# Phase velocity

periodic gravity waves on the surface of deep water. The red dot moves with the phase velocity, and is located at a fixed wave phase: the crest for the case shown.
The phase velocity (or phase speed) of a wave is the rate at which the phase of the wave propagates in space. This is the speed at which the phase of any one frequency component of the wave travels. For such a component, any given phase of the wave (for example, the crest) will appear to travel at the phase velocity. The phase speed is given in terms of the wavelength "λ" (lambda) and period "T" as

:$v_mathrm\left\{p\right\} = frac\left\{lambda\right\}\left\{T\right\}.$

Or, equivalently, in terms of the wave's angular frequency "&omega;" and wavenumber "k" by

:$v_mathrm\left\{p\right\} = frac\left\{omega\right\}\left\{k\right\}.$

In a dispersive medium, the phase velocity varies with frequency and is not necessarily the same as the group velocity of the wave, which is the rate that changes in amplitude (known as the "envelope" of the wave) will propagate.

The phase velocity of electromagnetic radiation may under certain circumstances (e.g. in the case of anomalous dispersion) exceed the speed of light in a vacuum, but this does not indicate any superluminal information or energy transfer. It was theoretically described by physicists such as Arnold Sommerfeld and Leon Brillouin. "See dispersion for a full discussion of wave velocities."

Matter wave phase

In quantum mechanics, particles also behave as waves with complex phases. By the de Broglie hypothesis, we see that

:$v_mathrm\left\{p\right\} = frac\left\{omega\right\}\left\{k\right\} = frac\left\{E/hbar\right\}\left\{p/hbar\right\} = frac\left\{E\right\}\left\{p\right\}$.

Using relativistic relations for energy and momentum, we have

:

where "E" is the total energy of the particle (i.e. rest energy plus kinetic energy in kinematic sense), "p" the momentum, $gamma$ the Lorentz factor, "c" the speed of light, and β the velocity as a fraction of c. The variable "v" can either be taken to be the velocity of the particle or the group velocity of the corresponding matter wave. See the article on "group velocity" for more detail. Since the particle velocity $v < c$ for a massive particle according to special relativity, phase velocity of matter waves always exceed "c", i.e.

:$v_mathrm\left\{p\right\} > c ,$,

and as we can see, it approaches "c" when the particle velocity is in the relativistic range. The superluminal phase velocity does not violate special relativity, for it doesn't carry any information. See the article on "signal velocity" for detail.

ee also

*Group velocity
*Wave propagation

References

* Tipler, Paul A. and Ralph A. Llewellyn (2003). "Modern Physics". 4th ed. New York; W. H. Freeman and Company. ISBN 0-7167-4345-0. 222-3 pp.
* Leon Brillouin "Wave Propagation And Group Velocity" Academic Press Inc., New York and London (1960) ISBN 0-1213-4968-3.
* Main, Iain G. (1988)."Vibrations and Waves in Physics". 2nd ed. New York; Cambridge University Press. ISBN 0-5212-7846-5. 214-6 pp.

* [http://gregegan.customer.netspace.net.au/APPLETS/20/20.html Subluminal] , a Java applet
* [http://publicliterature.org/tools/group_and_phase_velocity/ Group and Phase Velocity] - Java applet showing the difference between group and phase velocity.

Wikimedia Foundation. 2010.

### Look at other dictionaries:

• phase velocity — fazinis greitis statusas T sritis Standartizacija ir metrologija apibrėžtis Greitis, kuriuo sklinda monochromatinės (begalinės sinusinės) bangos fazė: c = ω/k = λf; čia ω – kampinis dažnis, k – kampinis bangos skaičius, λ – bangos ilgis, f –… …   Penkiakalbis aiškinamasis metrologijos terminų žodynas

• phase velocity — fazinis greitis statusas T sritis fizika atitikmenys: angl. phase velocity vok. Phasengeschwindigkeit, f rus. фазовая скорость, f pranc. vitesse de phase, f …   Fizikos terminų žodynas

• phase velocity — noun : the velocity of a wave motion as determined by the product of the wavelength and frequency called also wave velocity; compare group velocity, particle velocity * * * Physics. the velocity with which a simple harmonic wave is propagated,… …   Useful english dictionary

• phase velocity — Physics. the velocity with which a simple harmonic wave is propagated, equal to the wavelength divided by the period of vibration. Cf. group velocity, particle velocity. [1930 35] * * * …   Universalium

• phase velocity — noun the velocity of propagation of a pure sine wave of infinite extent and infinitesimal amplitude …   Wiktionary

• phase velocity — /feɪz vəˈlɒsəti/ (say fayz vuh losuhtee) noun Physics (of a wave) the velocity of the wavefronts …   Australian English dictionary

• phase velocity of electromagnetic waves — fazinis elektromagnetinių bangų greitis statusas T sritis Standartizacija ir metrologija apibrėžtis Greitis, kuriuo sklinda elektromagnetinių bangų fazė. atitikmenys: angl. phase speed of electromagnetic waves; phase velocity of electromagnetic… …   Penkiakalbis aiškinamasis metrologijos terminų žodynas

• phase velocity of electromagnetic waves — fazinis elektromagnetinių bangų greitis statusas T sritis fizika atitikmenys: angl. phase speed of electromagnetic waves; phase velocity of electromagnetic waves rus. фазовая скорость электромагнитных волн, f pranc. vitesse de phase des ondes… …   Fizikos terminų žodynas

• Phase (waves) — Phase in waves is the fraction of a wave cycle which has elapsed relative to an arbitrary point.[1] Contents 1 Formula 2 Phase shift 3 Phase difference …   Wikipedia

• Velocity-addition formula — In physics, a velocity addition formula is an equation that relates the velocities of moving objects in different reference frames. Contents 1 Galilean addition of velocities 2 Special theory of relativity 3 Derivation …   Wikipedia