Airy wave theory

In fluid dynamics, Airy wave theory (often referred to as linear wave theory) gives a linearised description of the propagation of gravity waves on the surface of a homogeneous fluid layer. The theory assumes that the fluid layer has a uniform mean depth, and that the fluid flow is inviscid, incompressible and irrotational. This theory was first published, in correct form, by George Biddell Airy in the 19th century.[1]

Airy wave theory is often applied in ocean engineering and coastal engineering for the modelling of random sea states – giving a description of the wave kinematics and dynamics of high-enough accuracy for many purposes.[2] [3] Further, several second-order nonlinear properties of surface gravity waves, and their propagation, can be estimated from its results.[4] Airy wave theory is also a good approximation for tsunami 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 the wave height to water depth (for waves in shallow water), and wave height to wavelength (for waves in deep water).


Sine wave amplitude
Wave characteristics.
Dispersion gravity 1
Dispersion of gravity waves on a fluid surface. 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.

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 layers at the boundaries of the fluid domain.[5]

Airy wave theory is often used in ocean engineering and coastal engineering. Especially for random waves, sometimes called wave turbulence, the evolution of the wave statistics – including the wave spectrum – is predicted well over not too long distances (in terms of wavelengths) and in not too shallow water. Diffraction is one of the wave effects which can be described with Airy wave theory. Further, by using the WKBJ approximation, wave shoaling and refraction can be predicted.[2]

Earlier attempts to describe surface gravity waves using potential flow were made by, among others, Laplace, Poisson, Cauchy and Kelland. But Airy was the first to publish the correct derivation and formulation in 1841.[1] Soon after, in 1847, the linear theory of Airy was extended by Stokes for non-linear wave motion – known as Stokes' wave theory – correct up to third order in the wave steepness.[6] Even before Airy's linear theory, Gerstner derived a nonlinear trochoidal wave theory in 1802, which however is not irrotational.[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 elevation η(x,t) of one wave component is sinusoidal, as a function of horizontal position x and time t:


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

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 dispersive waves – exhibiting frequency dispersion – meaning that each wavenumber has its own frequency and phase speed.

Note that in engineering the wave height H – the difference in elevation between crest and trough – is often used:

valid in the present case of linear periodic waves.

Orbital wave motion
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.

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 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.

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.

Mathematical formulation of the wave motion

Flow problem formulation

The waves propagate in the horizontal direction, with coordinate x, and a fluid domain bound above by a free surface at z = η(x,t), with z the vertical coordinate (positive in the upward direction) and t being time.[7] The level z = 0 corresponds with the mean surface elevation. The impermeable bed underneath the fluid layer is at z = -h. Further, the flow is assumed to be incompressible and irrotational – a good approximation of the flow in the fluid interior for waves on a liquid surface – and potential theory can be used to describe the flow. The velocity potential Φ(x,z,t) is related to the flow velocity components ux and uz in the horizontal (x) and vertical (z) directions by:

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

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

The bed being impermeable, leads to the kinematic bed boundary-condition:

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

At the free surface, for infinitesimal waves, the vertical motion of the flow has to be equal to the vertical velocity of the free surface. This leads to the kinematic free-surface boundary-condition:

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

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

Solution for a progressive monochromatic wave

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

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

with sinh and cosh the hyperbolic sine and hyperbolic cosine function, respectively. But η and Φ also have to satisfy the dynamic boundary condition, which results in non-trivial (non-zero) values for the wave amplitude a only if the linear dispersion relation is satisfied:

with tanh the hyperbolic tangent. So angular frequency ω and wavenumber k – or equivalently period T and wavelength λ – cannot be chosen independently, but are related. This means that wave propagation at a fluid surface is an eigenproblem. When ω and k satisfy the dispersion relation, the wave amplitude a can be chosen freely (but small enough for Airy wave theory to be a valid approximation).

