For example, imagine what happens if a stone is thrown 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 wave 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 more slowly and they die out as they emerge from the trailing boundary of the group.
Definition and interpretation 
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.
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.
Consider a wave packet as a function of position x and time t: α(x,t). Let A(k) be its Fourier transform at time t=0:
By the superposition principle, the wavepacket at any time t is:
where and . Then, after some algebra,
The factor in front of the integral has absolute value 1. Therefore,
i.e. the envelope of the wavepacket travels at velocity . This explains the group velocity formula.
Higher order terms in dispersion 
Part of the previous derivation is the assumption:
If the wavepacket has a relatively large frequency spread, or if the dispersion has sharp variations (such as due to a resonance), or if the packet travels over very long distances, this assumption is not valid. As a result, the envelope of the wave packet not only moves, but also distorts. Loosely speaking, different frequency-components of the wavepacket travel at different speeds, with the faster components moving towards the front of the wavepacket and the slower moving towards the back. Eventually, the wave packet gets stretched out.
The next-higher term in the Taylor series (related to the second derivative of ) is called group velocity dispersion.
Physical interpretation 
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.
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 velocities 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.
Other expressions 
For light, the refractive index n, vacuum wavelength λ0, and wavelength in the medium λ, are related by
with vp = ω/k the phase velocity.
The group velocity, therefore, satisfies:
In three dimensions 
For waves traveling through three dimensions, such as light waves, sound waves, and matter waves, the formulas for phase and group velocity are generalized in a straightforward way:
- One dimension:
- Three dimensions:
If the waves are propagating through an anisotropic (i.e., not rotationally symmetric) medium, for example a crystal, then the phase velocity vector and group velocity vector may point in different directions.
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
where is the mass of the particle and v its velocity.
Also in special relativity we find that
Group velocity (equal to an electron's speed) should not be confused with phase velocity (equal to the product of the electron's frequency multiplied by its wavelength).
Both in relativistic and non-relativistic quantum physics, we can 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.
See also 
- Wave propagation
- Dispersion (optics) for a full discussion of wave velocities
- Phase velocity
- Front velocity
- Group delay and phase delay
- Signal velocity
- Slow light
- Wave propagation speed
- Defining equation (physics)
- Brillouin, Léon (2003) , Wave Propagation in Periodic Structures: Electric Filters and Crystal Lattices, Dover, p. 75, ISBN 978-0-486-49556-9
- Lighthill, James (2001) , Waves in fluids, Cambridge University Press, p. 242, ISBN 978-0-521-01045-0
- Lighthill (1965)
- Hayes (1973)
- Griffiths, David J. (1995). Introduction to Quantum Mechanics. Prentice Hall. p. 48.
- David K. Ferry (2001). Quantum Mechanics: An Introduction for Device Physicists and Electrical Engineers (2nd ed.). CRC Press. pp. 18–19. ISBN 978-0-7503-0725-3.
- Gehring, George M.; Schweinsberg, Aaron; Barsi, Christopher; Kostinski, Natalie; Boyd, Robert W. (2006), "Observation of a Backward Pulse Propagation Through a Medium with a Negative Group Velocity", Science 312 (5775): 895–897, Bibcode:2006Sci...312..895G, doi:10.1126/science.1124524, PMID 16690861
- Dolling, Gunnar; Enkrich, Christian; Wegener, Martin; Soukoulis, Costas M.; Linden, Stefan (2006), "Simultaneous Negative Phase and Group Velocity of Light in a Metamaterial", Science 312 (5775): 892–894, Bibcode:2006Sci...312..892D, doi:10.1126/science.1126021, PMID 16690860
- Schweinsberg, A.; Lepeshkin, N. N.; Bigelow, M.S.; Boyd, R. W.; Jarabo, S. (2005), "Observation of superluminal and slow light propagation in erbium-doped optical fiber", Europhysics Letters 73 (2): 218–224, Bibcode:2006EL.....73..218S, doi:10.1209/epl/i2005-10371-0
- Bigelow, Matthew S.; Lepeshkin, Nick N.; Shin, Heedeuk; Boyd, Robert W. (2006), "Propagation of a smooth and discontinuous pulses through materials with very large or very small group velocities", Journal of Physics: Condensed Matter 18 (11): 3117–3126, Bibcode:2006JPCM...18.3117B, doi:10.1088/0953-8984/18/11/017
- Withayachumnankul, W.; Fischer, B. M.; Ferguson, B.; Davis, B. R.; Abbott, D. (2010), "A Systemized View of Superluminal Wave Propagation", Proceedings of the IEEE 98 (10): 1775–1786, doi:10.1109/JPROC.2010.2052910
- Brillouin, Léon (1960), Wave Propagation and Group Velocity, New York: Academic Press Inc., OCLC 537250
- Atmospheric and oceanic fluid dynamics: fundamentals and large-scale circulation, by Geoffrey K. Vallis, p239
Further reading 
- Tipler, Paul A. (2003), Modern Physics (4th ed.), New York: W. H. Freeman and Company, ISBN 0-7167-4345-0. Unknown parameter
|unused_data=ignored (help) 223 p.
- Biot, M. A. (1957), "General theorems on the equivalence of group velocity and energy transport", Physical Review 105 (4): 1129–1137, Bibcode:1957PhRv..105.1129B, doi:10.1103/PhysRev.105.1129
- Whitham, G. B. (1961), "Group velocity and energy propagation for three-dimensional waves", Communications on Pure and Applied Mathematics 14 (3): 675–691, doi:10.1002/cpa.3160140337
- Lighthill, M. J. (1965), "Group velocity", IMA Journal of Applied Mathematics 1 (1): 1–28, doi:10.1093/imamat/1.1.1
- Bretherton, F. P.; Garrett, C. J. R. (1968), "Wavetrains in inhomogeneous moving media", Proceedings of the Royal Society of London, Series A, Mathematical and Physical Sciences 302 (1471): 529–554, Bibcode:1968RSPSA.302..529B, doi:10.1098/rspa.1968.0034
- Hayes, W. D. (1973), "Group velocity and nonlinear dispersive wave propagation", Proceedings of the Royal Society of London, Series A, Mathematical and Physical Sciences 332 (1589): 199–221, Bibcode:1973RSPSA.332..199H, doi:10.1098/rspa.1973.0021
- Whitham, G. B. (1974), Linear and nonlinear waves, Wiley, ISBN 0471940909
- 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.
- Maarten Ambaum has a webpage with movie demonstrating the importance of group velocity to downstream development of weather systems.
|Velocities of waves|
|Phase velocity • Group velocity • Front velocity • Signal velocity|