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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, and forced by gravity and surface tension. As a result, water with a free surface is generally considered to be a dispersive medium.
Surface gravity waves, moving under the forcing by gravity, propagate faster for increasing wavelength. For a given wavelength, gravity waves in deeper water have a larger phase speed than in shallower water. In contrast with this, capillary waves 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 smallamplitude waves.
Contents
Frequency dispersion for surface gravity waves
This section is about frequency dispersion for waves on a fluid layer forced by gravity, and according to linear theory. For surface tension effects on frequency dispersion, see surface tension effects in Airy wave theory and capillary wave.
Wave propagation and dispersion
The simplest propagating wave of unchanging form is a sine wave. A sine wave with water surface elevation η( x, t ) is given by:^{[1]}
 <math>\eta(x,t) = a \sin \left( \theta(x,t) \right),\,</math>
where a is the amplitude (in metres) and θ = θ( x, t ) is the phase function (in radians), depending on the horizontal position ( x , in metres) and time ( t , in seconds):^{[2]}
 <math>\theta = 2\pi \left( \frac{x}{\lambda}  \frac{t}{T} \right) = k x  \omega t,</math> with <math>k = \frac{2\pi}{\lambda}</math> and <math>\omega = \frac{2\pi}{T},</math>
where:
 λ is the wavelength (in metres),
 T is the period (in seconds),
 k is the wavenumber (in radians per metre) and
 ω is the angular frequency (in radians per second).
Characteristic phases of a water wave are:
 the upward zerocrossing at θ = 0,
 the wave crest at θ = ½ π,
 the downward zerocrossing at θ = π and
 the wave trough at θ = 1½ π.
A certain phase repeats itself after an integer m multiple of 2π: sin(θ) = sin(θ+m•2π).
Essential for water waves, and other wave phenomena in physics, is that free propagating waves of nonzero amplitude only exist when the angular frequency ω and wavenumber k (or equivalently the wavelength λ and period T ) satisfy a functional relationship: the frequency dispersion relation^{[3]}^{[4]}
 <math>\omega^2 = \Omega^2(k).\,</math>
The dispersion relation has two solutions: ω = +Ω(k) and ω = −Ω(k), corresponding to waves travelling in the positive or negative x–direction. The dispersion relation will in general depend on several other parameters in addition to the wavenumber k. For gravity waves, according to linear theory, these are the acceleration by gravity g and the water depth h. The dispersion relation for these waves is:^{[5]}^{[4]}
<math>\omega^2 = g\, k\, \tanh(k\,h)</math> or <math> \displaystyle \lambda = \frac{g}{2\pi}\, T^2\, \tanh\left( 2\pi\, \frac{h}{\lambda} \right),</math>
an implicit equation with tanh denoting the hyperbolic tangent function.
An initial wave phase θ = θ_{0} propagates as a function of space and time. Its subsequent position is given by:
 <math>x = \frac{\lambda}{T}\, t + \frac{\lambda}{2\pi}\, \theta_0 = \frac{\omega}{k}\, t + \frac{\theta_0}{k}.</math>
This shows that the phase moves with the velocity:^{[1]}
 <math>c_p = \frac{\lambda}{T} = \frac{\omega}{k} = \frac{\Omega(k)}{k},</math>
which is called the phase velocity.
Phase velocity
A sinusoidal wave, of small surfaceelevation amplitude and with a constant wavelength, propagates with the phase velocity, also called celerity or phase speed. While the phase velocity is a vector and has an associated direction, celerity or phase speed refer only to the magnitude of the phase velocity. According to linear theory for waves forced by gravity, the phase speed depends on the wavelength and the water depth. For a fixed water depth, long waves (with large wavelength) propagate faster than shorter waves.
In the left figure, it can be seen that shallow water waves, with wavelengths λ much larger than the water depth h, travel with the phase velocity^{[1]}
 <math>c_p = \sqrt{gh} \qquad \scriptstyle \text{(shallow water),}\,</math>
with g the acceleration by gravity and c_{p} the phase speed. Since this shallowwater phase speed is independent of the wavelength, shallow water waves do not have frequency dispersion.
