# Phase velocity

File:Wave phase.gif
Phase velocity in 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

${\displaystyle v_{\mathrm {p} }={\frac {\lambda }{T}}.}$

Or, equivalently, in terms of the wave's angular frequency ω and wavenumber k by

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

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

${\displaystyle v_{\mathrm {p} }={\frac {\omega }{k}}={\frac {E/\hbar }{p/\hbar }}={\frac {E}{p}}}$.

Using relativistic relations for energy and momentum, we have

${\displaystyle v_{\mathrm {p} }={\frac {E}{p}}={\frac {\gamma mc^{2}}{\gamma mv}}={\frac {c^{2}}{v}}={\frac {c}{\beta }}}$

where E is the total energy of the particle (i.e. rest energy plus kinetic energy in kinematic sense), p the momentum, ${\displaystyle \gamma }$ 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 ${\displaystyle v for a massive particle according to special relativity, phase velocity of matter waves always exceed c, i.e.

${\displaystyle v_{\mathrm {p} }>c\,}$,

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.