '''linear wave theory''') gives a [[linear system|linearised]] description of the [[wave propagation|propagation]] of [[gravity wave]]s 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.<ref name=Craik2004>Craik (2004).</ref>
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| Airy wave theory is often applied in [[Offshore construction|ocean engineering]] and [[coastal engineering]] for the modelling of [[random]] [[sea state]]s – giving a description of the wave [[kinematics]] and [[dynamics (mechanics)|dynamics]] of high-enough accuracy for many purposes.<ref name=Goda>{{Cite book
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| | last=Goda
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| | first=Y.
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| | title=Random Seas and Design of Maritime Structures
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| | series=Advanced Series on Ocean Engineering | volume=15
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| | year=2000
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| | publisher=World Scientific Publishing Company
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| | location=Singapore
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| | isbn=981-02-3256-X
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| | oclc=45200228
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| }}</ref>
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| <ref>Dean & Dalrymple (1991).</ref> Further, several [[perturbation theory|second-order]] [[nonlinear]] properties of surface gravity waves, and their propagation, can be estimated from its results.<ref>Phillips (1977), §3.2, pp. 37–43 and §3.6, pp. 60–69.</ref> Airy wave theory is also a good approximation for [[tsunami]] waves in the ocean, before they steepen near the coast.
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|
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| This linear theory is often used to get a quick and rough estimate of wave characteristics and their effects. This approximation is accurate for small ratios of the [[wave height]] to water depth (for waves in [[waves in shallow water|shallow water]]), and wave height to wavelength (for waves in deep water).
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| == Description ==
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| [[File:Sine wave amplitude.svg|right|thumb|Wave characteristics.]]
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| [[File:Dispersion gravity 1.svg|right|thumb|Dispersion of gravity waves on a fluid surface. [[phase velocity|Phase]] and [[group velocity]] divided by ''√(gh)'' as a function of ''h/λ''. '''A''': phase velocity, '''B''': group velocity, '''C''': phase and group velocity ''√(gh)'' valid in shallow water. ''Drawn lines'': based on dispersion relation valid in arbitrary depth. ''Dashed lines'': based on dispersion relation valid in deep water.]]
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| Airy wave theory uses a [[potential flow]] (or [[velocity potential]]) 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, given 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 layer]]s at the boundaries of the fluid domain.<ref>{{cite journal
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| | first = M. J.
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| | last = Lighthill
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| | authorlink=M. J. Lighthill
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| | year = 1986
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| | title = Fundamentals concerning wave loading on offshore structures
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| | journal = J. Fluid Mech.
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| | volume = 173
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| | pages = 667–681
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| | doi = 10.1017/S0022112086001313
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| |bibcode = 1986JFM...173..667L }}</ref>
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| Airy wave theory is often used in [[Offshore construction|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.<ref name=Goda/>
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| Earlier attempts to describe surface gravity waves using potential flow were made by, among others, [[Pierre-Simon Laplace|Laplace]], [[Siméon Denis Poisson|Poisson]], [[Augustin Louis Cauchy|Cauchy]] and [[Philip Kelland|Kelland]]. But [[George Biddell Airy|Airy]] was the first to publish the correct derivation and formulation in 1841.<ref name=Craik2004/> Soon after, in 1847, the linear theory of Airy was extended by [[George Gabriel Stokes|Stokes]] for [[non-linear]] wave motion – known as [[Stokes wave|Stokes' wave theory]] – correct up to [[Perturbation theory#Perturbation orders|third order]] in the wave steepness.<ref name=Stokes1847>Stokes (1847).</ref> Even before Airy's linear theory, [[František Josef Gerstner|Gerstner]] derived a nonlinear [[trochoid]]al wave theory in 1804, which however is not [[irrotational]].<ref name=Craik2004/>
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| 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'':
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| :<math>\eta(x,t)\, =\, a\, \cos\, \left( kx\, -\, \omega t\right)</math>
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| where
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| *''a'' is the wave [[amplitude]] in metre,
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| *cos is the [[cosine]] function,
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| *''k'' is the [[angular wavenumber]] in [[radian]] per metre, related to the [[wavelength]] ''λ'' as
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| :<math>k\,=\,\frac{2\pi}{\lambda},\,</math>
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| *''ω'' is the [[angular frequency]] in radian per [[second]], related to the [[period (physics)|period]] ''T'' and [[frequency]] ''f'' by
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| :<math>\omega\,=\,\frac{2\pi}{T}\,=\,2\pi\,f.\,</math>
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| The waves propagate along the water surface with the [[phase speed]] ''c<sub>p</sub>'':
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| :<math>c_p\, =\, \frac{\omega}{k}\, =\, \frac{\lambda}{T}.</math>
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| 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. Surface gravity waves on a fluid are [[dispersion (water waves)|dispersive]] waves – exhibiting frequency dispersion – meaning that each wavenumber has its own frequency and phase speed.
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| Note that in engineering the [[wave height]] ''H'' – the difference in elevation between [[crest (physics)|crest]] and [[trough (physics)|trough]] – is often used:
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| :<math>H\, =\, 2\, a \qquad \text{and} \qquad a\, =\, \frac12\, H,</math>
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| valid in the present case of linear periodic waves.
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| [[File:Orbital wave motion.svg|left|thumb|400px|Orbital motion under linear waves. The yellow dots indicate the momentary position of fluid particles on their (orange) orbits. The black dots are the centres of the orbits.]]
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| 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 [[orbit|orbital motion]]. Within the framework of Airy wave theory, the orbits are closed curves: circles in deep water, and ellipses in finite depth—with the ellipses 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 reduced to 4% of its free-surface value at a depth of half a wavelength.
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| In a similar fashion, there is also a pressure oscillation underneath the free surface, with wave-induced pressure oscillations reducing with depth below the free surface – in the same way as for the orbital motion of fluid parcels.
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| == Mathematical formulation of the wave motion ==
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| === Flow problem formulation ===
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| The waves propagate in the horizontal direction, with [[Cartesian coordinate system|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]].<ref name=solution>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.</ref> The level ''z'' = 0 corresponds with the mean surface elevation. The [[Permeability (earth sciences)|impermeable]] bed underneath the fluid layer is at ''z'' = -''h''. Further, the flow is assumed to be [[incompressible flow|incompressible]] and [[irrotational flow|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''<sub>''x''</sub> and ''u''<sub>''z''</sub> in the horizontal (''x'') and vertical (''z'') directions by:
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| :<math>
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| u_x\, =\, \frac{\partial\Phi}{\partial x}
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| \quad \text{and} \quad
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| u_z\, =\, \frac{\partial\Phi}{\partial z}.
