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.

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

thumb|right|388px|Sinusoidal wave.

The simplest propagating wave of unchanging form is a sine wave. A sine wave with water surface elevation <math>\eta(x,t)</math> is given by:

:<math>\eta(x,t) = a \sin \left( \theta(x,t) \right),\,</math>

where a is the amplitude (in metres) and <math>\theta=\theta(x,t)</math> is the phase function (in radians), depending on the horizontal position (x, in metres) and time (t, in seconds):

:<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 zero-crossing at θ&nbsp;=&nbsp;0,
  • the wave crest at θ&nbsp;=&nbsp;'&nbsp;π,
  • the downward zero-crossing at θ&nbsp;=&nbsp;π and
  • the wave trough at θ&nbsp;=&nbsp;&nbsp;π.

A certain phase repeats itself after an integer m multiple of <math>2\pi</math>. <math>\sin(\theta)=\sin(\theta+2\pi m)</math>

Essential for water waves, and other wave phenomena in physics, is that free propagating waves of non-zero 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

:<math>\omega^2 = \Omega^2(k).\,</math>

The dispersion relation has two solutions: <math>\omega=+\Omega(k)</math> and <math>\omega=-\Omega(k)</math>, 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:

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 narrow-band wave field.

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<sub>g</sub> and group duration of τ<sub>g</sub>. The group velocity is:

:<math>c_g = \frac{\Lambda_g}{\tau_g}.</math>

[[File:Wave group space time.svg|thumb|388px|The number of waves per group as observed in space at a certain moment (upper blue line), is different from the number of waves per group seen in time at a fixed position (lower orange line), due to frequency dispersion.

{| class="wikitable collapsible collapsed" style="width:98%;"

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|For the shown case, a bichromatic group of gravity waves on the surface of deep water, the group velocity is half the phase velocity. In this example, there are waves between two wave group nodes in space, while there are waves between two wave group nodes in time.

|}]]

thumb|right|388px|[[North Pacific storm waves as seen from the NOAA M/V Noble Star, Winter 1989.]]

The number of waves in a wave group, measured in space at a certain moment is: Λ<sub>g</sub>&nbsp;/&nbsp;λ. While measured at a fixed location in time, the number of waves in a group is: τ<sub>g</sub>&nbsp;/&nbsp;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 <math>c_g=\frac{1}{2}c_p</math>, a wave group has twice as many waves in time as it has in space.

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:

Dispersion relation

In the table below, the dispersion relation <math>\omega^2 = [\Omega(k)]^2</math> between angular frequency ω = 2π / T and wave number k = 2π / λ is given, as well as the phase and group speeds. 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 Pierre-Simon 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).

The shallow water (with small h / λ) limit, ω<sup>2</sup> = gh k<sup>2</sup>, was derived by Joseph Louis Lagrange.

Surface tension effects

[[File:Dispersion capillary.svg|thumb|right|Dispersion of gravity-capillary waves on the surface of deep water. Phase and group velocity divided by <math>\scriptstyle \sqrt[4]{g\sigma/\rho}</math> as a function of inverse relative wavelength <math>\scriptstyle \frac{1}{\lambda}\sqrt{\sigma/(\rho g)}</math>.<br>Blue lines (A): phase velocity, Red lines (B): group velocity.<br>Drawn lines: dispersion relation for gravity-capillary waves.<br>Dashed lines: dispersion relation for deep-water gravity waves.<br>Dash-dot lines: dispersion relation valid for deep-water capillary waves.]]

In case of gravity–capillary waves, where surface tension affects the waves, the dispersion relation becomes:

Interfacial waves

thumb|right|Wave motion on the interface between two layers of [[viscosity|inviscid homogeneous fluids of different density, confined between horizontal rigid boundaries (at the top and bottom). The motion is forced by gravity. The upper layer has mean depth h and density ρ, while the lower layer has mean depth h and density ρ. The wave amplitude is a, the wavelength is denoted by λ.]]

For two homogeneous layers of fluids, of mean thickness h below the interface and ' above – under the action of gravity and bounded above and below by horizontal rigid walls – the dispersion relationship ω<sup>2</sup>&nbsp;=&nbsp;Ω<sup>2</sup>(k) for gravity waves is provided by:

:<math>

\Omega^2(k) = \frac{g\, k (\rho - \rho')}{\rho\, \coth( k h ) + \rho'\, \coth( k h')},

</math>

where again ρ and ' are the densities below and above the interface, while coth is the hyperbolic cotangent function. For the case ' is zero this reduces to the dispersion relation of surface gravity waves on water of finite depth h.

When the depth of the two fluid layers becomes very large (h→∞, '→∞), the hyperbolic cotangents in the above formula approaches the value of one. Then:

:<math>

\Omega^2(k) = \frac{\rho - \rho'}{\rho + \rho'}\, g\, k.

</math>

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 near-solitons, 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. 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 (where a is wave amplitude). To the third order, and for deep water, the dispersion relation is

:<math> \omega^2 = gk \left[1+(ka)^2\right],</math> so <math>c_p = \sqrt{\frac{g}{k\, \left[ 1 + \tfrac12\, (ka)^2 \right] + \mathcal{O}\left((ka)^4\right).</math>

This implies that large waves travel faster than small ones of the same frequency. This is only noticeable when the wave steepness 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 non-moving 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:

:<math>\left( \omega - \mathbf k \cdot \mathbf V \right)^2 = \Omega^2(k),</math>

where k is the wavenumber vector, related to k as: k = |k|. The dot 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

  • Dispersive partial differential equation
  • Capillary wave

Dispersive water-wave 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

References

  • , 2 Parts, 967 pages.
  • Originally published in 1879, the 6th extended edition appeared first in 1932.
  • Mathematical aspects of dispersive waves are discussed on the Dispersive Wiki.