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.
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, 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.
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.
Electromagnetic field behavior in dispersive isotropic negative phase velocity/ negative refractive index guided wave structures compatible with millimeter-wave monolithic integrated circuits.(Research Article)(Report)
Jan 01, 2007; A microstrip configuration has been loaded with a dispersive isotropic left- handed medium (LHM) substrate and studied regarding...
WIPO ASSIGNS PATENT TO NIDEC FOR "ROTOR PHASE/ VELOCITY ESTIMATION DEVICE FOR ALTERNATING-CURRENT MOTOR" (JAPANESE INVENTOR)
Oct 26, 2011; GENEVA, Oct. 26 -- Publication No. WO/2011/129423 was published on Oct. 20. Title of the invention: "ROTOR PHASE/VELOCITY...
Publication No. WO/2009/104576 Published on Aug. 27, Assigned to Meidensha for Rotor Phase, Velocity Detection Device
Sep 01, 2009; GENEVA, Sept. 1 -- Kazuya Ogura and Katsuyuki Watanabe have developed a rotor phase and velocity detection device. The patent has...