Table of wave quantities

In the table below, several flow quantities and parameters according to Airy wave theory are given.[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 the x = (x,y) plane. The wavenumber vector is k, and is perpendicular to the cams of the wave crests. Secondly, allowance is made for a mean flow velocity U, in the horizontal direction and uniform over (independent of) depth z. This introduces a Doppler shift in the dispersion relations. 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).

The table only gives the oscillatory parts of flow quantities – velocities, particle excursions and pressure – and not their mean value or drift. The oscillatory particle excursions ξx and ξz are the time integrals of the oscillatory flow velocities ux and uz respectively.

Water depth is classified into three regimes:[8]

  • 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),[9]
  • shallow water – for a water depth smaller than the wavelength divided by 20, h < ​120 λ, the phase speed of the waves is only dependent on water depth, and no longer a function of period or wavelength;[10] and
  • intermediate depth – all other cases, ​120 λ < h < ½ λ, where both water depth and period (or wavelength) have a significant influence on the solution of Airy wave theory.

In the limiting cases of deep and shallow water, simplifying approximations to the solution can be made. While for intermediate depth, the full formulations have to be used.

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 > ½ λ )
shallow water
( h < 0.05 λ )
intermediate depth
( all λ and h )
surface elevation m
wave phase rad
observed angular frequency rad/s
intrinsic angular frequency rad/s
unit vector in the wave propagation direction
dispersion relation rad/s
phase speed m/s
group speed m/s
horizontal velocity m/s
vertical velocity m/s
horizontal particle excursion m
vertical particle excursion m
pressure oscillation N/m2

Surface tension effects

Dispersion capillary
Dispersion of gravity–capillary waves on the surface of deep water. Phase and group velocity divided by as a function of inverse relative wavelength .
Blue lines (A): phase velocity cp, Red lines (B): group velocity cg.
Drawn lines: gravity–capillary waves.
Dashed lines: gravity waves.
Dash-dot lines: pure capillary waves.

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

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[12]

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

The group velocity ∂Ω/∂k of capillary waves – dominated by surface tension effects – is greater than the phase velocity Ω/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]

Interfacial waves

Surface waves are a special case of interfacial waves, on the 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 ω2 = Ω2(k) is given through:[11][14][15]

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

Interfacial wave rigid boundaries
Wave motion on the interface between two layers of 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 (related to the wavenumber k by: k = 2π / λ), the gravitational acceleration by g and the phase speed as cp (with cp = 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 ω2 = Ω2(k) for gravity waves is provided by:[16]

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.[17]

Second-order wave properties

Several second-order wave properties, i.e. quadratic in the wave amplitude a, can be derived directly from Airy wave theory. They are of importance in many practical applications, e.g. forecasts of wave conditions.[18] 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

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 J / m2
radiation stress or excess horizontal momentum flux due to the wave motion N / m
wave action J·s / m2
mean mass-flux due to the wave motion or the wave pseudo-momentum kg / (m·s)
mean horizontal mass-transport velocity m / s
Stokes drift m / s
wave-energy propagation J / (m2·s)
wave action conservation J / m2
wave-crest conservation rad / (m·s)   with  
mean mass conservation kg / (m2·s)
mean horizontal-momentum evolution N / m2

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: Whitham's averaged Lagrangian method.[19] 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 , including the splash-zone effects of the waves on horizontal mass transport, and not the mean 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.[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 the kinetic and potential energy density, integrated over the depth of the fluid layer and averaged over the wave phase. Simplest to derive is the mean potential energy density per unit horizontal area Epot of the surface gravity waves, which is the deviation of the potential energy due to the presence of the waves:[21]

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 Ekin of the wave motion is similarly found to be:[21]

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

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.[22][23] Adding potential and kinetic contributions, Epot and Ekin, the mean energy density per unit horizontal area E of the wave motion is:

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

with γ the surface tension.

