Phase velocity
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 period T as
Or, 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, imagine a basic sine wave, A cos (kx−ωt). Given time t, the source produces ωt/2π = ft oscillations. At the same time, the initial wave front propagates away from the source through the space to the distance x to fit the same amount of oscillations, kx = ωt. So that 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 = −dΦ/dt and k = +dΦ/dx, the wave velocity is v = dx/dt = ω/k.
Contents
Relation to group velocity, refractive index and transmission speed
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 Δω/Δk. 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} = ω/k and the group velocity v_{g} = dω/dk depend on the frequency and on the medium. The ratio between the phase speed v_{p} and the speed of light c is known as the refractive index, n = c/v_{p} = ck/ω. Taking the derivative of ω = ck/n with respect to k, we recover the group speed,
Noting that c/n = v_{p}, this shows that the group speed is equal to the phase speed only when the refractive index is a constant: dn/dk = 0, and in this case the phase speed and group speed are independent of frequency: ω/k=dω/dk=c/n. ^{2} 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.
See also
 Cherenkov radiation
 Dispersion (optics)
 Group velocity
 Propagation delay
 Shear wave splitting
 Wave propagation
 Wave propagation speed
 Planck constant
 Speed of light
 Matter wave#Phase velocity
References
Footnotes
 ^ 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.
 ^ ^{a} ^{b} ^{c} "Phase, Group, and Signal Velocity". Mathpages.com. Retrieved 20110724.
Other
 Brillouin, Léon (1960), Wave Propagation And Group Velocity, New York and London: Academic Press Inc., ISBN 0121349683
 Main, Iain G. (1988), Vibrations and Waves in Physics (2nd ed.), New York: Cambridge University Press, pp. 214–216, ISBN 0521278465
 Tipler, Paul A.; Llewellyn, Ralph A. (2003), Modern Physics (4th ed.), New York: W. H. Freeman and Company, pp. 222–223, ISBN 0716743450
External links
 Subluminal – A Java applet
 Simulation – A Java applet by Paul Falstad
 Phase vs. Group Velocity – Various Phase and Groupvelocity relations (animation)
Velocities of waves 

Phase velocity • Group velocity • Front velocity • Signal velocity 