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| </math>
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| Then, due to the [[Continuity equation#Fluid dynamics|continuity equation]] for an incompressible flow, the potential ''Φ'' has to satisfy the [[Laplace equation]]:
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| :<math>
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| (1) \qquad
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| \frac{\partial^2\Phi}{\partial x^2}\, +\,
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| \frac{\partial^2\Phi}{\partial z^2}\, =\, 0.
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| </math>
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| [[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 [[perturbation theory|zeroth-order solution]]) of the flow is. Here, we assume the base state is rest, implying the mean flow velocities are zero.
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| The bed being impermeable, leads to the [[kinematics|kinematic]] bed boundary-condition:
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| :<math>(2) \qquad \frac{\partial\Phi}{\partial z}\, =\, 0 \quad \text{ at } z\, =\, -h.</math>
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| In case of deep water – by which is meant [[Infinity|infinite]] water depth, from a mathematical point of view – the flow velocities have to go to zero in the [[limit (mathematics)|limit]] as the vertical coordinate goes to minus infinity: ''z'' → -∞.
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| 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:
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| :<math>(3) \qquad \frac{\partial\eta}{\partial t}\, =\, \frac{\partial\Phi}{\partial z} \quad \text{ at } z\, =\, \eta(x,t).</math>
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| 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 principle|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 such a constant pressure does not alter the flow. After linearisation, this gives the [[dynamics (physics)|dynamic]] free-surface boundary condition:
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| :<math>(4) \qquad \frac{\partial\Phi}{\partial t}\, +\, g\, \eta\, =\, 0 \quad \text{ at } z\, =\, \eta(x,t).</math>
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| 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.
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| === Solution for a progressive monochromatic wave ===
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| {{see also|Dispersion (water waves)}}
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| For a propagating wave of a single frequency – a [[monochromatic]] wave – the surface elevation is of the form:<ref name=solution/>
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| :<math>\eta\, =\, a\, \cos\, ( k x\, -\, \omega t ).</math>
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| 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:
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| :<math>\Phi\, =\, \frac{\omega}{k}\, a\, \frac{\cosh\, \bigl( k\, (z+h) \bigr)}{\sinh\, (k\, h)}\, \sin\, ( k x\, -\, \omega t),</math>
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| with sinh and cosh the [[hyperbolic sine]] and [[hyperbolic cosine]] function, respectively.
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| 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 (water waves)|dispersion relation]] is satisfied:
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| :<math>\omega^2\, =\, g\, k\, \tanh\, ( k h ),</math>
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| 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).
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| === Table of wave quantities ===
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| In the table below, several flow quantities and parameters according to Airy wave theory are given.<ref name=solution/> 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 [[crest (physics)|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. At an Earth-fixed location, the ''observed angular frequency'' (or ''absolute angular frequency'') is ''ω''. On the other hand, in a [[frame of reference]] moving with the mean velocity '''''U''''' (so the mean velocity as observed from this reference frame is zero), the angular frequency is different. It is called the ''intrinsic angular frequency'' (or ''relative angular frequency''), denoted as ''σ''. So in pure wave motion, with '''''U'''''='''''0''''', both frequencies ''ω'' and ''σ'' are equal. The wave number ''k'' (and wavelength ''λ'') are independent of the [[frame of reference]], and have no Doppler shift (for monochromatic waves).
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| The table only gives the oscillatory parts of flow quantities – velocities, particle excursions and pressure – and not their mean value or drift.
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| The oscillatory particle excursions '''''ξ'''''<sub>''x''</sub> and ''ξ''<sub>''z''</sub> are the time [[integral]]s of the oscillatory flow velocities '''''u'''''<sub>''x''</sub> and ''u''<sub>''z''</sub> respectively.
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| Water depth is classified into three regimes:<ref>Dean & Dalrymple (1991) pp. 64–65</ref>
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| * '''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),<ref>The error in the phase speed is less than 0.2% if depth ''h'' is taken to be infinite, for ''h'' > ½ ''λ''.</ref>
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| * '''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 [[Periodic function|period]] or wavelength;<ref>The error in the phase speed is less than 2% if wavelength effects are neglected for ''h'' <{{frac|1|20}} ''λ''.</ref> and
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| * '''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.
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| 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.