Wave action, wave energy flux and radiation stress

In general, there can be an energy transfer between the wave motion and the mean fluid motion. This means, that the wave energy density is not in all cases a conserved quantity (neglecting dissipative effects), but the total energy density – the sum of the energy density per unit area of the wave motion and the mean flow motion – is. However, there is for slowly varying wave trains, propagating in slowly varying bathymetry and mean-flow fields, a similar and conserved wave quantity, the wave action [19][24][25]

with the action flux and the group velocity vector. Action conservation forms the basis for many wind wave models and wave turbulence models.[26] It is also the basis of coastal engineering models for the computation of wave shoaling.[27] Expanding the above wave action conservation equation leads to the following evolution equation for the wave energy density:[28]


  • is the mean wave energy density flux,
  • is the radiation stress tensor and
  • is the mean-velocity shear-rate tensor.

In this equation in non-conservation form, the Frobenius inner product is the source term describing the energy exchange of the wave motion with the mean flow. Only in case the mean shear-rate is zero, the mean wave energy density is conserved. The two tensors and are in a Cartesian coordinate system of the form:[29]

with and the components of the wavenumber vector and similarly and the components in of the mean velocity vector .

Wave mass flux and wave momentum

The mean horizontal momentum per unit area induced by the wave motion – and also the wave-induced mass flux or mass transport – is:[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 of phase velocity for periodic nonlinear waves:[6]

  • Stokes first definition of wave celerity (S1) – with the mean Eulerian flow velocity equal to zero for all elevations z below the wave troughs, 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.[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 flow U in the opposite direction – called the 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.[33]

Mass and momentum evolution equations

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

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

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

where h(x,t) is the mean water-depth, slowly varying in space and time. Similarly, the mean horizontal momentum evolves as:[19][34]

with d the still-water depth (the sea bed is at z=–d), is the wave radiation-stress tensor, is the identity matrix and is the dyadic product:

Note that mean horizontal momentum is only conserved if the sea bed is horizontal (i.e. the still-water depth d is a constant), in agreement with Noether's theorem.

The system of equations is closed through the description of the waves. Wave energy propagation is described through the wave-action conservation equation (without dissipation and nonlinear wave interactions):[19][24]

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

with the angular frequency ω a function of the (angular) wavenumber k, related through the dispersion relation. For this to be possible, the wave field must be coherent. By taking the curl of the wave-crest conservation, it can be seen that an initially irrotational wavenumber field stays irrotational.

Stokes drift

When following a single particle in pure wave motion according to linear Airy wave theory, a first approximation gives closed elliptical orbits for water particles.[36] 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 above on second-order wave properties).[37] The Stokes drift velocity , which is the particle drift after one wave cycle divided by the period, can be estimated using the results of linear theory:[38]

