Group velocity and Phase velocity

Group velocityFrequency dispersion in 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 surface gravity waves, the water particle velocities are much smaller than thephase velocity, in most cases.This shows a wave with the group velocity andphase velocity going in different directions. Aswe can see, the group velocity is positive and thephase velocity is negative.The group velocity of a wave is the velocity with which the overallshape of the wave's amplitudes — known as the modulation orenvelope of the wave — propagates through space.For example, imagine what happens if a stone is thrown into themiddle of a very still pond. When the stone hits the surface of thewater, a circular pattern of waves appears. It soon turns into a circularring of waves with a quiescent center. The ever expanding ring ofwaves is the wave group , within which one can discern individualwavelets of differing wavelengths traveling at different speeds. Thelonger waves travel faster than the group as a whole, but they die out asthey approach the leading edge. The shorter waves travel slower andthey die out as they emerge from the trailing boundary of the group.Definition and interpretationDefinitionThe group velocity v gis defined by the equationwhere:ω is the wave's angular frequency (usually expressed in radians per second);k is the angular wavenumber (usually expressed in radians per meter).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.Physical interpretationThe 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.[1][2][3][4]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.[2] Materials that exhibit large anomalous dispersion allow the group velocity of the light to exceed c and/or become negative.[4]HistoryThe 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.[5]Other expressionsFor light, the refractive index n, vacuum wavelength λ, and wavelength in the medium λ, are related byThe group velocity, therefore, satisfies:Matter-wave group velocityAlbert 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 thatwhereE is the total energy of the particle,p is its momentum,is the reduced Planck constant.For a free non-relativistic particle it follows thatwhereis the mass of the particle andits velocity.Also in special relativity we find thatwhereis the rest mass of the particle,c is the speed of light in a vacuum,is the Lorentz factor.and v is the velocity of the particle regardless of wave behavior.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.ReferencesNotes[1]George M. Gehring, Aaron Schweinsberg, Christopher Barsi, Natalie Kostinski, Robert W. Boyd, “Observation of a Backward PulsePropagation Through a Medium with a Negative Group Velocity”, Science. 312, 895-897 (2006).[2]Gunnar Dolling, Christian Enkrich, Martin Wegener, Costas M. Soukoulis, Stefan Linden, “Simultaneous Negative Phase and Group Velocityof Light in a Metamaterial”, Science. 312, 892-894 (2006).[3] A. Schweinsberg, N. N. Lepeshkin, M.S. Bigelow, R.W. Boyd, S. Jarabo, “Observation of superluminal and slow light propagation inerbium-doped optical fiber”, Europhysics Letters. 73, 218-224 (2005).[4]Matthew S Bigelow, Nick N Lepeshkin, Heedeuk Shin, Robert W Boyd, “Propagation of a smooth and discontinuous pulses through materialswith very large or very small group velocities”, Journal of Physics: Condensed Matter. 18, 3117-3126 (2006)[5]Brillouin, Léon. Wave Propagation and Group Velocity. Academic Press Inc., New York (1960)Further reading•Tipler, Paul A. (2003), Modern Physics (4th ed ed.), New York: W. H. Freeman and Company, ISBN 0-7167-4345-0. 223 p.•Biot, M. A. (1957), "General theorems on the equivalence of group velocity and energy transport", Physical Review105 (4): 1129–1137, doi:10.1103/PhysRev.105.1129•Whitham, G. B. (1961), "Group velocity and energy propagation for three-dimensional waves", Communications on Pure and Applied Mathematics14 (3): 675–691, doi:10.1002/cpa.3160140337•Lighthill, M. J. (1965), "Group velocity", IMA Journal of Applied Mathematics1 (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,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,doi:10.1098/rspa.1973.0021•Whitham, G. B. (1974), Linear and nonlinear waves, Wiley, ISBN 0471940909External links•Greg Egan has an excellent Java applet on his web site (.au/APPLETS/ 20/20.html) that illustrates the apparent difference in group velocity from phase velocity.•Group and Phase Velocity (/tools/group_and_phase_velocity/) - Java applet with configurable group velocity and frequency.