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| {| class="wikitable" style="width:90%; text-align:center;"
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| |-
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| ! colspan="6" | Properties of gravity waves on the surface of deep water, shallow water and at intermediate depth, according to Airy wave theory<ref name=solution/>
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| |-
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| ! style="width:13%;"| quantity
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| ! style="width:7%;"| symbol
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| ! style="width:4%;"| units
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| ! style="width:22%;"| deep water<br>( ''h'' > ½ ''λ'' )
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| ! style="width:22%;"| shallow water<br>( ''h'' < 0.05 ''λ'' )
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| ! style="width:22%;"| intermediate depth<br>( all ''λ'' and ''h'' )
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| |- style="height:80px"
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| ! surface elevation
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| | <math>\eta(\boldsymbol{x},t)\,</math>
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| || m
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| | colspan="3" | <math>a\, \cos\, \theta(\boldsymbol{x},t)\,</math>
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| |- style="height:80px"
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| ! wave phase
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| | <math>\theta(\boldsymbol{x},t)\,</math>
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| || [[radian|rad]]
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| | colspan="3" | <math>\boldsymbol{k}\cdot\boldsymbol{x}\, -\, \omega\, t\,</math>
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| |- style="height:80px"
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| ! observed [[angular frequency]]
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| | <math>\omega\,</math>
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| || rad / [[second|s]]
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| | colspan="3" | <math> \left( \omega\, -\, \boldsymbol{k}\cdot\boldsymbol{U} \right)^2\, =\, \bigl( \Omega(k) \bigr)^2 \quad \text{ with } \quad k\,=\, |\boldsymbol{k}| \, </math>
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| |- style="height:80px"
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| ! intrinsic angular frequency
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| | <math>\sigma\,</math>
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| || rad / s
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| | colspan="3" | <math> \quad \sigma^2\, =\, \bigl( \Omega(k) \bigr)^2 \quad \text{ with } \quad \sigma\, =\, \omega\, -\, \boldsymbol{k}\cdot\boldsymbol{U}\, </math>
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| |- style="height:80px"
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| ! unit vector in the wave propagation direction
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| | <math> \boldsymbol{e}_k\,</math>
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| || –
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| | colspan="3" | <math>\frac{\boldsymbol{k}}{k}\,</math>
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| |- style="height:80px"
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| ! {{Anchor|Dispersion relation}}[[dispersion (water waves)|dispersion relation]]
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| | <math>\Omega(k)\,</math>
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| || rad / s
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| || <math>\Omega(k)\, =\, \sqrt{g\, k}</math>
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| || <math>\Omega(k)\, =\, k\, \sqrt{g\, h}\,</math>
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| || <math>\Omega(k)\, =\, \sqrt{g\, k\, \tanh\, (k\, h)}\,</math>
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| |- style="height:80px"
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| ! [[phase speed]]
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| | <math>c_p=\frac{\Omega(k)}{k}\,</math>
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| || m / s
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| || <math>\sqrt{\frac{g}{k}}\, =\, \frac{g}{\sigma}\,</math>
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| || <math>\sqrt{g h}</math>
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| || <math>\sqrt{\frac{g}{k}\, \tanh\, (k\, h)\,}</math>
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| |- style="height:80px"
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| ! [[group speed]]
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| | <math>c_g = \frac{\partial\Omega}{\partial k}</math>
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| || m / s
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| || <math>\frac{1}{2}\, \sqrt{\frac{g}{k}}\, =\, \frac{1}{2}\, \frac{g}{\sigma}\,</math>
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| || <math>\sqrt{g h}\,</math>
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| || <math>\frac{1}{2}\, c_p\, \left( 1\, +\, k\, h\, \frac{1\, -\, \tanh^2\, (k\, h)}{\tanh\, (k\, h)} \right)</math>
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| |- style="height:80px"
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| ! ratio
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| | <math> \frac{c_g}{c_p}\,</math>
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| || –
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| || <math>\frac{1}{2}\,</math>
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| || <math>1\,</math>
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| || <math>\frac{1}{2}\, \left( 1\, +\, k\, h\, \frac{1\, -\, \tanh^2\, (k\, h)}{\tanh\, (k\, h)} \right)</math>
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| |- style="height:80px"
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| ! horizontal velocity
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| | <math>\boldsymbol{u}_x(\boldsymbol{x},z,t)\,</math>
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| || m / s
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| || <math>\boldsymbol{e}_k\, \sigma\, a\; \text{e}^{\displaystyle k\, z}\, \cos\, \theta\,</math>
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| || <math>\boldsymbol{e}_k\, \sqrt{\frac{g}{h}}\, a\, \cos\, \theta\,</math>
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| || <math>\boldsymbol{e}_k\, \sigma\, a\, \frac{\cosh\, \bigl( k\, (z+h) \bigr)}{\sinh\, (k\, h)}\, \cos\, \theta\,</math>
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| |- style="height:80px"
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| ! vertical velocity
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| | <math>u_z(\boldsymbol{x},z,t)\,</math>
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| || m / s
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| || <math>\sigma\, a\; \text{e}^{\displaystyle k\, z}\, \sin\, \theta\,</math>
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| || <math>\sigma\, a\, \frac{z\, +\, h}{h}\, \sin\, \theta\,</math>
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| || <math>\sigma\, a\, \frac{\sinh\, \bigl( k\, (z+h) \bigr)}{\sinh\, (k\, h)}\, \sin\, \theta\,</math>
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| |- style="height:80px"
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| ! horizontal particle excursion
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| | <math>\boldsymbol{\xi}_x(\boldsymbol{x},z,t)\,</math>
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| || m
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| || <math>-\boldsymbol{e}_k\, a\; \text{e}^{\displaystyle k\, z}\, \sin\, \theta\,</math>
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| || <math>-\boldsymbol{e}_k\, \frac{1}{k\, h}\, a\, \sin\, \theta\,</math>
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| || <math>-\boldsymbol{e}_k\, a\, \frac{\cosh\, \bigl( k\, (z+h) \bigr)}{\sinh\, (k\, h)}\, \sin\, \theta\,</math>
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| |- style="height:80px"
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| ! vertical particle excursion
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| | <math>\xi_z(\boldsymbol{x},z,t)\,</math>
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| || m
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| || <math>a\; \text{e}^{\displaystyle k\, z}\, \cos\, \theta\,</math>
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| || <math>a\, \frac{z\, +\, h}{h}\, \cos\, \theta\,</math>
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| || <math>a\, \frac{\sinh\, \bigl( k\, (z+h) \bigr)}{\sinh\, (k\, h)}\, \cos\, \theta\,</math>
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| |- style="height:80px"
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| ! [[pressure]] oscillation
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| | <math>p(\boldsymbol{x},z,t)\,</math>
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| || [[Newton (unit)|N]] / m<sup>2</sup>
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| || <math>\rho\, g\, a\; \text{e}^{\displaystyle k\, z}\, \cos\, \theta\,</math>
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| || <math>\rho\, g\, a\, \cos\, \theta\,</math>
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| || <math>\rho\, g\, a\, \frac{\cosh\, \bigl( k\, (z+h) \bigr)}{\cosh\, (k\, h)}\, \cos\, \theta\,</math>
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| |}
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| == Surface tension effects ==
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| {{main|Capillary wave}}
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| [[File:Dispersion capillary.svg|thumb|right|Dispersion of gravity–capillary waves on the surface of deep water. Phase and group velocity divided by <math>\scriptstyle \sqrt[4]{g\sigma/\rho}</math> as a function of inverse relative wavelength <math>\scriptstyle \frac{1}{\lambda}\sqrt{\sigma/(\rho g)}</math>.<br>Blue lines (A): phase velocity ''c<sub>p</sub>'', Red lines (B): group velocity ''c<sub>g</sub>''.<br>Drawn lines: gravity–capillary waves.<br>Dashed lines: deep-water gravity waves.<br>Dash-dot lines: deep–water pure capillary waves.]]
