Phase velocity

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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 frequency ω and wave vector k by

$v_\mathrm{p} = \frac{\omega}{k}$

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 exceed the speed of light in a vacuum, but this does not indicate any superluminal information or energy transfer. ^{[citation needed]}

See dispersion for a full discussion of wave velocities.

[edit] Matter wave phase

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

$v_\mathrm{p} = \frac{\omega}{k} = \frac{E}{p} = \frac{(\gamma - 1) m c^2}{\gamma m v} = \left( \frac{\gamma - 1}{\gamma \beta} \right) c = \left( \frac{\gamma - 1}{\gamma {\beta}^2} \right) v$

where E is the kinetic energy of the particle, p is its momentum, $γ$ is the Lorentz factor, c is the speed of light, and $β$ is 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. In the extremely relativistic range, the phase velocity equation simply reduces to