Influid dynamics,Airy wave theory (often referred to aslinear wave theory) gives alinearised description of thepropagation ofgravity 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 19th century.^[1]

Airy wave theory is often applied inocean engineering andcoastal engineering for the modelling ofrandomsea states – giving a description of the wavekinematics anddynamics of high-enough accuracy for many purposes.^[2][3] Further, severalsecond-ordernonlinear properties of surface gravity waves, and their propagation, can be estimated from its results.^[4] Airy wave theory is also a good approximation fortsunami waves in the ocean, before they steepen near the coast.

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 thewave height to water depth (for waves inshallow water), and wave height to wavelength (for waves in deep water).

Airy wave theory uses apotential flow (orvelocity 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 takeviscosity,vorticity,turbulence 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 layers at the boundaries of the fluid domain.^[5]

Airy wave theory is often used inocean 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.^[2]

Earlier attempts to describe surface gravity waves using potential flow were made by, among others,Laplace,Poisson,Cauchy andKelland. ButAiry was the first to publish the correct derivation and formulation in 1841.^[1] Soon after, in 1847, the linear theory of Airy was extended byStokes fornon-linear wave motion – known asStokes' wave theory – correct up tothird order in the wave steepness.^[6] Even before Airy's linear theory,Gerstner derived a nonlineartrochoidal wave theory in 1802, which however is notirrotational.^[1]

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 elevationη(x,t) of one wave component issinusoidal, as a function of horizontal positionx and timet:

${\displaystyle \eta (x,t)=a\cos \left(kx-\omega t\right)}$

where

• a is the waveamplitude in metres,
• cos is thecosine function,
• k is theangular wavenumber inradians per metre, related to thewavelengthλ byk =⁠2π/λ⁠,
• ω is theangular frequency in radians per second, related to theperiodT andfrequencyf byω =⁠2π/T⁠ = 2πf.

The waves propagate along the water surface with thephase speedc_p:

${\displaystyle c_p=\frac{\omega }{k}=\frac{\lambda }{T}.}$

The angular wavenumberk and frequencyω are not independent parameters (and thus also wavelengthλ and periodT are not independent), but are coupled. Surface gravity waves on a fluid aredispersive waves – exhibiting frequency dispersion – meaning that each wavenumber has its own frequency and phase speed.

Note that in engineering thewave heightH – the difference in elevation betweencrest andtrough – is often used:

${\displaystyle H=2a\text{and}a=\frac{1}{2}H,}$

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 closed curves: circles in deep water and ellipses in finite depth—with the circles dying out before reaching the bottom of the fluid layer, and the ellipses 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 reduced to 4% of its free-surface value at a depth of half a wavelength.

In a similar fashion, there is also apressure 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.

The waves propagate in the horizontal direction, withcoordinatex, and a fluid domain bound above by a free surface atz =η(x,t), withz the vertical coordinate (positive in the upward direction) andt being time.^[7] The levelz = 0 corresponds with the mean surface elevation. Theimpermeable bed underneath the fluid layer is atz = −h. Further, the flow is assumed to beincompressible andirrotational – 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. Thevelocity potentialΦ(x,z,t) is related to theflow velocity componentsu_x andu_z in the horizontal (x) and vertical (z) directions by:

${\displaystyle u_x=\frac{\mathrm{\partial }\mathrm{\Phi }}{\mathrm{\partial }x}\text{and}{u}_z=\frac{\mathrm{\partial }\mathrm{\Phi }}{\mathrm{\partial }z}.}$

Then, due to thecontinuity equation for an incompressible flow, the potentialΦ has to satisfy theLaplace equation:

$${\displaystyle \frac{{\mathrm{\partial }}^2\mathrm{\Phi }}{\mathrm{\partial }{x}^2}+\frac{{\mathrm{\partial }}^2\mathrm{\Phi }}{\mathrm{\partial }{z}^2}=0.}$$

1

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 (orzeroth-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 thekinematic bed boundary-condition:

$${\displaystyle \frac{\mathrm{\partial }\mathrm{\Phi }}{\mathrm{\partial }z}=0\text{ at }z=-h.}$$

2

In case of deep water – by which is meantinfinite water depth, from a mathematical point of view – the flow velocities have to go to zero in thelimit as the vertical coordinate goes to minus infinity:z → −∞.

