- 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 small-amplitude waves.
- 1 Frequency dispersion for surface gravity waves
- 2 History
- 3 Surface tension effects
- 4 Nonlinear effects
- 5 Waves on a mean current: Doppler shift
- 6 See also
- 7 Notes
- 8 References
- 9 External links
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.
Wave propagation and dispersion
- with and
- 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 = 0,
- the wave crest at θ = ½ π,
- the downward zero-crossing 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 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
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 and the water depth.
An initial wave phase θ = θ0 propagates as a function of space and time. Its subsequent position is given by:
This shows that the phase moves with the velocity:
which is called the phase velocity.
A sinusoidal wave, of small surface-elevation 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.
with g the acceleration by gravity and cp the phase speed. Since this shallow-water 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 cp is independent of the water depth:
Since the phase speed satisfies cp = λ/T = λf, wavelength and period (or frequency) are related. For instance in deep water:
The dispersion characteristics for intermediate depth are given below.
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 cg different from the phase velocity cp, 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 shallow-water phase velocity. This is because shallow water waves are not dispersive. In deep water, the group velocity is equal to half the phase velocity: cg = ½ cp.
In case of a the 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 Λg and group duration of τg. The group velocity is:
The number of waves in a wave group, measured in space at a certain moment is: Λg / λ. While measured at a fixed location in time, the number of waves in a group is: τg / T. So the ratio of the number of waves measured in space to those measured in time is:
So in deep water, with cg = ½ cp, a wave group has twice as many waves in time as it has in space.
- a the wave amplitude of each frequency component in metres,
- k1 and k2 the wave number of each wave component, in radians per metre, and
- ω1 and ω2 the angular frequency of each wave component, in radians per second.
Both ω1 and k1, as well as ω2 and k2, have to satisfy the dispersion relation:
The part between square brackets is the slowly-varying amplitude of the group, with group wave number ½ ( k1 - k2 ) and group angular frequency ½ ( ω1 - ω2 ). As a result, the group velocity is, for the limit k1 → k2 :
Wave groups can only be discerned in case of a narrow-banded signal, with the wave-number difference k1 - k2 small compared to the mean wave number ½ (k1 + k2).
Multi-component 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.
In the table below, the dispersion relation ω2 = [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.
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 )
( h < 0.05 λ )
( all λ and h )
dispersion relation rad / s phase velocity m / s group velocity m / s ratio - wavelength m for given period T, the solution of:
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 deep-water group velocity is half the phase velocity. In shallow water, for wavelengths larger than twenty times the water depth, as found quite often near the coast, the group velocity is equal to the phase velocity.
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, ω2 = gh k2, was derived by Joseph Louis Lagrange.
Surface tension effects
with σ the surface tension (in N/m).
Amplitude dispersion effects appear for instance in the solitary wave (or soliton): a single hump of water traveling with constant velocity in shallow water with a horizontal bed. 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:
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 soliton solutions only exist for positive values of H, solitary gravity waves of depression do not exist.
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
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 non-moving medium is:
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.
Other articles on dispersion
Dispersive water-wave models
- Airy wave theory
- Benjamin–Bona–Mahony equation
- Boussinesq approximation (water waves)
- Camassa-Holm equation
- Davey–Stewartson equations
- Kadomtsev–Petviashvili equation or KP equation
- Korteweg–de Vries equation (or KdV equation)
- Luke's variational principle
- Nonlinear Schrödinger equation
- Shallow water equations
- ^ a b c d See Lamb (1994), §229, pp. 366–369.
- ^ See Whitham (1974), p.11.
- ^ This dispersion relation is for a non-moving homogeneous medium, so in case of water waves for a constant water depth and no mean current.
- ^ a b See Phillips (1977), p. 37.
- ^ 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
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.
- ^ 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.
- ^ See Phillips (1977), p. 102.
- ^ See Dean and Dalrymple (1991), page 65.
- ^ See Craik (2004).
- ^ See Lamb (1994), §250, pp. 417–420.
- ^ See Phillips (1977), p. 24.
- 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 978 981 02 0420 4, OCLC 22907242
- Dingemans, M.W. (1997), Water wave propagation over uneven bottoms, Advanced Series on Ocean Engineering, 13, World Scientific, Singapore, ISBN 981 02 0427 2, OCLC 36126836 , 2 Parts, 967 pages.
- Lamb, H. (1994), Hydrodynamics (6th ed.), Cambridge University Press, ISBN 978 0 521 45868 9, 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 0 08 339932 8
- Lighthill, M.J. (1978), Waves in fluids, Cambridge University Press, ISBN 0 521 29233 6, OCLC 2966533
- Phillips, O.M. (1977), The dynamics of the upper ocean (2nd ed.), Cambridge University Press, ISBN 0 521 29801 6, OCLC 7319931
- Whitham, G. B. (1974), Linear and nonlinear waves, Wiley-Interscience, ISBN 0 471 94090 9, OCLC 815118
- Mathematical aspects of dispersive waves are discussed on the Dispersive Wiki.
Wikimedia Foundation. 2010.
Look at other dictionaries:
Boussinesq approximation (water waves) — In fluid dynamics, the Boussinesq approximation for water waves is an approximation valid for weakly non linear and fairly long waves. The approximation is named after Joseph Boussinesq, who first derived them in response to the observation by… … Wikipedia
Dispersion — may refer to: In physics: The dependence of wave velocity on frequency or wavelength: Dispersion (optics), for light waves Dispersion (water waves) Acoustic dispersion, for sound waves Dispersion relation, the mathematical description of… … Wikipedia
Dispersion relation — The refraction of a light in a prism is due to dispersion. In physics and electrical engineering, dispersion most often refers to frequency dependent effects in wave propagation. Note, however, that there are several other uses of the word… … Wikipedia
dispersion — /di sperr zheuhn, sheuhn/, n. 1. Also, dispersal. an act, state, or instance of dispersing or of being dispersed. 2. Optics. a. the variation of the index of refraction of a transparent substance, as glass, with the wavelength of light, with the… … Universalium
Color of water — For the book by James McBride, see The Color of Water. Water portal … Wikipedia
Deep ocean water — (DOW) is the name for cold, salty water found deep below the surface of Earth s oceans. Ocean water differs in temperature and salinity, with warm, relatively non salty water found at the surface, and very cold salty water found deeper below the… … Wikipedia
Capillary wave — A capillary wave is a wave travelling along the interface between two fluids, whose dynamics are dominated by the effects of surface tension.Capillary waves are common in nature and home, and are often referred to as ripple. The wavelength of… … Wikipedia
List of wave topics — This is a list of wave topics.0 ndash;9*21 cm lineA*Abbe prism *absorption spectrum *acoustics *Airy disc *Airy wave theory *Alfvén wave *Alpha waves *amphidromic point *amplitude *amplitude modulation *analog sound vs. digital sound *animal… … Wikipedia
Korteweg–de Vries equation — In mathematics, the Korteweg–de Vries equation (KdV equation for short) is a mathematical model of waves on shallow water surfaces. It is particularly notable as the prototypical example of an exactly solvable model, that is, a non linear partial … Wikipedia
Dispersive partial differential equation — In mathematics, a dispersive partial differential equation or dispersive PDE is a partial differential equation that is dispersive. In this context, dispersion means that waves of different wavelength propagate at different phase velocities.… … Wikipedia