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The 'phase velocity' of a wave is the rate at which the phase of the wave propagates in space. This is the velocity at which the phase of any one frequency component of the wave will propagate. You could pick one particular phase of the wave (for example the crest) and it would appear to travel at the phase velocity. The phase velocity is given in terms of the wave's angular frequency ω and wave vector ''k'' by
:
Note that the phase velocity 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.''

Contents

Matter wave phase

External links

See also

References

Matter wave phase

In quantum mechanics, particles also behave as waves with complex phases. By the de Broglie hypothesis, we see that
:.
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 for a massive particle according to special relativity, phase velocity of matter waves always exceed ''c'', i.e.
:$v\_mathrm\{p\}\; >\; c\; ,$,
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.

External links

★ Subluminal, a Java applet

★ Group and Phase Velocity - Java applet showing the difference between group and phase velocity.

See 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.