At the free surface, forinfinitesimal 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:

$${\displaystyle \frac{\mathrm{\partial }\eta }{\mathrm{\partial }t}=\frac{\mathrm{\partial }\mathrm{\Phi }}{\mathrm{\partial }z}\text{ at }z=\eta (x,t).}$$

3

If the free surface elevationη(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 byBernoulli'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 thedynamic free-surface boundary condition:

$${\displaystyle \frac{\mathrm{\partial }\mathrm{\Phi }}{\mathrm{\partial }t}+g\eta =0\text{ at }z=\eta (x,t).}$$

4

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

For a propagating wave of a single frequency – amonochromatic wave – the surface elevation is of the form:^[7]

${\displaystyle \eta =a\cos (kx-\omega t).}$

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:

${\displaystyle \mathrm{\Phi }=\frac{\omega }{k}a\frac{\cosh k(z+h)}{\sinh kh}\sin (kx-\omega t),}$

withsinh andcosh thehyperbolic sine andhyperbolic cosine function, respectively. Butη andΦ also have to satisfy the dynamic boundary condition, which results in non-trivial (non-zero) values for the wave amplitudea only if the lineardispersion relation is satisfied:

${\displaystyle {\omega }^2=gk\tanh kh,}$

withtanh thehyperbolic tangent. So angular frequencyω and wavenumberk – or equivalently periodT and wavelengthλ – cannot be chosen independently, but are related. This means that wave propagation at a fluid surface is aneigenproblem. Whenω andk satisfy the dispersion relation, the wave amplitudea can be chosen freely (but small enough for Airy wave theory to be a valid approximation).

In the table below, several flow quantities and parameters according to Airy wave theory are given.^[7] 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 thex = (x,y) plane. Thewavenumber vector isk, and is perpendicular to the cams of thewave crests. Secondly, allowance is made for a mean flow velocityU, in the horizontal direction and uniform over (independent of) depthz. This introduces aDoppler shift in the dispersion relations. At an Earth-fixed location, theobserved angular frequency (orabsolute angular frequency) isω. On the other hand, in aframe of reference moving with the mean velocityU (so the mean velocity as observed from this reference frame is zero), the angular frequency is different. It is called theintrinsic angular frequency (orrelative angular frequency), denotedσ. So in pure wave motion, withU =0, both frequenciesω andσ are equal. The wave numberk (and wavelengthλ) are independent of theframe of reference, and have no Doppler shift (for monochromatic waves).

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 excursionsξ_x andξ_z are the timeintegrals of the oscillatory flow velocitiesu_x andu_z respectively.

Water depth is classified into three regimes:^[8]

• deep water – for a water depth larger than half thewavelength,h >⁠1/2⁠λ, thephase speed of the waves is hardly influenced by depth (this is the case for most wind waves on the sea and ocean surface),^[9]
• shallow water – for a water depth smaller than 5% of the wavelength,h <⁠1/20⁠λ, the phase speed of the waves is only dependent on water depth, and no longer a function ofperiod or wavelength;^[10] and
• intermediate depth – all other cases,⁠1/20⁠λ <h <⁠1/2⁠λ, 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.

