NationMaster - Encyclopedia: 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) propagate through space. The group velocity is defined by the equation: This article is about waves in the most general scientific sense. ... The velocity of an object is its speed in a particular direction. ...

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

where:

v_g 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. 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. However, superluminal communication is not possible in this case, 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, or have negative group velocity, making the pulse appear to propagate backwards. It has been suggested that this article or section be merged into Angular velocity. ... Wavenumber in most physical sciences is a wave property inversely related to wavelength, having SI units of reciprocal meters (mâˆ’1). ... The ASCII codes for the word Wikipedia represented in binary, the numeral system most commonly used for encoding computer information. ... The signal velocity of a wave is the speed at which a pulse travels through a medium. ... This article is about waves in the most general scientific sense. ... For other uses, see Laser (disambiguation). ... A line showing the speed of light on a scale model of Earth and the Moon The speed of light in a vacuum is an important physical constant denoted by the letter c for constant or the Latin word celeritas meaning swiftness. It is the speed of all electromagnetic radiation... Superluminal communication is the term used to describe the hypothetical process by which one might send information at faster-than-light (FTL) speeds. ...

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. Partial plot of a function f. ... The relation between the energy of a system and its corresponding momentum is known as its dispersion relation. ... In mathematics, two quantities are called proportional if they vary in such a way that one of the quantities is a constant multiple of the other, or equivalently if they have a constant ratio. ... The phase velocity of a wave is the rate at which the phase of the wave propagates in space. ... Optical fibers An optical fiber (or fibre) is a glass or plastic fiber designed to guide light along its length by confining as much light as possible in a propagating form. ...

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. The phase velocity of a wave is the rate at which the phase of the wave propagates in space. ... William Rowan Hamilton Sir William Rowan Hamilton (August 4, 1805 â€“ September 2, 1865) was an Irish mathematician, physicist, and astronomer who made important contributions to the development of optics, dynamics, and algebra. ... 1839 (MDCCCXXXIX) was a common year starting on Tuesday (see link for calendar). ... John William Strutt, 3rd Baron Rayleigh (12 November 1842 â€“ 30 June 1919) was an English physicist who (with William Ramsay) discovered the element argon, an achievement that earned him the Nobel Prize for Physics in 1904. ... 1877 (MDCCCLXXVII) was a common year starting on Monday (see link for calendar). ...

1 Matter wave group
2 See also
3 References
4 External links

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 concluded, 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 Albert Einstein( ) (March 14, 1879 â€“ April 18, 1955) was a German-born theoretical physicist who is widely considered to have been one of the greatest physicists of all time. ... In physics, wave-particle duality holds that light and matter exhibit properties of both waves and of particles. ... 1905 (MCMV) was a common year starting on Sunday (link will display the full calendar). ... Louis-Victor-Pierre-Raymond, 7th duc de Broglie, generally known as Louis de Broglie (August 15, 1892â€“March 19, 1987), was a French physicist and Nobel Prize laureate. ... In physics, the de Broglie hypothesis is the statement that all matter has a wave-like nature (wave-particle duality). ...

$v_g = frac{partial omega}{partial k} = frac{partial (E/hbar)}{partial (p/hbar)} = frac{partial E}{partial p}$

where

E is the total energy of the particle,

p is its momentum,

$hbar$ is Planck's constant.

Using special relativity, we find that In classical physics, the total energy of an object is the sum of its potential energy and its kinetic energy. ... In classical mechanics, momentum (pl. ... A commemoration plaque for Max Planck on his discovery of Plancks constant, in front of Humboldt University, Berlin. ... The special theory of relativity was proposed in 1905 by Albert Einstein in his article On the Electrodynamics of Moving Bodies. Some three centuries earlier, Galileos principle of relativity had stated that all uniform motion was relative, and that there was no absolute and well-defined state of rest...

$begin{align} v_g &= frac{partial E}{partial p} = frac{partial}{partial p} left( sqrt{p^2c^2+m_0^2c^4} right) &= frac{pc^2}{sqrt{p^2c^2+m_0^2 c^4}} &= frac{gamma m_0vc^2}{sqrt{{gamma}^2m_0^2v^2c^2+m_0^2c^4}} = frac{gamma vc}{sqrt{{gamma}^2v^2+c^2}} &= frac{vc}{sqrt{v^2+ left( c / gamma right)^2}} = frac{vc}{sqrt{v^2 + left( sqrt{c^2 - v^2} right)^2}} &= v. end{align}$

where

$m 0$ is the rest mass,

c is the speed of light in a vacuum,

γ

is the Lorentz factor.

and 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 molecules. The term mass in special relativity is used in a couple of different ways, occasionally leading to a great deal of confusion. ... A line showing the speed of light on a scale model of Earth and the Moon The speed of light in a vacuum is an important physical constant denoted by the letter c for constant or the Latin word celeritas meaning swiftness. It is the speed of all electromagnetic radiation... It has been suggested that Lorentz term be merged into this article or section. ... Fig. ... In science, a molecule is the smallest particle of a pure chemical substance that still retains its chemical composition and properties. ...

References

Brillouin, Léon. Wave Propagation and Group Velocity. Academic Press Inc., New York (1960).
Tipler, Paul A. and Ralph A. Llewellyn (2003). Modern Physics. 4th ed. New York; W. H. Freeman and Company. ISBN 0-7167-4345-0. 223 p.

External links

Greg Egan has an excellent Java applet on his web site that illustrates the apparent difference in group velocity from phase velocity.
Group and Phase Velocity - Java applet with configurable group velocity and frequency.

Categories: Radio frequency propagation | Optics | Wave mechanics | Physical quantity Greg Egan (August 20, 1961, Perth, Western Australia) is an Australian computer programmer and science fiction author. ... The phase velocity of a wave is the rate at which the phase of the wave propagates in space. ...

Results from FactBites:

Group velocity - Wikipedia, the free encyclopedia (496 words)
	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.
	The group velocity is often thought of as the velocity at which energy or information is conveyed along a wave.
	This "group velocity dispersion" is an important effect in the propagation of signals through optical fibers and in the design of short pulse lasers.

Phase velocity - Wikipedia, the free encyclopedia (305 words)
	The phase velocity of a wave is the rate at which the phase of the wave propagates in space.
	Note that the phase velocity 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 exceed the speed of light in a vacuum, but this does not indicate any superluminal information or energy transfer.

More results at FactBites »

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