Group vs. Phase Velocity?

Question

What is the difference between phase velocity and group velocity?
Does this ideas apply to all types of waves or for specific waves?
What makes the group velocity faster than the speed of light (I already know the derivation but i was thinking what are the possible reasons why this phenomena always happen)?

Answer 1

You have this one backwards, it's the phase velocity of matter waves that travel faster than the speed of light, while the group wave of matter waves travel slower. For light in a vacuum, both the phase and group velocities are the same, while light in a medium of a positive index of refraction behaves like matter waves. Group velocity is the speed of the "bulge" that is produced when a Gaussian spread of frequencies are added together. Even if the phase velocity of the waves of each of those frequencies were superluminal, the group velocity never exceeds the speed of light. Specifically, the group velocity v is related to enegy and momentum by the dispersion equation:

v = dE/dp

where E is energy, and p is momentum. For light in a vacuum, where E = pc, this becomes simply v = c.

Answer 2

The group and phase velocities are prosperities of any wave-packet. By wave-packet, I mean a wave that is somewhat localized in a given region in space. For example, the ripples on pond when you toss a rock in it. The ripples do not spread across the whole pond, but are localized as a set of rings propagating away from the impact sight.

A packet can only be formed by a superposition of many plane waves, called Fourier harmonics. The phase velocity is the velocity at which each harmonic propagates. The group velocity is the velocity at which the "center of mass" of packet as a whole propagates.

By "center of mass," I mean the expectation value of the coordinate. If the wave is only one-dimensional, the expectation value is
\[
= \int dx x |\psi|^2,
\]
where \psi is the wave-function of the packet.

If the wave is "non-dispersive," the phase velocity and group velocities are equal, and do not depend on the wave-length of the wave. Light in a vacuum and sound (at small amplitudes) are two examples of non-dispersive waves. A wave packet, in a non-dispersive medium, does not change its shape as it propagates.

"Dispersive" waves, on the other hand, are waves where the velocity does depend on the wavelength. The phase and group velocities are not equal when the wave is dispersive. Examples of these are de Broglie waves, and surface waves on water. When a wave is dispersive, a wave-packet will change shape as it propagates.

The group velocity can be very different from the phase velocity, only when the phase velocity has a very large derivative. In physical systems, the derivative of the phase velocity is usually only large when the medium is strongly absorbing. As a result, the wave will be damped before the effect can be measured.

In the experiments where light packets move faster than light, a second laser is used to "burn a hole" in the absorbing material so that light can propagate where it normally would be absorbed.

Answer 3

http://en.wikipedia.org/wiki/Group_velocity