Properties of gravity waves on the surface of deep water, shallow water and at intermediate depth, according to Airy wave theory[7]
quantity	symbol	units	deep water (h >⁠1/2⁠λ)	shallow water (h <⁠1/20⁠λ)	intermediate depth (allλ andh)
surface elevation	${\displaystyle \eta (\mathbf{x},t)}$	m	${\displaystyle a\cos \theta (\mathbf{x},t)}$
wave phase	${\displaystyle \theta (\mathbf{x},t)}$	rad	${\displaystyle \mathbf{k}\cdot \mathbf{x}-\omega t}$
observedangular frequency	${\displaystyle \omega }$	rad·s−1	${\displaystyle {\left(\omega -\mathbf{k}\cdot \mathbf{U}\right)}^2=(\mathrm{\Omega }(k){)}^2\text{ with }k=|\mathbf{k}|}$
intrinsic angular frequency	${\displaystyle \sigma }$	rad·s−1	${\displaystyle {\sigma }^2=(\mathrm{\Omega }(k){)}^2\text{ with }\sigma =\omega -\mathbf{k}\cdot \mathbf{U}}$
unit vector in the wave propagation direction	${\displaystyle {\mathbf{e}}_k}$	–	${\displaystyle \frac{\mathbf{k}}{k}}$
dispersion relation	${\displaystyle \mathrm{\Omega }(k)}$	rad·s−1	${\displaystyle \mathrm{\Omega }(k)=\sqrt{gk}}$	${\displaystyle \mathrm{\Omega }(k)=k\sqrt{gh}}$	${\displaystyle \mathrm{\Omega }(k)=\sqrt{gk\tanh kh}}$
phase speed	${\displaystyle c_p=\frac{\mathrm{\Omega }(k)}{k}}$	m·s−1	${\displaystyle \sqrt{\frac{g}{k}}=\frac{g}{\sigma }}$	${\displaystyle \sqrt{gh}}$	${\displaystyle \sqrt{\frac{g}{k}\tanh kh}}$
group speed	${\displaystyle c_g=\frac{\mathrm{\partial }\mathrm{\Omega }}{\mathrm{\partial }k}}$	m·s−1	${\displaystyle \frac{1}{2}\sqrt{\frac{g}{k}}=\frac{1}{2}\frac{g}{\sigma }}$	${\displaystyle \sqrt{gh}}$	${\displaystyle \frac{1}{2}{c}_p\left(1+kh\frac{1-{\tanh }^{2}kh}{\tanh kh}\right)}$
ratio	${\displaystyle \frac{c_g}{c_p}}$	–	${\displaystyle \frac{1}{2}}$	${\displaystyle 1}$	${\displaystyle \frac{1}{2}\left(1+kh\frac{1-{\tanh }^{2}kh}{\tanh kh}\right)}$
horizontal velocity	${\displaystyle {\mathbf{u}}_x(\mathbf{x},z,t)}$	m·s−1	${\displaystyle {\mathbf{e}}_k\sigma a{e}^{kz}\cos \theta }$	${\displaystyle {\mathbf{e}}_k\sqrt{\frac{g}{h}}a\cos \theta }$	${\displaystyle {\mathbf{e}}_k\sigma a\frac{\cosh k(z+h)}{\sinh kh}\cos \theta }$
vertical velocity	${\displaystyle u_z(\mathbf{x},z,t)}$	m·s−1	${\displaystyle \sigma a{e}^{kz}\sin \theta }$	${\displaystyle \sigma a\frac{z+h}{h}\sin \theta }$	${\displaystyle \sigma a\frac{\sinh k(z+h)}{\sinh kh}\sin \theta }$
horizontal particle excursion	${\displaystyle {\mathit{\xi }}_x(\mathbf{x},z,t)}$	m	${\displaystyle -{\mathbf{e}}_{k}a{e}^{kz}\sin \theta }$	${\displaystyle -{\mathbf{e}}_k\frac{1}{kh}a\sin \theta }$	${\displaystyle -{\mathbf{e}}_{k}a\frac{\cosh k(z+h)}{\sinh kh}\sin \theta }$
vertical particle excursion	${\displaystyle {\xi }_z(\mathbf{x},z,t)}$	m	${\displaystyle a{e}^{kz}\cos \theta }$	${\displaystyle a\frac{z+h}{h}\cos \theta }$	${\displaystyle a\frac{\sinh k(z+h)}{\sinh kh}\cos \theta }$
pressure oscillation	${\displaystyle p(\mathbf{x},z,t)}$	N·m−2	${\displaystyle \rho ga{e}^{kz}\cos \theta }$	${\displaystyle \rho ga\cos \theta }$	${\displaystyle \rho ga\frac{\cosh k(z+h)}{\cosh kh}\cos \theta }$

Due tosurface tension, the dispersion relation changes to:^[11]

${\displaystyle {\mathrm{\Omega }}^2(k)=\left(g+\frac{\gamma }{\rho }{k}^2\right)k\tanh kh,}$

withγ the surface tension in newtons per metre. All above equations for linear waves remain the same, if the gravitational accelerationg is replaced by^[12]

${\displaystyle \tilde{g}=g+\frac{\gamma }{\rho }{k}^2.}$

As a result of surface tension, the waves propagate faster. Surface tension only has influence for short waves, with wavelengths less than a fewdecimeters in case of a water–air interface. For very short wavelengths – 2 mm or less, in case of the interface between air and water – gravity effects are negligible. Note that surface tension can be altered bysurfactants.

Thegroup velocity⁠∂Ω/∂k⁠ of capillary waves – dominated by surface tension effects – is greater than thephase velocity⁠Ω/k⁠. This is opposite to the situation of surface gravity waves (with surface tension negligible compared to the effects of gravity) where the phase velocity exceeds the group velocity.^[13]

Surface waves are a special case of interfacial waves, on theinterface between two fluids of differentdensity.