•Maarten Ambaum has a webpage with movie (/~sws97mha/Downstream/) demonstrating the importance of group velocity to downstream development of weather systems.Article Sources and Contributors5Article Sources and ContributorsGroup velocity Source: /w/index.php?oldid=419518173 Contributors: 1pezguy, 203.37.81.xxx, Aboctok, Airflow rus, Ale2006, Anaxial, Anythingyouwant, Arcturus, Avalcarce, AxelBoldt, Brandon, Brews ohare, CYD, Catslash, Charles Matthews, Cholewa, Conversion script, Crowsnest, DavidGrayson, Dicklyon, Dougher, DrBob, Dreish, Dstrozzi, Długosz, Edgerck, Evand, F G Sedgwick, Fantasi, Fresheneesz, GBMorris, Giftlite, Gnomepower, Graham87, Hao2lian, Icairns, Jalexbnbl, Jbrundell, Johner, JustinWick, Karl Palmen, KasugaHuang,L3370x, Ligulem, LorenzoB, LungZeno, Mange01, MatthewMastracci, Mgiganteus1, Mh26, Michael Hardy, Nandesuka, Noegenesis, Oblivious, Ojigiri, Oli Filth, Sbyrnes321, Scot.parker,SebastianHelm, Soccerdude 13, Ssd, Sumit Dutta, Sverdrup, Tantalate, Tarquin, Teply, The Flying Spaghetti Monster, Trelvis, VeryVerily, Voltagedrop, 百家姓之四, 56 anonymous editsImage Sources, Licenses and ContributorsImage:Wave group.gif Source: /w/index.php?title=File:Wave_group.gif License: GNU Free Documentation License Contributors: User:KraaiennestImage:StandingWavePhysics.gif Source: /w/index.php?title=File:StandingWavePhysics.gif License: Creative Commons Zero Contributors: L3370x LicenseCreative Commons Attribution-Share Alike 3.0 Unported/licenses/by-sa/3.0/Phase velocityFrequency dispersion in 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 thegroup velocity. The red dot overtakes two greendots, when moving from the left to the right ofthe figure.New waves seem to emerge at the backof a wave group, grow in amplitude until they areat the center of the group, and vanish at the wavegroup front.For surface gravity waves, the waterparticle velocities are much smaller than thephase velocity, in most cases.This shows a wave with the Group velocity andPhase velocity going in different directions. Wecan observe how the group velocity is positive,while the phase velocity is negative.The phase velocity of a wave is the rate at which the phase of the wavepropagates in space. This is the speed at which the phase of any onefrequency component of the wave travels. For such a component, anygiven phase of the wave (for example, the crest) will appear to travel atthe phase velocity. The phase velocity is given in terms of thewavelength λ (lambda) and period Tas Or, equivalently, in terms of the wave's angular frequency ω and wavenumber kbyIn a dispersive medium, the phase velocity varies with frequency and is not necessarily the same as the group velocity of the wave, which is the rate at which changes in amplitude (known as the envelope of the wave)propagate.The phase velocity of electromagnetic radiation may, under certain circumstances, (for example anomalous dispersion) exceed the speed of light in a vacuum, but this does not indicate any superluminal information or energy transfer. It was theoretically described by physicists such as Arnold Sommerfeld and Léon Brillouin. See dispersion for a full discussion of wave velocities.Matter wave phaseIn quantum mechanics, particles also behave as waves with complex phases. By the de Broglie hypothesis, we see thatUsing relativistic relations for energy and momentum, we havewhere E is the total energy of the particle (i.e. rest energy plus kinetic energy in kinematic sense), p the momentum, the Lorentz factor, c the speed of light, and β the speed as a fraction of c. The variable v can either be taken to be the speed of the particle or the group velocity of the corresponding matter wave. Since the particle speed for any particle that has mass (according to special relativity), the phase velocity of matter waves always exceeds c, i.e.and as we can see, it approaches c when the particle speed is in the relativistic range. The superluminal phase velocity does not violate special relativity, as it carries no information. See the article on signal velocity for details.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. 222-3 pp.•Léon Brillouin "Wave Propagation And Group Velocity" Academic Press Inc., New York and London (1960) ISBN 0-12-134968-3.•Main, Iain G. (1988).Vibrations and Waves in Physics. 2nd ed. New York; Cambridge University Press. ISBN 0-521-27846-5. 214-6 pp.External links•Subluminal [1], a Java applet•Simulation [2], a Java applet by Paul Falstad•Group and Phase Velocity [3] - Java applet showing the difference between group and phase velocity.References[1].au/APPLETS/20/20.html[2]/dispersion/[3]/tools/group_and_phase_velocity/Article Sources and Contributors3Article Sources and ContributorsPhase velocity Source: /w/index.php?oldid=420295047 Contributors: 207.139.221.xxx, Abdull, Aborghgr, Antikon, BD2412, Berto, BorgHunter, CYD, Conversionscript, Crowsnest, Dcamp314, Dirac66, DrBob, Długosz, Ehinson56, Endogenous -i, Fantasi, Frokor, Gareth Owen, Giftlite, Jalanpalmer, Jonas Olson, Karl Palmen, KasugaHuang, L-H, L3370x, LaurelESH, Light current, Ligulem, LorenzoB, Lum08, Mange01, Mejor Los Indios, Mfwitten, Michael Hardy, Neparis, OKwikifan, Oblivious, RMFan1, Supermood00d, Sverdrup, Tantalate, Teply, The Anome, Thrapper, Tim Starling, Trelvis, Trovatore, USPatent, Voltagedrop, Whkoh, 百家姓之四, 45 anonymous editsImage Sources, Licenses and ContributorsImage:Wave group.gif Source: /w/index.php?title=File:Wave_group.gif License: GNU Free Documentation License Contributors: User:KraaiennestImage:StandingWavePhysics.gif Source: /w/index.php?title=File:StandingWavePhysics.gif License: Creative Commons Zero Contributors: L3370xLicenseCreative Commons Attribution-Share Alike 3.0 Unported/licenses/by-sa/3.0/。