| Group Velocity |
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: where: vg ω k 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. For example, it is possible to design experiments where the group velocity of Laser light pulses sent through specially prepared materials significantly exceeds the Speed Of Light in vacuum (though Superluminal Communication is not possible, since the signal velocity remains less than the speed of light). It is also possible to reduce the group velocity to zero, stopping the 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 Fiber s 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 . MATTER WAVE GROUP 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 figured, should always equal the group velocity of the corresponding wave. De Broglie figured that if the duality equations already known for light were the same for any particle, then his hypothesis would hold. This means that : where ''E'' is the Kinetic Energy of the particle, ''p'' is its Momentum , and is Dirac's Constant . Using Special Relativity , we find that : where ''m'' is the Rest Mass , ''c'' is the Speed Of Light in a vacuum, and is the Lorentz Factor . The variable ''v'' is the velocity of the particle regardless of wave behavior. Quantum Mechanics has very accurately demonstrated this hypothesis, and the relation has been shown explicitly for particles as large as certain Molecules . SEE ALSO
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