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

See also


  1. ^ a b c Craik (2004).
  2. ^ a b Goda, Y. (2000). Random Seas and Design of Maritime Structures. Advanced Series on Ocean Engineering. 15. Singapore: World Scientific Publishing Company. ISBN 978-981-02-3256-6. OCLC 45200228.
  3. ^ Dean & Dalrymple (1991).
  4. ^ Phillips (1977), §3.2, pp. 37–43 and §3.6, pp. 60–69.
  5. ^ Lighthill, M. J. (1986). "Fundamentals concerning wave loading on offshore structures". J. Fluid Mech. 173: 667–681. Bibcode:1986JFM...173..667L. doi:10.1017/S0022112086001313.
  6. ^ a b Stokes (1847).
  7. ^ a b c d 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.
  8. ^ Dean & Dalrymple (1991) pp. 64–65
  9. ^ The error in the phase speed is less than 0.2% if depth h is taken to be infinite, for h > ½ λ.
  10. ^ The error in the phase speed is less than 2% if wavelength effects are neglected for h <​120 λ.
  11. ^ a b Phillips (1977), p. 37.
  12. ^ Lighthill (1978), p. 223.
  13. ^ Phillips (1977), p. 175.
  14. ^ Lamb, H. (1994), §267, page 458–460.
  15. ^ Dingemans (1997), Section 2.1.1, p. 45.
  16. ^ Turner, J. S. (1979), Buoyancy effects in fluids, Cambridge University Press, p. 18, ISBN 978-0521297264
  17. ^ Apel, J. R. (1987), Principles of ocean physics, Academic Press, pp. 231–239, ISBN 9780080570747
  18. ^ See for example: the High seas forecasts of NOAA's National Weather service.
  19. ^ a b c d e Whitham, G.B. (1974). Linear and nonlinear waves. Wiley-Interscience. ISBN 978-0-471-94090-6. OCLC 815118., p. 559.
  20. ^ Phillips (1977), p. 23–25.
  21. ^ a b Phillips (1977), p. 39.
  22. ^ a b Phillips (1977), p. 38.
  23. ^ Lord Rayleigh (J. W. Strutt) (1877). "On progressive waves". Proceedings of the London Mathematical Society. 9: 21–26. doi:10.1112/plms/s1-9.1.21. Reprinted as Appendix in: Theory of Sound 1, MacMillan, 2nd revised edition, 1894.
  24. ^ a b Phillips (1977), p. 26.
  25. ^ Bretherton, F. P.; Garrett, C. J. R. (1968). "Wavetrains in inhomogeneous moving media". Proceedings of the Royal Society of London, Series A. 302 (1471): 529–554. Bibcode:1968RSPSA.302..529B. doi:10.1098/rspa.1968.0034.
  26. ^ Phillips (1977), pp. 179–183.
  27. ^ Phillips (1977), pp. 70–74.
  28. ^ Phillips (1977), p. 66.
  29. ^ Phillips (1977), p. 68.
  30. ^ Phillips (1977), pp. 39–40 & 61.
  31. ^ Phillips (1977), p. 40.
  32. ^ Phillips (1977), p. 70.
  33. ^ McIntyre, M. E. (1978). "On the 'wave-momentum' myth". Journal of Fluid Mechanics. 106: 331–347. Bibcode:1981JFM...106..331M. doi:10.1017/S0022112081001626.
  34. ^ a b c Phillips (1977), pp. 61–63.
  35. ^ Phillips (1977), p. 23.
  36. ^ LeBlond, P.H.; Mysak, L A. (1981). Waves in the Ocean. Elsevier Oceanography Series. 20. Elsevier. pp. 85 & 110–111. ISBN 978-0-444-41926-2.
  37. ^ Craik, A.D.D. (1988). Wave interactions and fluid flows. Cambridge University Press. p. 105. ISBN 978-0-521-36829-2.
  38. ^ a b Phillips (1977), p. 44.



Further reading

External links


Airy may refer to:

Sir George Biddell Airy (1801–1892), British Astronomer Royal from 1835 to 1881, for whom the following features, phenomena, and theories are named:

Airy (lunar crater)

Airy (Martian crater)

Airy-0, a smaller crater within the previous one on Mars, and which defines the prime meridian of the planet

Airy wave theory, a linear theory describing the propagation of "gravity waves" on the surface of a fluid

Airy disk, a diffraction pattern in optics

Airy beam, a non-spreading, transversely accelerating optical wavepacket

Airy function, a mathematical function

Airy points, support points chosen to minimize the distortion of the length of a physical standard (such as the International Prototype Meter)

Anna Airy (1882–1964), British artist

Airy (software), a video-downloading utility

Airy, a character in the video game Bravely Default

Bahama Banks

The Bahama Banks are the submerged carbonate platforms that make up much of the Bahama Archipelago. The term is usually applied in referring to either the Great Bahama Bank around Andros Island, or the Little Bahama Bank of Grand Bahama Island and Great Abaco, which are the largest of the platforms, and the Cay Sal Bank north of Cuba. The islands of these banks are politically part of the Bahamas. Other banks are the three banks of the Turks and Caicos Islands, namely the Caicos Bank of the Caicos Islands, the bank of the Turks Islands, and wholly submerged Mouchoir Bank. Further southeast are the equally wholly submerged Silver Bank and Navidad Bank north of the Dominican Republic.

Cnoidal wave

In fluid dynamics, a cnoidal wave is a nonlinear and exact periodic wave solution of the Korteweg–de Vries equation. These solutions are in terms of the Jacobi elliptic function cn, which is why they are coined cnoidal waves. They are used to describe surface gravity waves of fairly long wavelength, as compared to the water depth.