Using another normalization for the same frequency dispersion relation, the figure on the right shows that in deep water, with water depth h larger than half the wavelength λ (so for h/λ > 0.5), the phase velocity c_{p} is independent of the water depth:^{[1]}
 <math>c_p = \sqrt{\frac{g\lambda}{2\pi}} = \frac{g}{2\pi} T \qquad \scriptstyle \text{(deep water),}</math>
with T the wave period (the reciprocal of the frequency f, T=1/f ). So in deep water the phase speed increases with the wavelength, and with the period.
Since the phase speed satisfies c_{p} = λ/T = λf, wavelength and period (or frequency) are related. For instance in deep water:
 <math>\lambda = \frac{g}{2\pi}T^2 \qquad \scriptstyle \text{(deep water).}</math>
The dispersion characteristics for intermediate depth are given below.
Group velocity
Interference of two sinusoidal waves with slightly different wavelengths, but the same amplitude and propagation direction, results in a beat pattern, called a wave group. As can be seen in the animation, the group moves with a group velocity c_{g} different from the phase velocity c_{p}, due to frequency dispersion.
The group velocity is depicted by the red lines (marked B) in the two figures above. In shallow water, the group velocity is equal to the shallowwater phase velocity. This is because shallow water waves are not dispersive. In deep water, the group velocity is equal to half the phase velocity: c_{g} = ½ c_{p}.^{[6]}
The group velocity also turns out to be the energy transport velocity. This is the velocity with which the mean wave energy is transported horizontally in a narrowband wave field.^{[7]}^{[8]}
In the case of a group velocity different from the phase velocity, a consequence is that the number of waves counted in a wave group is different when counted from a snapshot in space at a certain moment, from when counted in time from the measured surface elevation at a fixed position. Consider a wave group of length Λ_{g} and group duration of τ_{g}. The group velocity is:^{[9]}
 <math>c_g = \frac{\Lambda_g}{\tau_g}.</math>
The number of waves in a wave group, measured in space at a certain moment is: Λ_{g} / λ. While measured at a fixed location in time, the number of waves in a group is: τ_{g} / T. So the ratio of the number of waves measured in space to those measured in time is:
 <math>
\tfrac{\text{No. of waves in space}}{\text{No. of waves in time}} = \frac{\Lambda_g / \lambda}{\tau_g / T} = \frac{\Lambda_g}{\tau_g} \cdot \frac{T}{\lambda} = \frac{c_g}{c_p}.
</math>
So in deep water, with c_{g} = ½ c_{p},^{[10]} a wave group has twice as many waves in time as it has in space.^{[11]}
The water surface elevation η(x,t), as a function of horizontal position x and time t, for a bichromatic wave group of full modulation can be mathematically formulated as:^{[10]}
 <math>
\eta = a\, \sin \left( k_1 x  \omega_1 t \right)
+ a\, \sin \left( k_2 x  \omega_2 t \right),
</math> with:
 a the wave amplitude of each frequency component in metres,
 k_{1} and k_{2} the wave number of each wave component, in radians per metre, and
 ω_{1} and ω_{2} the angular frequency of each wave component, in radians per second.
Both ω_{1} and k_{1}, as well as ω_{2} and k_{2}, have to satisfy the dispersion relation:
 <math>\omega_1^2 = \Omega^2(k_1)\,</math> and <math>\omega_2^2 = \Omega^2(k_2).\,</math>
Using trigonometric identities, the surface elevation is written as:^{[9]}
 <math>
\eta= \left[ 2\, a\, \cos \left( \frac{k_1  k_2}{2} x  \frac{\omega_1  \omega_2}{2} t \right) \right]\;
\cdot\; \sin \left( \frac{k_1 + k_2}{2} x  \frac{\omega_1 + \omega_2}{2} t \right).