Consider two fluids separated by an interface, and without further boundaries. Then their dispersion relationω² = Ω²(k) is given through^[11][14][15]

${\displaystyle {\mathrm{\Omega }}^2(k)=|k|\left(\frac{\rho -{\rho }^{\prime }}{\rho +{\rho }^{\prime }}g+\frac{\gamma }{\rho +{\rho }^{\prime }}{k}^2\right),}$

whereρ andρ′ are the densities of the two fluids, below (ρ) and above (ρ′) the interface, respectively. Furtherγ 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 aRayleigh–Taylor instability develops.

For two homogeneous layers of fluids, of mean thicknessh below the interface andh′ above – under the action of gravity and bounded above and below by horizontal rigid walls – the dispersion relationshipω² = Ω²(k) for gravity waves is provided by:^[16]

${\displaystyle {\mathrm{\Omega }}^2(k)=\frac{gk(\rho -{\rho }^{\prime })}{\rho \coth kh+{\rho }^{\prime }\coth k{h}^{\prime }},}$

where againρ andρ′ are the densities below and above the interface, whilecoth is thehyperbolic cotangent function. For the caseρ′ is zero this reduces to the dispersion relation of surface gravity waves on water of finite depthh.

In this case the dispersion relation allows for two modes: abarotropic mode where the free surfaceamplitude is large compared with the amplitude of the interfacial wave, and abaroclinic mode where the opposite is the case – the interfacial wave is higher than and inantiphase with the free surface wave. The dispersion relation for this case is of a more complicated form.^[17]

Severalsecond-order wave properties, ones that arequadratic in the wave amplitudea, can be derived directly from Airy wave theory. They are of importance in many practical applications, such asforecasts of wave conditions.^[18] Using aWKBJ approximation, second-order wave properties also find their applications in describing waves in case of slowly varyingbathymetry, 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.

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.

Second-order quantities and their dynamics, using results of Airy wave theory
quantity	symbol	units	formula
mean wave-energy density per unit horizontal area	${\displaystyle E}$	J·m−2	${\displaystyle E=\frac{1}{2}\rho g{a}^2}$
radiation stress or excess horizontalmomentumflux due to the wave motion	${\displaystyle S_{xx}}$	N·m−1	${\displaystyle S_{xx}=\left(2\frac{c_g}{c_p}-\frac{1}{2}\right)E}$
wave action	${\displaystyle \mathcal{A}}$	J·s·m−2	${\displaystyle \mathcal{A}=\frac{E}{\sigma }=\frac{E}{\omega -kU}}$
mean mass-flux due to the wave motion or the wave pseudo-momentum	${\displaystyle M}$	kg·m−1·s−1	${\displaystyle M=\frac{E}{c_p}=k\frac{E}{\sigma }}$
mean horizontal mass-transport velocity	${\displaystyle \tilde{U}}$	m·s−1	${\displaystyle \tilde{U}=U+\frac{M}{\rho h}=U+\frac{E}{\rho h{c}_p}}$
Stokes drift	${\displaystyle {\overline{u}}_S}$	m·s−1	${\displaystyle {\overline{u}}_S=\frac{1}{2}\sigma k{a}^2\frac{\cosh 2k(z+h)}{{\sinh }^{2}kh}}$
wave-energy propagation		J·m−2·s−1	${\displaystyle \frac{\mathrm{\partial }E}{\mathrm{\partial }t}+\frac{\mathrm{\partial }}{\mathrm{\partial }x}((U+c_g)E)+S_{xx}\frac{\mathrm{\partial }U}{\mathrm{\partial }x}=0}$
wave action conservation		J·m−2	${\displaystyle \frac{\mathrm{\partial }\mathcal{A}}{\mathrm{\partial }t}+\frac{\mathrm{\partial }}{\mathrm{\partial }x}((U+c_g)\mathcal{A})=0}$
wave-crest conservation		rad·m−1·s−1	${\displaystyle \frac{\mathrm{\partial }k}{\mathrm{\partial }t}+\frac{\mathrm{\partial }\omega }{\mathrm{\partial }x}=0\text{with}\omega =\mathrm{\Omega }(k)+kU}$
mean mass conservation		kg·m−2·s−1	${\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }t}(\rho h)+\frac{\mathrm{\partial }}{\mathrm{\partial }x}\left(\rho h\tilde{U}\right)=0}$
mean horizontal-momentum evolution		N·m−2	${\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }t}\left(\rho h\tilde{U}\right)+\frac{\mathrm{\partial }}{\mathrm{\partial }x}\left(\rho h{\tilde{U}}^2+\frac{1}{2}\rho g{h}^2+S_{xx}\right)=\rho gh\frac{\mathrm{\partial }d}{\mathrm{\partial }x}}$