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| Due to [[surface tension]], the dispersion relation changes to:<ref name=Phillips_37>Phillips (1977), p. 37.</ref>
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| | |
| :<math> \Omega^2(k)\, =\, \left( g\, +\, \frac{\gamma}{\rho}\, k^2 \right)\, k\; \tanh\, ( k\, h ),</math>
| |
| | |
| with ''γ'' the surface tension, with [[SI]] units in N/m. All above equations for linear waves remain the same, if the gravitational acceleration ''g'' is replaced by<ref>Lighthill (1978), p. 223.</ref>
| |
| | |
| :<math>\tilde{g}\, =\, g\, +\, \frac{\gamma}{\rho}\, k^2.</math>
| |
| | |
| 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 – two millimeter in case of the interface between air and water – gravity effects are negligible.
| |
| | |
| == Interfacial waves ==
| |
| | |
| Surface waves are a special case of interfacial waves, on the [[interface (chemistry)|interface]] between two fluids of different [[density]].
| |
| | |
| === Two layers of infinite depth ===
| |
| | |
| Consider two fluids separated by an interface, and without further boundaries. Then their dispersion relation ''ω<sup>2</sup>'' = Ω<sup>2</sup>(''k'') is given through:<ref name=Phillips_37/><ref>[[Horace Lamb|Lamb, H.]] (1994), §267, page 458–460.</ref><ref>Dingemans (1997), Section 2.1.1, p. 45.</ref>
| |
| | |
| :<math>
| |
| \Omega^2(k)\, =\, |k|\, \left( \frac{\rho-\rho'}{\rho+\rho'} g\, +\, \frac{\gamma}{\rho+\rho'}\, k^2 \right),
| |
| </math>
| |
| | |
| where ''ρ'' and ''ρ‘'' are the densities of the two fluids, below (''ρ'') and above (''ρ‘'') the interface, respectively. Further ''γ'' is the surface tension on the interface.
| |
| | |
| For interfacial waves to exist, the lower layer has to be heavier than the upper one, ''ρ'' > ''ρ‘''. Otherwise, the interface is unstable and a [[Rayleigh–Taylor instability]] develops.
| |
| | |
| === Two layers between horizontal rigid planes ===
| |
| [[File:Interfacial wave rigid boundaries.svg|thumb|right|Wave motion on the interface between two layers of [[viscosity|inviscid]] homogeneous fluids of different density, confined between horizontal rigid boundaries (at the top and bottom). The motion is forced by gravity. The upper layer has mean depth ''h‘'' and density ''ρ‘'', while the lower layer has mean depth ''h'' and density ''ρ''. The wave amplitude is ''a'', the wavelength is denoted by ''λ'' ({{nowrap|related to the wavenumber ''k'' by: ''k'' {{=}} 2π / ''λ''}}), the gravitational acceleration by ''g'' and the [[phase speed]] as ''c''<sub>p</sub> ({{nowrap|with ''c''<sub>p</sub> {{=}} ''Ω''(''k'') / ''k''}}).]]
| |
| | |
| For two homogeneous layers of fluids, of mean thickness ''h'' below the interface and ''h′'' above – under the action of gravity and bounded above and below by horizontal rigid walls – the dispersion relationship ''ω<sup>2</sup>'' = Ω<sup>2</sup>(''k'') for gravity waves is provided by:<ref>{{Citation | publisher = Cambridge University Press | isbn = 0521297265 | last = Turner | first = J. S. | title = Buoyancy effects in fluids | year = 1979 | page = 18 }}</ref>
| |
| | |
| :<math>
| |
| \Omega^2(k) = \frac{g\, k (\rho - \rho')}{\rho\, \coth( k h ) + \rho'\, \coth( k h')},
| |
| </math>
| |
| | |
| where again ''ρ'' and ''ρ′'' are the densities below and above the interface, while coth is the [[hyperbolic cotangent]] function. For the case ''ρ′'' is zero this reduces to the dispersion relation of surface gravity waves on water of finite depth ''h''.
| |
| | |
| === Two layers bounded above by a free surface ===
| |
| In this case the dispersion relation allows for two modes: a [[barotropic]] mode where the free surface [[amplitude]] is large compared with the amplitude of the interfacial wave, and a [[baroclinic]] mode where the opposite is the case – the interfacial wave is higher than and in [[antiphase]] with the free surface wave. The dispersion relation for this case is of a more complicated form.<ref>{{Citation | publisher = Academic Press | isbn = 9780080570747 | last = Apel | first = J. R. | title = Principles of ocean physics | year = 1987 | pages = 231–239 }}</ref>
| |
| | |
| == Second-order wave properties ==
| |
| | |
| Several [[perturbation theory|second-order]] wave properties, ''i.e.'' [[quadratic function|quadratic]] in the wave amplitude ''a'', can be derived directly from Airy wave theory. They are of importance in many practical applications, ''e.g.'' [[forecasting|forecast]]s of wave conditions.<ref>See for example: the [http://www.weather.gov/om/marine/zone/hsmz.htm ''High seas forecasts''] of [[NOAA]]'s National Weather service.</ref> Using a [[WKBJ approximation]], second-order wave properties also find their applications in describing waves in case of slowly varying [[bathymetry]], and mean-flow variations of currents and surface elevation. As well as in the description of the wave and mean-flow interactions due to time and space-variations in amplitude, frequency, wavelength and direction of the wave field itself.
| |
| | |
| === Table of second-order wave properties {{anchor|Table of second-order wave properties}} ===
| |
| | |
| In the table below, several second-order wave properties – as well as the dynamical equations they satisfy in case of slowly varying conditions in space and time – are given. More details on these can be found below. The table gives results for wave propagation in one horizontal spatial dimension. Further on in this section, more detailed descriptions and results are given for the general case of propagation in two-dimensional horizontal space.