</math>
The part between square brackets is the slowly varying amplitude of the group, with group wave number ½ ( k_{1} − k_{2} ) and group angular frequency ½ ( ω_{1} − ω_{2} ). As a result, the group velocity is, for the limit k_{1} → k_{2} :^{[9]}^{[10]}
 <math>c_g = \lim_{k_1\, \to\, k_2} \frac{\omega_1  \omega_2}{k_1  k_2}
= \lim_{k_1\, \to\, k_2} \frac{\Omega(k_1)  \Omega(k_2)}{k_1  k_2} = \frac{\text{d}\Omega(k)}{\text{d}k}.</math>
Wave groups can only be discerned in case of a narrowbanded signal, with the wavenumber difference k_{1} − k_{2} small compared to the mean wave number ½ (k_{1} + k_{2}).
Multicomponent wave patterns
The effect of frequency dispersion is that the waves travel as a function of wavelength, so that spatial and temporal phase properties of the propagating wave are constantly changing. For example, under the action of gravity, water waves with a longer wavelength travel faster than those with a shorter wavelength.
While two superimposed sinusoidal waves, called a bichromatic wave, have an envelope which travels unchanged, three or more sinusoidal wave components result in a changing pattern of the waves and their envelope. A sea state – that is: real waves on the sea or ocean – can be described as a superposition of many sinusoidal waves with different wavelengths, amplitudes, initial phases and propagation directions. Each of these components travels with its own phase velocity, in accordance with the dispersion relation. The statistics of such a surface can be described by its power spectrum.^{[12]}
Dispersion relation
In the table below, the dispersion relation ω^{2} = [Ω(k)]^{2} between angular frequency ω = 2π / T and wave number k = 2π / λ is given, as well as the phase and group speeds.^{[9]}
Frequency dispersion of gravity waves on the surface of deep water, shallow water and at intermediate depth, according to linear wave theory  

quantity  symbol  units  deep water ( h > ½ λ ) 
shallow water ( h < 0.05 λ ) 
intermediate depth ( all λ and h ) 
dispersion relation  <math>\displaystyle\Omega(k)</math>  rad / s  <math>\sqrt{gk}=\sqrt{\frac{2\pi\,g}{\lambda}}</math>  <math>k \sqrt{g h} = \frac{2\pi}{\lambda} \sqrt{g h}</math>  <math>
\begin{align} &\sqrt{ gk\, \tanh\left( kh \right)}\, \\[1.2ex] &=\sqrt{\frac{2\pi g}{\lambda}\tanh\left(\frac{2\pi h}{\lambda}\right)}\, \end{align} </math> 
phase velocity  <math>\displaystyle c_p=\frac{\lambda}{T}=\frac{\omega}{k}</math>  m / s  <math>\sqrt{\frac{g}{k}} = \frac{g}{\omega} = \frac{g}{2\pi} T</math>  <math>\sqrt{g h}</math>  <math>\sqrt{\frac{g}{k}\tanh\left( k h \right)}</math> 
group velocity  <math>\displaystyle c_g= \frac{\partial\Omega}{\partial k}</math>  m / s  <math>\frac{1}{2}\sqrt{\frac{g}{k}} = \frac{1}{2}\frac{g}{\omega} = \frac{g}{4\pi} T</math>  <math>\sqrt{g h}</math>  <math>\frac{1}{2} c_p \left( 1 + \frac{2 k h}{\sinh\left(2 k h \right)} \right)</math> 
ratio  <math> \displaystyle \frac{c_g}{c_p}</math>    <math>\displaystyle\frac{1}{2}</math>  <math>\displaystyle 1</math>  <math>\frac{1}{2} \left( 1 + \frac{2 k h}{\sinh\left( 2 k h \right)} \right)</math> 
wavelength  <math>\displaystyle\lambda</math>  m  <math>\frac{g}{2\pi} T^2</math>  <math>T \sqrt{g h}</math>  for given period T, the solution of: <math>\displaystyle \left(\frac{2\pi}{T}\right)^2=\frac{2\pi g}{\lambda}\tanh\left(\frac{2\pi h}{\lambda}\right)</math> 
Deep water corresponds with water depths larger than half the wavelength, which is the common situation in the ocean. In deep water, longer period waves propagate faster and transport their energy faster. The deepwater group velocity is half the phase velocity. In shallow water, for wavelengths larger than twenty times the water depth,^{[13]} as found quite often near the coast, the group velocity is equal to the phase velocity.