The cnoidal wave solutions were derived by Korteweg and de Vries, in their 1895 paper in which they also propose their dispersive long-wave equation, now known as the Korteweg–de Vries equation. In the limit of infinite wavelength, the cnoidal wave becomes a solitary wave.

The Benjamin–Bona–Mahony equation has improved short-wavelength behaviour, as compared to the Korteweg–de Vries equation, and is another uni-directional wave equation with cnoidal wave solutions. Further, since the Korteweg–de Vries equation is an approximation to the Boussinesq equations for the case of one-way wave propagation, cnoidal waves are approximate solutions to the Boussinesq equations.

Cnoidal wave solutions can appear in other applications than surface gravity waves as well, for instance to describe ion acoustic waves in plasma physics.

Dispersion (water waves)

In fluid dynamics, dispersion of water waves generally refers to frequency dispersion, which means that waves of different wavelengths travel at different phase speeds. Water waves, in this context, are waves propagating on the water surface, with gravity and surface tension as the restoring forces. As a result, water with a free surface is generally considered to be a dispersive medium.

For a certain water depth, surface gravity waves – i.e. waves occurring at the air–water interface and gravity as the only force restoring it to flatness – propagate faster with increasing wavelength. On the other hand, for a given (fixed) wavelength, gravity waves in deeper water have a larger phase speed than in shallower water. In contrast with the behavior of gravity waves, capillary waves (i.e. only forced by surface tension) propagate faster for shorter wavelengths.

Besides frequency dispersion, water waves also exhibit amplitude dispersion. This is a nonlinear effect, by which waves of larger amplitude have a different phase speed from small-amplitude waves.

Fluid dynamics

In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids—liquids and gases. It has several subdisciplines, including aerodynamics (the study of air and other gases in motion) and hydrodynamics (the study of liquids in motion). Fluid dynamics has a wide range of applications, including calculating forces and moments on aircraft, determining the mass flow rate of petroleum through pipelines, predicting weather patterns, understanding nebulae in interstellar space and modelling fission weapon detonation.

Fluid dynamics offers a systematic structure—which underlies these practical disciplines—that embraces empirical and semi-empirical laws derived from flow measurement and used to solve practical problems. The solution to a fluid dynamics problem typically involves the calculation of various properties of the fluid, such as flow velocity, pressure, density, and temperature, as functions of space and time.

Before the twentieth century, hydrodynamics was synonymous with fluid dynamics. This is still reflected in names of some fluid dynamics topics, like magnetohydrodynamics and hydrodynamic stability, both of which can also be applied to gases.

Index of wave articles

This is a list of Wave topics.

List of submarine volcanoes

A list of active and extinct submarine volcanoes and seamounts located under the world's oceans. There are estimated to be 40,000 to 55,000 seamounts in the global oceans. Almost all are not well-mapped and many may not have been identified at all. Most are unnamed and unexplored. This list is therefore confined to seamounts that are notable enough to have been named and/or explored.

List of things named after George Airy

This is a list of things named after George Biddell Airy, a 19th-century mathematician and astronomer.

Mild-slope equation

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

The mild-slope equation models the propagation and transformation of water waves, as they travel through waters of varying depth and interact with lateral boundaries such as cliffs, beaches, seawalls and breakwaters. As a result, it describes the variations in wave amplitude, or equivalently wave height. From the wave amplitude, the amplitude of the flow velocity oscillations underneath the water surface can also be computed. These quantities—wave amplitude and flow-velocity amplitude—may subsequently be used to determine the wave effects on coastal and offshore structures, ships and other floating objects, sediment transport and resulting geomorphology changes of the sea bed and coastline, mean flow fields and mass transfer of dissolved and floating materials. Most often, the mild-slope equation is solved by computer using methods from numerical analysis.