| |
| | |
| {| class="wikitable" style="width:90%; text-align:center;"
| |
| ! colspan="4" | Second-order quantities and their dynamics, using results of Airy wave theory
| |
| |-
| |
| ! style="width:25%;"| quantity
| |
| ! style="width:10%;"| symbol
| |
| ! style="width:10%;"| units
| |
| ! style="width:45%;"| formula
| |
| |- style="height:80px"
| |
| ! mean wave-energy density per unit horizontal area
| |
| | <math>E\,</math>
| |
| || [[joule|J]] / m<sup>2</sup>
| |
| | <math>E\, =\, \frac12\, \rho\, g\, a^2\,</math>
| |
| |- style="height:80px"
| |
| ! [[radiation stress]] or excess horizontal [[momentum]] [[flux]] due to the wave motion
| |
| | <math>S_{xx}\,</math>
| |
| || N / m
| |
| | <math>S_{xx}\, =\, \left( 2\, \frac{c_g}{c_p}\, -\, \frac12 \right)\, E\,</math>
| |
| |- style="height:80px"
| |
| ! [[wave action (continuum mechanics)|wave action]]
| |
| | <math>\mathcal{A}\,</math>
| |
| || J·s / m<sup>2</sup>
| |
| | <math>\mathcal{A}\, =\, \frac{E}{\sigma}\, =\, \frac{E}{\omega\, -\, k\, U}\,</math>
| |
| |- style="height:80px"
| |
| ! mean mass-flux due to the wave motion or the wave pseudo-momentum
| |
| | <math>M\,</math>
| |
| || kg / (m·s)
| |
| | <math>M\, =\, \frac{E}{c_p}\, =\, k\, \frac{E}{\sigma}\,</math>
| |
| |- style="height:80px"
| |
| ! mean horizontal mass-transport velocity
| |
| | <math>\tilde{U}\,</math>
| |
| || m / s
| |
| | <math>\tilde{U}\, =\, U\, +\, \frac{M}{\rho\, h}\, =\, U\, +\, \frac{E}{\rho\, h\, c_p}\,</math>
| |
| |- style="height:80px"
| |
| ! [[Stokes drift]]
| |
| | <math>\bar{u}_S\,</math>
| |
| || m / s
| |
| | <math>\bar{u}_S\, =\, \frac12\, \sigma\, k\, a^2\, \frac{\cosh\, 2\,k\,(z+h)}{\sinh^2\, (k\,h)}\,</math>
| |
| |- style="height:80px"
| |
| ! wave-energy propagation
| |
| |
| |
| || J / (m<sup>2</sup>·s)
| |
| | <math>\frac{\partial E}{\partial t}\, +\, \frac{\partial}{\partial x} \Bigl( (U\, +\, c_g)\, E \Bigr)\, +\, S_{xx}\, \frac{\partial U}{\partial x}\, =\,0\,</math>
| |
| |- style="height:80px"
| |
| ! wave action conservation
| |
| |
| |
| || J / m<sup>2</sup>
| |
| | <math>\frac{\partial \mathcal{A}}{\partial t}\, +\, \frac{\partial}{\partial x} \Bigl( (U\, +\, c_g)\, \mathcal{A} \Bigr)\, =\, 0\,</math>
| |
| |- style="height:80px"
| |
| ! wave-[[crest (physics)|crest]] conservation
| |
| |
| |
| || rad / (m·s)
| |
| | <math>\frac{\partial k}{\partial t}\, +\, \frac{\partial \omega}{\partial x}\, =\, 0\,</math> with <math>\omega\, =\, \Omega(k)\, +\, k\, U\,</math>
| |
| |- style="height:80px"
| |
| ! mean mass conservation
| |
| |
| |
| || kg / (m<sup>2</sup>·s)
| |
| | <math>\frac{\partial}{\partial t}\Bigl( \rho\, h \Bigr)\, +\, \frac{\partial}{\partial x} \Bigl( \rho\, h\, \tilde{U} \Bigr)\, =\, 0\,</math>
| |
| |- style="height:80px"
| |
| ! mean horizontal-momentum evolution
| |
| |
| |
| || N / m<sup>2</sup>
| |
| | <math>\frac{\partial}{\partial t}\Bigl( \rho\, h\, \tilde{U} \Bigr)\, +\, \frac{\partial}{\partial x} \left( \rho\, h\, \tilde{U}^2\, +\, \frac12\, \rho\, g\, h^2\, +\, S_{xx} \right)\, =\, \rho\, g\, h\, \frac{\partial d}{\partial x}\,</math>
| |
| |}
| |
| | |
| The last four equations describe the evolution of slowly varying wave trains over [[bathymetry]] in interaction with the [[mean flow]], and can be derived from a variational principle: [[Gerald B. Whitham|Whitham]]'s average [[Lagrangian]] method.<ref name=Whitham_559>{{Cite book | first=G.B. | last=Whitham | authorlink=Gerald B. Whitham | title=Linear and nonlinear waves | publisher = Wiley-Interscience | year=1974 | isbn=0-471-94090-9 | oclc=815118 }}, p. 559.</ref> In the mean horizontal-momentum equation, ''d''(''x'') is the still water depth, ''i.e.'' the bed underneath the fluid layer is located at ''z'' = –''d''. Note that the mean-flow velocity in the mass and momentum equations is the ''mass transport velocity'' <math>\tilde{U}</math>, including the splash-zone effects of the waves on horizontal mass transport, and not the mean [[Lagrangian and Eulerian coordinates|Eulerian]] velocity (e.g. as measured with a fixed flow meter).