History
The full linear dispersion relation was first found by PierreSimon Laplace, although there were some errors in his solution for the linear wave problem. The complete theory for linear water waves, including dispersion, was derived by George Biddell Airy and published in about 1840. A similar equation was also found by Philip Kelland at around the same time (but making some mistakes in his derivation of the wave theory).^{[14]}
The shallow water (with small h / λ) limit, ω^{2} = gh k^{2}, was derived by Joseph Louis Lagrange.
Surface tension effects
In case of gravity–capillary waves, where surface tension affects the waves, the dispersion relation becomes:^{[4]}
 <math>
\omega^2 = \left( g k + \frac{\sigma}{\rho} k^3 \right) \tanh (kh),
</math> with σ the surface tension (in N/m).
For a water–air interface (with σ = 0.074 N/m and ρ = 1000 kg/m³) the waves can be approximated as pure capillary waves – dominated by surfacetension effects – for wavelengths less than Script error: No such module "convert".. For wavelengths above Script error: No such module "convert". the waves are to good approximation pure surface gravity waves with very little surfacetension effects.^{[15]}
Nonlinear effects
Shallow water
Amplitude dispersion effects appear for instance in the solitary wave: a single hump of water traveling with constant velocity in shallow water with a horizontal bed. Note that solitary waves are nearsolitons, but not exactly – after the interaction of two (colliding or overtaking) solitary waves, they have changed a bit in amplitude and an oscillatory residual is left behind.^{[16]} The single soliton solution of the Korteweg–de Vries equation, of wave height H in water depth h far away from the wave crest, travels with the velocity:
 <math>c_p = c_g = \sqrt{g(h+H)}.</math>
So for this nonlinear gravity wave it is the total water depth under the wave crest that determines the speed, with higher waves traveling faster than lower waves. Note that solitary wave solutions only exist for positive values of H, solitary gravity waves of depression do not exist.
Deep water
The linear dispersion relation – unaffected by wave amplitude – is for nonlinear waves also correct at the second order of the perturbation theory expansion, with the orders in terms of the wave steepness k A (where A is wave amplitude). To the third order, and for deep water, the dispersion relation is^{[17]}
 <math> \omega^2 = gk \left[1+(kA)^2\right]. </math>
This implies that large waves travel faster than small ones of the same frequency. This is only noticeable when the wave steepness k A is large.
Waves on a mean current: Doppler shift
Water waves on a mean flow (so a wave in a moving medium) experience a Doppler shift. Suppose the dispersion relation for a nonmoving medium is:
 <math>\omega^2 = \Omega^2(k),\,</math>
with k the wavenumber. Then for a medium with mean velocity vector V, the dispersion relationship with Doppler shift becomes:^{[18]}
 <math>\left( \omega  \bold k \cdot \bold V \right)^2 = \Omega^2(k),</math>
where k is the wavenumber vector, related to k as: k = k. The inner product k•V is equal to: k•V = kV cos α, with V the length of the mean velocity vector V: V = V. And α the angle between the wave propagation direction and the mean flow direction. For waves and current in the same direction, k•V=kV.
See also
Other articles on dispersion
Dispersive waterwave models
 Airy wave theory
 Benjamin–Bona–Mahony equation
 Boussinesq approximation (water waves)
 Cnoidal wave
 Camassa–Holm equation
 Davey–Stewartson equation
 Kadomtsev–Petviashvili equation (also known as KP equation)
 Korteweg–de Vries equation (also known as KdV equation)
 Luke's variational principle
 Nonlinear Schrödinger equation
 Shallow water equations
 Stokes' wave theory
 Trochoidal wave
 Wave turbulence
 Whitham equation
Notes
 ^ ^{a} ^{b} ^{c} ^{d} See Lamb (1994), §229, pp. 366–369.