A first form of the mild-slope equation was developed by Eckart in 1952, and an improved version—the mild-slope equation in its classical formulation—has been derived independently by Juri Berkhoff in 1972. Thereafter, many modified and extended forms have been proposed, to include the effects of, for instance: wave–current interaction, wave nonlinearity, steeper sea-bed slopes, bed friction and wave breaking. Also parabolic approximations to the mild-slope equation are often used, in order to reduce the computational cost.

In case of a constant depth, the mild-slope equation reduces to the Helmholtz equation for wave diffraction.

Oceanic plateau

An oceanic or submarine plateau is a large, relatively flat elevation that is higher than the surrounding relief with one or more relatively steep sides.There are 184 oceanic plateaus covering an area of 18,486,600 km2 (7,137,700 sq mi), or about 5.11% of the oceans. The South Pacific region around Australia and New Zealand contains the greatest number of oceanic plateaus (see map).

Oceanic plateaus produced by large igneous provinces are often associated with hotspots, mantle plumes, and volcanic islands — such as Iceland, Hawaii, Cape Verde, and Kerguelen. The three largest plateaus, the Caribbean, Ontong Java, and Mid-Pacific Mountains, are located on thermal swells. Other oceanic plateaus, however, are made of rifted continental crust, for example Falkland Plateau, Lord Howe Rise, and parts of Kerguelen, Seychelles, and Arctic ridges.

Plateaus formed by large igneous provinces were formed by the equivalent of continental flood basalts such as the Deccan Traps in India and the Snake River Plain in the United States.

In contrast to continental flood basalts, most igneous oceanic plateaus erupt through young and thin (6–7 km (3.7–4.3 mi)) mafic or ultra-mafic crust and are therefore uncontaminated by felsic crust and representative for their mantle sources.

These plateaus often rise 2–3 km (1.2–1.9 mi) above the surrounding ocean floor and are more buoyant than oceanic crust. They therefore tend to withstand subduction, more-so when thick and when reaching subduction zones shortly after their formations. As a consequence, they tend to "dock" to continental margins and be preserved as accreted terranes. Such terranes are often better preserved than the exposed parts of continental flood basalts and are therefore a better record of large-scale volcanic eruptions throughout Earth's history. This "docking" also means that oceanic plateaus are important contributors to the growth of continental crust. Their formations often had a dramatic impact on global climate, such as the most recent plateaus formed, the three, large, Cretaceous oceanic plateaus in the Pacific and Indian Ocean: Ontong Java, Kerguelen, and Caribbean.

Physical oceanography

Physical oceanography is the study of physical conditions and physical processes within the ocean, especially the motions and physical properties of ocean waters.

Physical oceanography is one of several sub-domains into which oceanography is divided. Others include biological, chemical and geological oceanography.

Physical oceanography may be subdivided into descriptive and dynamical physical oceanography.Descriptive physical oceanography seeks to research the ocean through observations and complex numerical models, which describe the fluid motions as precisely as possible.

Dynamical physical oceanography focuses primarily upon the processes that govern the motion of fluids with emphasis upon theoretical research and numerical models. These are part of the large field of Geophysical Fluid Dynamics (GFD) that is shared together with meteorology. GFD is a sub field of Fluid dynamics describing flows occurring on spatial and temporal scales that are greatly influenced by the Coriolis force.

Radiation stress

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

The radiation stress tensor describes the additional forcing due to the presence of the waves, which changes the mean depth-integrated horizontal momentum in the fluid layer. As a result, varying radiation stresses induce changes in the mean surface elevation (wave setup) and the mean flow (wave-induced currents).

For the mean energy density in the oscillatory part of the fluid motion, the radiation stress tensor is important for its dynamics, in case of an inhomogeneous mean-flow field.

The radiation stress tensor, as well as several of its implications on the physics of surface gravity waves and mean flows, were formulated in a series of papers by Longuet-Higgins and Stewart in 1960–1964.

Radiation stress derives its name from the analogous effect of radiation pressure for electromagnetic radiation.

Stokes wave

In fluid dynamics, a Stokes wave is a non-linear and periodic surface wave on an inviscid fluid layer of constant mean depth.