| |
| | |
| === Wave energy density ===
| |
| | |
| Wave energy is a quantity of primary interest, since it is a primary quantity that is transported with the wave trains.<ref>Phillips (1977), p. 23–25.</ref> 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 energy|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''<sub>pot</sub> of the surface gravity waves, which is the deviation of the potential energy due to the presence of the waves:<ref name=Phillips_39>Phillips (1977), p. 39.</ref>
| |
| | |
| :<math>E_\text{pot}\, =\, \overline{\int_{-h}^{\eta} \rho\,g\,z\;\text{d}z}\, -\, \int_{-h}^0 \rho\,g\,z\; \text{d}z\,
| |
| =\, \overline{\frac12\,\rho\,g\,\eta^2}\,
| |
| =\, \frac14\, \rho\,g\,a^2,
| |
| </math>
| |
| | |
| 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''<sub>kin</sub> of the wave motion is similarly found to be:<ref name=Phillips_39/>
| |
| | |
| :<math>
| |
| E_\text{kin}\, =\, \overline{\int_{-h}^0 \frac12\, \rho\, \left[\, \left| \boldsymbol{U}\, +\, \boldsymbol{u}_x \right|^2\, +\, u_z^2\, \right]\; \text{d}z}\,
| |
| -\, \int_{-h}^0 \frac12\, \rho\, \left| \boldsymbol{U} \right|^2\; \text{d}z\,
| |
| =\, \frac14\, \rho\, \frac{\sigma^2}{k\, \tanh\, (k\, h)}\,a^2,
| |
| </math>
| |
| with ''σ'' the intrinsic frequency, see the [[#Table of wave quantities|table of wave quantities]]. Using the dispersion relation, the result for surface gravity waves is:
| |
| | |
| :<math>E_\text{kin}\, =\, \frac14\, \rho\, g\, a^2.</math>
| |
| | |
| 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]].<ref name=Phillips_38/><ref>{{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.</ref> Adding potential and kinetic contributions, ''E''<sub>pot</sub> and ''E''<sub>kin</sub>, the mean energy density per unit horizontal area ''E'' of the wave motion is:
| |
| | |
| :<math>E\, =\, E_\text{pot}\, +\, E_\text{kin}\, =\, \frac12\, \rho\, g\, a^2.</math>
| |
| | |
| In case of surface tension effects not being negligible, their contribution also adds to the potential and kinetic energy densities, giving<ref name=Phillips_38>Phillips (1977), p. 38.</ref>
| |
| | |
| :<math>
| |
| E_\text{pot}\, =\, E_\text{kin}\, =\, \frac14\, \left( \rho\, g\, +\, \gamma\, k^2 \right)\, a^2,
| |
| \qquad \text{so} \qquad
| |
| E\, =\, E_\text{pot}\, +\, E_\text{kin}\, =\, \frac12\, \left( \rho\, g\, +\, \gamma\, k^2 \right)\, a^2,
| |
| </math>
| |
| | |
| 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 [[dissipation|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 (continuum mechanics)|wave action]] <math>\mathcal{A}=E/\sigma:</math><ref name=Whitham_559/><ref name=Phillips_26>Phillips (1977), p. 26.</ref><ref>{{Cite journal | 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 |bibcode = 1968RSPSA.302..529B }}</ref>
| |
| | |
| :<math>\frac{\partial \mathcal{A}}{\partial t}\, +\, \nabla\cdot\left[ \left(\boldsymbol{U}+\boldsymbol{c}_g\right)\, \mathcal{A}\right]\, =\, 0,</math>
| |
| | |
| with <math>\left(\boldsymbol{U}+\boldsymbol{c}_g\right)\, \mathcal{A}</math> the action [[flux]] and <math>\boldsymbol{c}_g=c_g\,\boldsymbol{e}_k</math> the [[group velocity]] vector. Action conservation forms the basis for many [[wind wave model]]s and [[wave turbulence]] models.<ref>Phillips (1977), pp. 179–183.</ref> It is also the basis of [[coastal engineering]] models for the computation of [[wave shoaling]].<ref>Phillips (1977), pp. 70–74.</ref> Expanding the above wave action conservation equation leads to the following evolution equation for the wave energy density:<ref name=Phillips_66>Phillips (1977), p. 66.</ref>
| |
| | |
| :<math>\frac{\partial E}{\partial t}\, +\, \nabla\cdot\left[\left( \boldsymbol{U}+\boldsymbol{c}_g\right)\, E \right]\, +\, \mathbb{S}:\left(\nabla\boldsymbol{U}\right)\, =\, 0,</math>
| |
| | |
| with:
| |
| *<math>\left( \boldsymbol{U}+\boldsymbol{c}_g\right)\, E</math> is the mean wave energy density flux,
| |
| *<math>\mathbb{S}</math> is the [[radiation stress]] [[tensor]] and
| |
| *<math>\nabla\boldsymbol{U}</math> is the mean-velocity [[shear rate|shear-rate]] tensor.
| |
| In this equation in non-conservation form, the [[Frobenius inner product]] <math>\mathbb{S}:(\nabla\boldsymbol{U})</math> 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, <math>\nabla\boldsymbol{U}=\mathsf{0},</math> the mean wave energy density <math>E</math> is conserved. The two tensors <math>\mathbb{S}</math> and <math>\nabla\boldsymbol{U}</math> are in a [[Cartesian coordinate system]] of the form:<ref>Phillips (1977), p. 68.</ref>
| |
| | |
| :<math>
| |
| \begin{align}
| |
| \mathbb{S}\, &=\, \begin{pmatrix} S_{xx} & S_{xy} \\ S_{yx} & S_{yy} \end{pmatrix}\,
| |
| =\, \mathbb{I}\, \left( \frac{c_g}{c_p} - \frac12 \right)\, E\,
| |
| +\, \frac{1}{k^2}\, \begin{pmatrix} k_x\, k_x & k_x\, k_y \\[2ex] k_y\, k_x & k_y\, k_y \end{pmatrix}\, \frac{c_g}{c_p}\, E,
| |
| \\
| |
| \mathbb{I}\, &=\, \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}
| |
| \quad \text{and}
| |
| \\
| |
| \nabla \boldsymbol{U}\, &=\,
| |
| \begin{pmatrix}
| |
| \displaystyle \frac{\partial U_x}{\partial x} & \displaystyle \frac{\partial U_y}{\partial x}
| |
| \\[2ex]
| |
| \displaystyle \frac{\partial U_x}{\partial y} & \displaystyle \frac{\partial U_y}{\partial y}
| |
| \end{pmatrix},
| |
| \end{align}
| |
| </math>
| |
| | |
| with <math>k_x</math> and <math>k_y</math> the components of the wavenumber vector <math>\boldsymbol{k}</math> and similarly <math>U_x</math> and <math>U_y</math> the components in of the mean velocity vector <math>\boldsymbol{U}</math>.
