# Group velocity

The group velocity of a wave is the velocity with which the variations in the shape of the wave's amplitude (known as the modulation or envelope of the wave) propagates through space. The group velocity is defined by the equation:

$\displaystyle v_g \equiv \frac{\partial \omega}{\partial k}$

where:

vg is the group velocity
ω is the wave's angular frequency
k is the wave number

The group velocity is often thought of as the velocity at which energy or information is conveyed along a wave. In most cases this is accurate, and the group velocity can be thought of as the signal velocity of the waveform. However, if the wave is travelling through an absorptive medium, this does not always hold. It is possible to design experiments where the group velocity of laser light pulses sent through specially prepared materials significantly exceeds the signal velocity. (It is also possible to stop the laser pulse.)

The function ω(k), which gives ω as a function of k, is known as the dispersion relation. If ω is directly proportional to k, then the group velocity is exactly equal to the phase velocity. Otherwise, the envelope of the wave will become distorted as it propagates. This "group velocity dispersion" is an important effect in the propagation of signals through optical fibers and in the design of short pulse lasers.

The idea of a group velocity distinct from a wave's phase velocity was first proposed by W.R. Hamilton in 1839, and the first full treatment was by Rayleigh in his "Theory of Sound" in 1877.