Group velocity


Group velocity

[
Frequency dispersion in bichromatic groups of gravity waves on the surface of deep water. The red dot moves with the phase velocity, and the green dots propagate with the group velocity. In this deep-water case, the phase velocity is twice the group velocity. The red dot overtakes two green dots, when moving from the left to the right of the figure.
New waves seem to emerge at the back of a wave group, grow in amplitude until they are at the center of the group, and vanish at the wave group front.
For gravity surface-waves, the water particle velocities are much smaller than the phase velocity, in most cases.] 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

:v_g equiv frac{partial omega}{partial k},

where::"ω" is the wave's angular frequency;:"k" is the wave number.

Note: The above definition of group velocity is only useful for wavepackets, which is a pulse that is localized in both real space and frequency space. Because waves at different frequencies propagate at differing phase velocities in dispersive media, for a large frequency range (a narrow envelope in space) the observed pulse would change shape while traveling, making group velocity an unclear or useless quantity.

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 laser light pulses sent through specially prepared materials to significantly exceed 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. However, in all these cases, photons continue to propagate at the expected speed of light in the medium. George M. Gehring, Aaron Schweinsberg, Christopher Barsi, Natalie Kostinski, Robert W. Boyd, “Observation of a Backward Pulse Propagation Through a Medium with a Negative Group Velocity”, Science. 312, 895-897 (2006).] Gunnar Dolling, Christian Enkrich, Martin Wegener, Costas M. Soukoulis, Stefan Linden, “Simultaneous Negative Phase and Group Velocity of Light in a Metamaterial”, Science. 312, 892-894 (2006).] A. Schweinsberg, N. N. Lepeshkin, M.S. Bigelow, R.W. Boyd, S. Jarabo, “Observation of superluminal and slow light propagation in erbium-doped optical fiber”, Europhysics Letters. 73, 218-224 (2005).] Matthew S Bigelow, Nick N Lepeshkin, Heedeuk Shin, Robert W Boyd, “Propagation of a smooth and discontinuous pulses through materials with very large or very small group velocities”, Journal of Physics: Condensed Matter. 18, 3117-3126 (2006)]

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 high-power, short-pulse lasers.

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. Materials that exhibit large anomalous dispersion allow the group velocity of the light to exceed c and/or become negative.

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 [Brillouin, Léon. "Wave Propagation and Group Velocity". Academic Press Inc., New York (1960)] .

Matter wave group velocity

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

: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 the reduced Planck constant.

For a free non-relativistic particle it follows that

:egin{align} v_g &= frac{partial E}{partial p} = frac{partial}{partial p} left( frac{1}{2}frac{p^2}{m} ight)\\ &= frac{p}{m}\\ &= v.end{align}

where:"m" is the mass of the particle and :and "v" its velocity.

Also in special relativity we find that

:egin{align} v_g &= frac{partial E}{partial p} = frac{partial}{partial p} left( sqrt{p^2c^2+m^2c^4} ight)\\ &= frac{pc^2}{sqrt{p^2c^2 + m^2c^4\\ &= frac{p}{msqrt{(p/(mc))^2+1\\ &= frac{p}{mgamma}\\ &= frac{mvgamma}{mgamma}\\ &= v.end{align}

where:"m" is the rest mass of the particle,:"c" is the speed of light in a vacuum,:gamma is the Lorentz factor. :and "v" is the velocity of the particle regardless of wave behavior.

As a result we can, both in relativistic and non-relativistic quantum physics, identify the group velocity of a particle's wave function with the particle velocity.

Quantum mechanics has very accurately demonstrated this hypothesis, and the relation has been shown explicitly for particles as large as molecules.Fact|date=April 2008

ee also

*Dispersion (optics) for a full discussion of wave velocities
*Phase velocity
*Slow light

References


* 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 [http://gregegan.customer.netspace.net.au/APPLETS/20/20.html his web site] that illustrates the apparent difference in group velocity from phase velocity.
* [http://publicliterature.org/tools/group_and_phase_velocity/ Group and Phase Velocity] - Java applet with configurable group velocity and frequency.
* Maarten Ambaum has a [http://www.met.rdg.ac.uk/~sws97mha/Downstream/ webpage with movie] demonstrating the importance of group velocity to downstream development of weather systems.


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