| |
| | |
| === Wave mass flux and wave momentum ===
| |
| | |
| The mean horizontal [[momentum]] per unit area <math>\boldsymbol{M}</math> induced by the wave motion – and also the wave-induced [[mass flux]] or mass [[transport phenomena (engineering & physics)|transport]] – is:<ref name=Phillips_mass>Phillips (1977), pp. 39–40 & 61.</ref>
| |
| | |
| :<math>
| |
| \boldsymbol{M}\, =\,
| |
| \overline{\int_{-h}^\eta \rho\, \left( \boldsymbol{U}+\boldsymbol{u}_x\right)\; \text{d}z}\,
| |
| -\, \int_{-h}^0 \rho\, \boldsymbol{U}\; \text{d}z\,
| |
| =\, \frac{E}{c_p}\, \boldsymbol{e}_k,
| |
| </math>
| |
| | |
| which is an exact result for periodic progressive water waves, also valid for [[nonlinear]] waves.<ref>Phillips (1977), p. 40.</ref> However, its validity strongly depends on the way how wave momentum and mass flux are defined. [[George Gabriel Stokes|Stokes]] already identified two possible definitions of [[phase velocity]] for periodic nonlinear waves:<ref name=Stokes1847/>
| |
| *''Stokes first definition of wave [[phase velocity|celerity]]'' (S1) – with the mean [[Lagrangian and Eulerian coordinates|Eulerian flow velocity]] equal to zero for all elevations ''z'' below the wave [[Crest (physics)|trough]]s, and
| |
| *''Stokes second definition of wave celerity'' (S2) – with the mean mass transport equal to zero.
| |
| The above relation between wave momentum '''''M''''' and wave energy density ''E'' is valid within the framework of Stokes' first definition.
| |
| | |
| However, for waves perpendicular to a coast line or in closed laboratory [[wave channel]], the second definition (S2) is more appropriate. These wave systems have zero mass flux and momentum when using the second definition.<ref>Phillips (1977), p. 70.</ref> In contrast, according to Stokes' first definition (S1), there is a wave-induced mass flux in the wave propagation direction, which has to be balanced by a mean flow '''''U''''' in the opposite direction – called the [[undertow (wave action)|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.<ref>{{Cite journal | 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 |bibcode = 1981JFM...106..331M }}</ref>
| |
| | |
| ==== 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 <math>\tilde{\boldsymbol{U}}</math> defined as:<ref name=Phillips_61_63>Phillips (1977), pp. 61–63.</ref>
| |
| | |
| :<math>\tilde{\boldsymbol{U}}\, =\, \boldsymbol{U}\, +\, \frac{\boldsymbol{M}}{\rho\,h}.</math>
| |
| | |
| Note that for deep water, when the mean depth ''h'' goes to infinity, the mean Eulerian velocity <math>\boldsymbol{U}</math> and mean transport velocity <math>\tilde{\boldsymbol{U}}</math> become equal.
| |
| | |
| The equation for mass conservation is:<ref name=Phillips<ref name=Whitham_559/><ref name=Phillips_61_63/>
| |
| | |
| :<math>
| |
| \frac{\partial}{\partial t}\left( \rho\, h\, \right)\,
| |
| +\, \nabla \cdot \left( \rho\, h\,\tilde{\boldsymbol{U}} \right)\,
| |
| =\, 0,
| |
| </math>
| |
| | |
| where ''h''('''''x''''',''t'') is the mean water-depth, slowly varying in space and time.
| |
| Similarly, the mean horizontal momentum evolves as:<ref name=Whitham_559/><ref name=Phillips_61_63/>
| |
| | |
| :<math>
| |
| \frac{\partial}{\partial t}\left( \rho\, h\, \tilde{\boldsymbol{U}}\right)\,
| |
| +\, \nabla \cdot \left( \rho\, h\, \tilde{\boldsymbol{U}} \otimes \tilde{\boldsymbol{U}}\, +\, \frac12\,\rho\,g\,h^2\,\mathbb{I}\, +\, \mathbb{S} \right)\,
| |
| =\, \rho\, g\, h\, \nabla d,
| |
| </math>
| |
| with ''d'' the still-water depth (the sea bed is at ''z''=–''d''), <math>\mathbb{S}</math> is the wave radiation-stress [[tensor]], <math>\mathbb{I}</math> is the [[identity matrix]] and <math>\otimes</math> is the [[dyadic product]]:
| |
| | |
| :<math>
| |
| \tilde{\boldsymbol{U}} \otimes \tilde{\boldsymbol{U}}\, =\,
| |
| \begin{pmatrix}
| |
| \tilde{U}_x\, \tilde{U}_x & \tilde{U}_x\, \tilde{U}_y
| |
| \\[2ex]
| |
| \tilde{U}_y\, \tilde{U}_x & \tilde{U}_y\, \tilde{U}_y
| |
| \end{pmatrix}.
| |
| </math>
| |
| 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):<ref name=Whitham_559/><ref name="Phillips_26" />
| |
| | |
| :<math>
| |
| \frac{\partial}{\partial t} \left( \frac{E}{\sigma}\, \right)
| |
| +\, \nabla \cdot \left[ \left( \boldsymbol{U} +\boldsymbol{c}_g \right)\, \frac{E}{\sigma} \right]\,
| |
| =\, 0.
| |
| </math>
| |
| | |
| The wave kinematics are described through the wave-crest conservation equation:<ref>Phillips (1977), p. 23.</ref>
| |
| | |
| :<math>\frac{\partial \boldsymbol{k}}{\partial t}\, +\, \nabla \omega\, =\, \boldsymbol{0},</math>
| |
| | |
| with the angular frequency ''ω'' a function of the (angular) [[wavenumber]] '''''k''''', related through the [[dispersion (water waves)|dispersion relation]]. For this to be possible, the wave field must be [[coherence (physics)|coherent]]. By taking the [[curl (mathematics)|curl]] of the wave-crest conservation, it can be seen that an initially [[irrotational]] wavenumber field stays irrotational.