The last four equations describe the evolution of slowly varying wave trains overbathymetry in interaction with themean flow, and can be derived from a variational principle:Whitham'saveraged Lagrangian method.^[19] In the mean horizontal-momentum equation,d(x) is the still water depth, that is, the bed underneath the fluid layer is located atz = −d. Note that the mean-flow velocity in the mass and momentum equations is themass transport velocityŨ, including the splash-zone effects of the waves on horizontal mass transport, and not the meanEulerian velocity (for example, as measured with a fixed flow meter).

Wave energy is a quantity of primary interest, since it is a primary quantity that is transported with the wave trains.^[20] 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 thekinetic 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 areaE_pot of the surface gravity waves, which is the deviation of the potential energy due to the presence of the waves:^[21]

The overbar denotes 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 areaE_kin of the wave motion is similarly found to be:^[21]

withσ the intrinsic frequency, see thetable of wave quantities. Using the dispersion relation, the result for surface gravity waves is:

${\displaystyle E_{\text{kin}}=\frac{1}{4}\rho g{a}^2.}$

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 aconservative system.^[22][23] Adding potential and kinetic contributions,E_pot andE_kin, the mean energy density per unit horizontal areaE of the wave motion is:

${\displaystyle E=E_{\text{pot}}+E_{\text{kin}}=\frac{1}{2}\rho g{a}^2.}$

In case of surface tension effects not being negligible, their contribution also adds to the potential and kinetic energy densities, giving^[22]

${\displaystyle E_{\text{pot}}=E_{\text{kin}}=\frac{1}{4}\left(\rho g+\gamma k^2\right)a^2,}$

so

${\displaystyle E=E_{\text{pot}}+E_{\text{kin}}=\frac{1}{2}\left(\rho g+\gamma k^2\right)a^2,}$

withγ thesurface tension.

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 (neglectingdissipative 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 varyingbathymetry and mean-flow fields, a similar and conserved wave quantity, thewave actionA =⁠E/σ⁠:^[19][24][25]

${\displaystyle \frac{\mathrm{\partial }\mathcal{A}}{\mathrm{\partial }t}+\mathrm{\nabla }\cdot \left[\left(\mathbf{U}+{\mathbf{c}}_g\right)\mathcal{A}\right]=0,}$

with(U +c_g)A the actionflux andc_g =c_ge_k thegroup velocity vector. Action conservation forms the basis for manywind wave models andwave turbulence models.^[26] It is also the basis ofcoastal engineering models for the computation ofwave shoaling.^[27] Expanding the above wave action conservation equation leads to the following evolution equation for the wave energy density:^[28]

${\displaystyle \frac{\mathrm{\partial }E}{\mathrm{\partial }t}+\mathrm{\nabla }\cdot \left[\left(\mathbf{U}+{\mathbf{c}}_g\right)E\right]+\mathit{S}:\left(\mathrm{\nabla }\mathbf{U}\right)=0,}$

with:

• (U +c_g)E is the mean wave energy density flux,
• S is theradiation stresstensor and
• ∇U is the mean-velocityshear rate tensor.

In this equation in non-conservation form, theFrobenius inner productS : (∇U) is the source term describing the energy exchange of the wave motion with the mean flow. Only in the case that the mean shear-rate is zero,∇U =0, the mean wave energy densityE is conserved. The two tensorsS and∇U are in aCartesian coordinate system of the form:^[29]

withk_x andk_y the components of the wavenumber vectork and similarlyU_x andU_y the components in of the mean velocity vectorU.

The mean horizontalmomentum per unit areaM induced by the wave motion – and also the wave-inducedmass flux or masstransport – is:^[30]

which is an exact result for periodic progressive water waves, also valid for nonlinear waves.^[31] However, its validity strongly depends on the way how wave momentum and mass flux are defined.Stokes already identified two possible definitions ofphase velocity for periodic nonlinear waves:^[6]

• Stokes first definition of wavecelerity (S1) – with the meanEulerian flow velocity equal to zero for all elevationsz' below the wavetroughs, and
• Stokes second definition of wave celerity (S2) – with the mean mass transport equal to zero.

The above relation between wave momentumM and wave energy densityE is valid within the framework of Stokes' first definition.

