group velocity
(physics) The velocity of the envelope of a group of interfering waves having slightly different frequencies and phase velocities.
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(physics) The velocity of the envelope of a group of interfering waves having slightly different frequencies and phase velocities.
The velocity of propagation of a group of waves forming a wave packet; also, the velocity of energy flow in a traveling wave or wave packet. The pure sine waves used to define phase velocity vp do not ever really exist, for they would require infinite extent. What do exist are groups of waves, wave packets, which are combined disturbances of a group of sine waves having a range of frequencies and wavelengths. Good approximations to pure sine waves exist, provided the extent of the media is very large in comparison with the wavelength of the sine wave. In nondispersive media, pure sine waves of different frequencies alltravel at the same speed vp, and any wave packet retains its shapeas it propagates. In this case, the group velocity vg is the same as vp. But if there is dispersion, the wave packet changes shape as it moves, because each different frequency which makes up the packet moves with a different phase velocity. If vp is frequency-dependent, then vg is not equal to vp. See also Phase velocity; Sine wave; Wave motion.
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) propagate through space. For example, imagine what happens if you throw a stone into the middle of a very still pond. When the stone hits the surface of the water, a circular pattern of waves appears. It soon turns into a circular ring of waves with a quiescent center. The ever expanding ring of waves is the group, within which one can discern individual wavelets of differing wavelengths traveling at different speeds. The longer waves travel faster than the group as a whole, but they die out as they approach the leading edge. The shorter waves travel slower and they die out as they emerge from the trailing boundary of the group.
The group velocity vg is defined by the equation

where:
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. Since the 1980s, various experiments have verified
that it is possible for the group velocity of
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
Anomalous dispersion happens in areas of rapid spectral variation with respect to the refractive index. Therefore, negative values of the group velocity will occur in these areas. Anomalous dispersion plays a fundamental role in achieving backward propagating and superluminal light. Anomalous dispersion can also be used to produce group and phase speeds that are in different directions [ 2 ]. Materials that exhibit large anomalous dispersion allow the group velocity of the light to exceed c and/or become negative [ 4 ].
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.
Albert Einstein first explained the wave-particle duality of light in 1905. Louis de Broglie hypothesized that any particle should also exhibit such a duality. The velocity of a particle, he concluded then (but may be questioned today, see above), should always equal the group velocity of the corresponding wave. De Broglie deduced that if the duality equations already known for light were the same for any particle, then his hypothesis would hold. This means that

where
is Dirac's constant.Using special relativity, we find that
where
Quantum mechanics has very accurately demonstrated this hypothesis, and the relation has been shown explicitly for particles as large as molecules.
| Velocities of Waves |
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| Phase velocity | Group velocity | Front velocity | Velocity of energy transfer | Signal velocity | Information velocity |
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