| |
| | |
| === Stokes drift ===
| |
| {{main|Stokes drift}}
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| When following a single particle in pure wave motion <math>(\boldsymbol{U}=\boldsymbol{0}),</math> according to linear Airy wave theory, a first approximation gives closed elliptical orbits for water particles.<ref name=LeBlond>{{Cite book | publisher = Elsevier | series = Elsevier Oceanography Series |volume = 20 | isbn = 978-0-444-41926-2 | last1 = LeBlond | first1 = P.H. | last2 = Mysak | first2 = L A. | title = Waves in the Ocean | year = 1981 | pages=85 & 110–111}}</ref> However, for nonlinear waves, particles exhibit a [[Stokes drift]] for which a second-order expression can be derived from the results of Airy wave theory (see the [[#Table of second-order wave properties|table above on second-order wave properties]]).<ref>{{Cite book | publisher = Cambridge University Press | isbn = 978-0-521-36829-2 | last = Craik | first = A.D.D. | title = Wave interactions and fluid flows | year = 1988 | page=105 }}</ref> The Stokes drift velocity <math>\bar{\boldsymbol{u}}_S</math>, which is the particle drift after one wave cycle divided by the [[Periodic function|period]], can be estimated using the results of linear theory:<ref name=Phillips_44>Phillips (1977), p. 44.</ref>
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| :<math>\bar{\boldsymbol{u}}_S\, =\, \frac12\, \sigma\, k\, a^2\, \frac{\cosh\, 2\,k\,(z+h)}{\sinh^2\, (k\,h)}\, \boldsymbol{e}_k,</math>
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| so it varies as a function of elevation. The given formula is for Stokes first definition of wave celerity. When <math>\rho\,\bar{\boldsymbol{u}}_S</math> is [[integral|integrated]] over depth, the expression for the mean wave momentum <math>\boldsymbol{M}</math> is recovered.<ref name=Phillips_44/>
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| == See also ==
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| * [[Boussinesq approximation (water waves)]] – [[nonlinear]] theory for waves in [[waves and shallow water|shallow water]].
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| * [[Capillary wave]] – surface waves under the action of [[surface tension]]
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| * [[Cnoidal wave]] – nonlinear periodic waves in shallow water, solutions of the [[Korteweg–de Vries equation]]
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| * [[Mild-slope equation]] – refraction and diffraction of surface waves over varying depth
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| * [[Ocean surface wave]] – real water waves as seen in the ocean and sea
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| * [[Stokes wave]] – nonlinear periodic waves in non-shallow water
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| * [[Wave power]] – using ocean and sea waves for power generation.
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| == Notes ==
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| {{reflist|colwidth=30em}}
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| == References ==
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| === Historical ===
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| *{{Cite journal | 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=[[Hugh James Rose]], et al. }}. Also: "Trigonometry, On the Figure of the Earth, Tides and Waves", 396 pp.
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| *{{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 }}<br>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 }}
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| === Further reading ===
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| | |
| * {{cite journal
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| | first=A. D. D.
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| | last=Craik
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| | year=2004
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| | title=The origins of water wave theory
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| | journal=Annual Review of Fluid Mechanics
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| | volume=36
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| | pages=1–28
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| | doi=10.1146/annurev.fluid.36.050802.122118
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| |bibcode = 2004AnRFM..36....1C }}
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| * {{cite book
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| | title=Water wave mechanics for engineers and scientists
| |
| | first=R. G.
| |
| | last=Dean
| |
| | coauthors=Dalrymple, R. A.
| |
| | year=1991
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| | series=Advanced Series on Ocean Engineering
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| | volume=2
| |
| | publisher=World Scientific
| |
| | location=Singapore
| |
| | isbn=978-981-02-0420-4
| |
| | oclc=22907242
| |
| }}
| |
| * {{cite book
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| | title=Water wave propagation over uneven bottoms
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| | 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
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| }} Two parts, 967 pages.
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| * {{cite book
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| | first=H.
| |
| | last=Lamb
| |
| | authorlink=Horace Lamb
| |
| | year=1994
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| | title=Hydrodynamics
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| | publisher=Cambridge University Press
| |
| | edition=6th
| |
| | isbn=978-0-521-45868-9
| |
| | oclc=30070401
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| }} Originally published in 1879, the 6th extended edition appeared first in 1932.
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| * {{cite book
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| | title=Fluid mechanics
| |
| | first=L. D.
| |
| | last=Landau
| |
| | authorlink=Lev Landau
| |
| | coauthors=[[Evgeny Lifshitz|Lifshitz, E. M.]]
| |
| | year=1986
| |
| | publisher=Pergamon Press
| |
| | series=Course of Theoretical Physics
| |
| | volume=6
| |
| | edition=2nd revised
| |
| | isbn=0-08-033932-8
| |
| | oclc=15017127
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| }}
| |
| * {{cite book
| |
| | first = M. J.
| |
| | last = Lighthill
| |
| | authorlink = M. J. Lighthill
| |
| | title = Waves in fluids
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| | publisher = Cambridge University Press
| |
| | year = 1978
| |
| | isbn = 0-521-29233-6
| |
| | oclc = 2966533
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| }} 504 pp.
| |
| * {{cite book
| |
| | first=O. M.
| |
| | last=Phillips
| |
| | title=The dynamics of the upper ocean
| |
| | publisher=Cambridge University Press
| |
| | year=1977
| |
| | edition=2nd
| |
| | isbn=0-521-29801-6
| |
| | oclc=7319931
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| }}
| |
| * {{citation
| |
| | first1 = J. V.
| |
| | last1 = Wehausen
| |
| | author1-link = John V. Wehausen
| |
| | first2 = E. V.
| |
| | last2 = Laitone
| |
| | lastauthoramp = yes
| |
| | editor1-first = S. | editor1-last = Flügge | editor1-link = Siegfried Flügge
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| | editor2-first = C. | editor2-last = Truesdell | editor2-link = Clifford Truesdell
| |
| | volume = 9
| |
| | pages = 653–667, §27
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| | title = Surface Waves
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| | journal = Encyclopaedia of Physics
| |
| | year = 1960
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| | publisher = Springer Verlag
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| | url = http://coe.berkeley.edu/SurfaceWaves/
| |
| | oclc = 612422741
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| }}
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| == External links ==
| |
| {{commons category|Ocean surface waves}}
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| {{commons category|Water waves}}
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| * [http://www.wikiwaves.org/Linear_Theory_of_Ocean_Surface_Waves Linear theory of ocean surface waves] on WikiWaves.
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| * [http://web.mit.edu/fluids-modules/www/potential_flows/LecturesHTML/lec19bu/node1.html Water waves] at [[Massachusetts Institute of Technology|MIT]].
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| {{physical oceanography}}
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| {{DEFAULTSORT:Airy Wave Theory}}
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| [[Category:Water waves]]
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| [[Category:Wave mechanics]]
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| [[Category:Physical oceanography]]
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