However, for waves perpendicular to a coast line or in closed laboratorywave channel, the second definition (S2) is more appropriate. These wave systems have zero mass flux and momentum when using the second definition.^[32] 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 flowU in the opposite direction – called theundertow.

So in general, there are quite some subtleties involved. Therefore also the term pseudo-momentum of the waves is used instead of wave momentum.^[33]

For slowly varyingbathymetry, wave and mean-flow fields, the evolution of the mean flow can de described in terms of the mean mass-transport velocityŨ defined as:^[34]

${\displaystyle \tilde{\mathbf{U}}=\mathbf{U}+\frac{\mathbf{M}}{\rho h}.}$

Note that for deep water, when the mean depthh goes to infinity, the mean Eulerian velocityU and mean transport velocityŨ become equal.

The equation for mass conservation is:^[19][34]

${\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }t}\left(\rho h\right)+\mathrm{\nabla }\cdot \left(\rho h\tilde{\mathbf{U}}\right)=0,}$

whereh(x,t) is the mean water depth, slowly varying in space and time.

Similarly, the mean horizontal momentum evolves as:^[19][34]

${\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }t}\left(\rho h\tilde{\mathbf{U}}\right)+\mathrm{\nabla }\cdot \left(\rho h\tilde{\mathbf{U}}\otimes \tilde{\mathbf{U}}+\frac{1}{2}\rho g{h}^2\mathit{I}+\mathit{S}\right)=\rho gh\mathrm{\nabla }d,}$

withd the still-water depth (the sea bed is atz = –d),S is the wave radiation-stresstensor,I is theidentity matrix and⊗ is thedyadic product:

Note that mean horizontalmomentum is only conserved if the sea bed is horizontal (the still-water depthd 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):^[19][24]

${\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }t}\left(\frac{E}{\sigma }\right)+\mathrm{\nabla }\cdot \left[\left(\mathbf{U}+{\mathbf{c}}_g\right)\frac{E}{\sigma }\right]=0.}$

The wave kinematics are described through the wave-crest conservation equation:^[35]

${\displaystyle \frac{\mathrm{\partial }\mathbf{k}}{\mathrm{\partial }t}+\mathrm{\nabla }\omega =\mathbf{0},}$

with the angular frequencyω a function of the (angular)wavenumberk, related through thedispersion relation. For this to be possible, the wave field must becoherent. By taking thecurl of the wave-crest conservation, it can be seen that an initiallyirrotational wavenumber field stays irrotational.

When following a single particle in pure wave motion (U =0), according to linear Airy wave theory, a first approximation gives closed elliptical orbits for water particles.^[36] However, for nonlinear waves, particles exhibit aStokes drift for which a second-order expression can be derived from the results of Airy wave theory (see thetable above on second-order wave properties).^[37] The Stokes drift velocityū_S, which is the particle drift after one wave cycle divided by theperiod, can be estimated using the results of linear theory:^[38]

${\displaystyle {\overline{\mathbf{u}}}_S=\frac{1}{2}\sigma k{a}^2\frac{\cosh 2k(z+h)}{{\sinh }^{2}kh}{\mathbf{e}}_k,}$

so it varies as a function of elevation. The given formula is for Stokes first definition of wave celerity. Whenρū_S isintegrated over depth, the expression for the mean wave momentumM is recovered.^[38]

• Boussinesq approximation (water waves) – nonlinear theory for waves inshallow water.
• Capillary wave – surface waves under the action ofsurface tension
• Cnoidal wave – nonlinear periodic waves in shallow water, solutions of theKorteweg–de Vries equation
• Mild-slope equation – refraction and diffraction of surface waves over varying depth
• Ocean surface wave – real water waves as seen in the ocean and sea
• Stokes wave – nonlinear periodic waves in non-shallow water
• Wave power – using ocean and sea waves for power generation.

• Airy, G. B. (1841). "Tides and waves". InHugh James Rose; et al. (eds.).Encyclopædia Metropolitana. Mixed Sciences. Vol. 3 (published 1817–1845). Also: "Trigonometry, On the Figure of the Earth, Tides and Waves", 396 pp.
• Stokes, G. G. (1847). "On the theory of oscillatory waves".Transactions of the Cambridge Philosophical Society.8:441–455.
  Reprinted in:Stokes, G. G. (1880).Mathematical and Physical Papers, Volume I. Cambridge University Press. pp. 197–229.

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• Linear theory of ocean surface wavesArchived 2022-01-30 at theWayback Machine on WikiWaves.
• Water waves atMIT.

• Water waves
• Wave mechanics

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