 ^ See Whitham (1974), p.11.
 ^ This dispersion relation is for a nonmoving homogeneous medium, so in case of water waves for a constant water depth and no mean current.
 ^ ^{a} ^{b} ^{c} See Phillips (1977), p. 37.
 ^ See e.g. Dingemans (1997), p. 43.
 ^ See Phillips (1977), p. 25.
 ^ Reynolds, O. (1877), "On the rate of progression of groups of waves and the rate at which energy is transmitted by waves", Nature 16: 343–44, Bibcode:1877Natur..16R.341., doi:10.1038/016341c0
Lord Rayleigh (J. W. Strutt) (1877), "On progressive waves", Proceedings of the London Mathematical Society 9: 21–26, doi:10.1112/plms/s19.1.21 Reprinted as Appendix in: Theory of Sound 1, MacMillan, 2nd revised edition, 1894.  ^ See Lamb (1994), §237, pp. 382–384.
 ^ ^{a} ^{b} ^{c} ^{d} See Dingemans (1997), section 2.1.2, pp. 46–50.
 ^ ^{a} ^{b} ^{c} See Lamb (1994), §236, pp. 380–382.
 ^ Henderson, K. L.; Peregrine, D. H.; Dold, J. W. (1999), "Unsteady water wave modulations: fully nonlinear solutions and comparison with the nonlinear Schrödinger equation", Wave Motion 29 (4): 341–361, doi:10.1016/S01652125(98)000456
 ^ See Phillips (1977), p. 102.
 ^ See Dean and Dalrymple (1991), page 65.
 ^ See Craik (2004).
 ^ See Lighthill (1978), pp. 224–225.
 ^ See e.g.: Craig, W.; Guyenne, P.; Hammack, J.; Henderson, D.; Sulem, C. (2006), "Solitary water wave interactions", Physics of Fluids 18 (057106): 25 pp., Bibcode:2006PhFl...18e7106C, doi:10.1063/1.2205916
 ^ See Lamb (1994), §250, pp. 417–420.
 ^ See Phillips (1977), p. 24.
References
 Craik, A.D.D. (2004), "The origins of water wave theory", Annual Review of Fluid Mechanics 36: 1–28, Bibcode:2004AnRFM..36....1C, doi:10.1146/annurev.fluid.36.050802.122118
 Dean, R.G.; Dalrymple, R.A. (1991), Water wave mechanics for engineers and scientists, Advanced Series on Ocean Engineering 2, World Scientific, Singapore, ISBN 9789810204204, OCLC 22907242
 Dingemans, M.W. (1997), Water wave propagation over uneven bottoms, Advanced Series on Ocean Engineering 13, World Scientific, Singapore, ISBN 9810204272, OCLC 36126836, 2 Parts, 967 pages.
 Lamb, H. (1994), Hydrodynamics (6th ed.), Cambridge University Press, ISBN 9780521458689, OCLC 30070401 Originally published in 1879, the 6th extended edition appeared first in 1932.
 Landau, L.D.; Lifshitz, E.M. (1987), Fluid Mechanics, Course of theoretical physics 6 (2nd ed.), Pergamon Press, ISBN 0080339328
 Lighthill, M.J. (1978), Waves in fluids, Cambridge University Press, 504 pp., ISBN 0521292336, OCLC 2966533
 Phillips, O.M. (1977), The dynamics of the upper ocean (2nd ed.), Cambridge University Press, ISBN 0521298016, OCLC 7319931
 Whitham, G. B. (1974), Linear and nonlinear waves, WileyInterscience, ISBN 0471940909, OCLC 815118
External links
 Mathematical aspects of dispersive waves are discussed on the Dispersive Wiki.