This type of modelling has its origins in the mid 19th century when Sir George Stokes – using a perturbation series approach, now known as the Stokes expansion – obtained approximate solutions for non-linear wave motion.

Stokes' wave theory is of direct practical use for waves on intermediate and deep water. It is used in the design of coastal and offshore structures, in order to determine the wave kinematics (free surface elevation and flow velocities). The wave kinematics are subsequently needed in the design process to determine the wave loads on a structure. For long waves (as compared to depth) – and using only a few terms in the Stokes expansion – its applicability is limited to waves of small amplitude. In such shallow water, a cnoidal wave theory often provides better periodic-wave approximations.

While, in the strict sense, Stokes wave refers to progressive periodic waves of permanent form, the term is also used in connection with standing waves and even for random waves.

Undersea mountain range

Undersea mountain ranges are mountain ranges that are mostly or entirely underwater, and specifically under the surface of an ocean. If originated from current tectonic forces, they are often referred to as a mid-ocean ridge. In contrast, if formed by past above-water volcanism, they are known as a seamount chain. The largest and best known undersea mountain range is a mid-ocean ridge, the Mid-Atlantic Ridge. It has been observed that, "similar to those on land, the undersea mountain ranges are the loci of frequent volcanic and earthquake activity".


In fluid dynamics, a wake may either be:

the region of recirculating flow immediately behind a moving or stationary blunt body, caused by viscosity, which may be accompanied by flow separation and turbulence, or

the wave pattern on the water surface downstream of an object in a flow, or produced by a moving object (e.g. a ship), caused by density differences of the fluids above and below the free surface and gravity (or surface tension).

Wave base

The wave base, in physical oceanography, is the maximum depth at which a water wave's passage causes significant water motion. For water depths deeper than the wave base, bottom sediments and the seafloor are no longer stirred by the wave motion above.

Wave shoaling

In fluid dynamics, wave shoaling is the effect by which surface waves entering shallower water change in wave height. It is caused by the fact that the group velocity, which is also the wave-energy transport velocity, changes with water depth. Under stationary conditions, a decrease in transport speed must be compensated by an increase in energy density in order to maintain a constant energy flux. Shoaling waves will also exhibit a reduction in wavelength while the frequency remains constant.

In shallow water and parallel depth contours, non-breaking waves will increase in wave height as the wave packet enters shallower water. This is particularly evident for tsunamis as they wax in height when approaching a coastline, with devastating results.

Wave tank

A wave tank is a laboratory setup for observing the behavior of surface waves. The typical wave tank is a box filled with liquid, usually water, leaving open or air-filled space on top. At one end of the tank an actuator generates waves; the other end usually has a wave-absorbing surface. A similar device is the ripple tank, which is flat and shallow and used for observing patterns of surface waves from above.

Wind wave

In fluid dynamics, wind waves, or wind-generated waves, are water surface waves that occur on the free surface of the oceans and other bodies (like lakes, rivers, canals, puddles or ponds). They result from the wind blowing over an area of fluid surface. Waves in the oceans can travel thousands of miles before reaching land. Wind waves on Earth range in size from small ripples, to waves over 100 ft (30 m) high.When directly generated and affected by local waters, a wind wave system is called a wind sea. After the wind ceases to blow, wind waves are called swells. More generally, a swell consists of wind-generated waves that are not significantly affected by the local wind at that time. They have been generated elsewhere or some time ago. Wind waves in the ocean are called ocean surface waves.

Wind waves have a certain amount of randomness: subsequent waves differ in height, duration, and shape with limited predictability. They can be described as a stochastic process, in combination with the physics governing their generation, growth, propagation, and decay—as well as governing the interdependence between flow quantities such as: the water surface movements, flow velocities and water pressure. The key statistics of wind waves (both seas and swells) in evolving sea states can be predicted with wind wave models.

Although waves are usually considered in the water seas of Earth, the hydrocarbon seas of Titan may also have wind-driven waves.

Ocean zones
Sea level


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