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 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 velocity is given in terms of the "wavelength λ (lambda) and "time period T as

Equivalently, in terms of the wave's "angular frequency ω, which specifies angular change per unit of time, and "wavenumber (or angular wave number) k, which represents the proportionality between the angular frequency ω and the linear speed (speed of propagation) ν_{p},

To understand where this equation comes from, consider a basic "sine wave, *A* cos (*kx*−*ωt*). After time t, the source has produced *ωt/2π = ft* oscillations. After the same time, the initial wave front has propagated away from the source through space to the distance x to fit the same number of oscillations, *kx* = *ωt*.

Thus the propagation velocity *v* is *v* = *x*/*t* = *ω*/*k*. The wave propagates faster when higher frequency oscillations are distributed less densely in space.^{[2]} Formally, *Φ* = *kx*−*ωt* is the phase. Since *ω* = −d*Φ*/d*t* and *k* = +d*Φ*/d*x*, the wave velocity is *v* = d*x*/d*t* = *ω*/*k*.

Since a pure sine wave cannot convey any information, some change in amplitude or frequency, known as "modulation, is required. By combining two sines with slightly different frequencies and wavelengths,

the amplitude becomes a sinusoid with *phase* speed Δ*ω*/Δ*k*. It is this modulation that represents the signal content. Since each amplitude *envelope* contains a group of internal waves, this speed is usually called the "group velocity, *v*_{g}.^{[2]}

In a given medium, the frequency is some function *ω*(*k*) of the wave number, so in general, the phase velocity *v _{p}* =

Taking the derivative of *ω* = *ck*/*n* with respect to k, yields the "group velocity,

Noting that *c*/*n* = *v _{p}*, indicates that the group speed is equal to the phase speed only when the refractive index is a constant d

Otherwise, both the phase velocity and the group velocity vary with frequency, and the medium is called "dispersive; the relation *ω*=*ω*(*k*) is known as the "dispersion relation of the medium.

The phase velocity of "electromagnetic radiation may – under certain circumstances (for example "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 "Léon Brillouin. See "dispersion for a full discussion of wave velocities.

**^**Nemirovsky, Jonathan; Rechtsman, Mikael C; Segev, Mordechai (9 April 2012). "Negative radiation pressure and negative effective refractive index via dielectric birefringence" (PDF).*Optics Express*.**20**(8): 8907–8914. "Bibcode:2012OExpr..20.8907N. "doi:10.1364/OE.20.008907. "PMID 22513601. Archived from the original (PDF) on 16 October 2013.- ^
^{a}^{b}^{c}"Phase, Group, and Signal Velocity". Mathpages.com. Retrieved 2011-07-24.

- Crawford jr., Frank S. (1968).
*Waves (Berkeley Physics Course, Vol. 3)*, McGraw-Hill, "ISBN "978-0070048607 Free online version - "Brillouin, Léon (1960),
*Wave Propagation And Group Velocity*, New York and London: Academic Press Inc., "ISBN "0-12-134968-3 - Main, Iain G. (1988),
*Vibrations and Waves in Physics*(2nd ed.), New York: Cambridge University Press, pp. 214–216, "ISBN "0-521-27846-5 - Tipler, Paul A.; Llewellyn, Ralph A. (2003),
*Modern Physics*(4th ed.), New York: W. H. Freeman and Company, pp. 222–223, "ISBN "0-7167-4345-0

- Subluminal – A Java applet
- Simulation – A Java applet by Paul Falstad
- Phase vs. Group Velocity – Various Phase- and Group-velocity relations (animation)