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| [[Image:Wave group.gif|frame|[[Dispersion (water waves)|Frequency dispersion]] in groups of [[gravity wave]]s 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.<br>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.<br>For surface gravity waves, the water particle velocities are much smaller than the phase velocity, in most cases.]]
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| [[Image:Wave opposite-group-phase-velocity.gif|thumb|frame|right|This shows a wave with the group velocity and phase velocity going in different directions.<ref name=nemirovsky2012negative>{{cite journal|last=Nemirovsky|first=Jonathan|coauthors=Rechtsman, Mikael C and Segev, Mordechai|title=Negative radiation pressure and negative effective refractive index via dielectric birefringence|journal=Optics Express|date=9 April 2012|volume=20|issue=8|pages=8907–8914|doi=10.1364/OE.20.008907|url=http://physics.technion.ac.il/~msegev/publications/NRP_Opt_Exp.pdf}}</ref> The group velocity is positive (i.e., the [[Envelope (waves)|envelope]] of the wave moves rightward), while the phase velocity is negative (i.e., the peaks and troughs move leftward).]]
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| The '''group velocity''' of a [[wave]] is the [[velocity]] with which the overall shape of the waves' amplitudes — known as the ''modulation'' or ''[[Envelope (waves)|envelope]]'' of the wave — propagates through space.
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| 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.
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| The shorter waves travel more slowly and they die out as they emerge from the trailing boundary of the group.
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| == Definition and interpretation ==
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| === Definition ===
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| [[File:Wave packet.svg|thumb|Solid line: A [[wave packet]]. Dashed line: The ''envelope'' of the wave packet. The envelope moves at the group velocity.]]
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| The group velocity ''v<sub>g</sub>'' is defined by the equation:<ref>{{Citation | publisher = Dover | isbn = 978-0-486-49556-9 | last = Brillouin | first = Léon | authorlink = Léon Brillouin | title = Wave Propagation in Periodic Structures: Electric Filters and Crystal Lattices | year = 2003 | origyear = 1946 | page=75 }}</ref><ref>{{Citation | publisher = Cambridge University Press | isbn = 978-0-521-01045-0 | last = Lighthill | first = James | authorlink=James Lighthill | title = Waves in fluids | year = 2001 | origyear=1978 | page=242 }}</ref><ref>{{harvtxt|Lighthill|1965}}</ref><ref>{{harvtxt|Hayes|1973}}</ref>
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| :<math>v_g \ \equiv\ \frac{\partial \omega}{\partial k}\,</math>
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| where ''ω'' is the wave's [[angular frequency]] (usually expressed in [[radians per second]]), and ''k'' is the [[angular wavenumber]] (usually expressed in radians per meter).
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| The [[function (mathematics)|function]] ''ω''(''k''), which gives ''ω'' as a function of ''k'', is known as the [[dispersion relation]].
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| * If ''ω'' is [[proportionality (mathematics)|directly proportional]] to ''k'', then the group velocity is exactly equal to the [[phase velocity]]. A wave of any shape will travel undistorted at this velocity.
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| * If ''ω'' is a linear function of ''k'', but not directly proportional (''ω''=''ak''+''b''), then the group velocity and phase velocity are different. The envelope of a [[wave packet]] (see figure on right) will travel at the group velocity, while the individual peaks and troughs within the envelope will move at the phase velocity.
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| * If ''ω'' is not a linear function of ''k'', the envelope of a wave packet will become distorted as it travels. This distortion is directly related to group velocity, as follows. Since a wave packet contains a range of different frequencies, the group velocity ∂ω/∂k is a range of different values (because ''ω'' is not a linear function of ''k''). Therefore the envelope does not move at a single velocity, but a range of different velocities, so the envelope gets distorted. See further discussion [[#Higher order terms in dispersion|below]].
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| ===Derivation===
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| One derivation of the formula for group velocity is as follows.<ref name=Griffiths>{{cite book | author=Griffiths, David J. | title=Introduction to Quantum Mechanics | publisher=[[Prentice Hall]] | year=1995 | page=48}}</ref><ref>
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| {{cite book
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| | title = Quantum Mechanics: An Introduction for Device Physicists and Electrical Engineers
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| | edition = 2nd
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| | author = David K. Ferry
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| | publisher = CRC Press
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| | year = 2001
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| | isbn = 978-0-7503-0725-3
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| | pages = 18–19
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| | url = http://books.google.com/books?id=imvYBULWPMQC&pg=PA18
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| }}</ref>
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| 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:
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| :<math> \alpha(x,0)= \int_{-\infty}^\infty dk \, A(k) e^{ikx},</math>
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| By the [[superposition principle]], the wavepacket at any time ''t'' is:
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| :<math> \alpha(x,t)= \int_{-\infty}^\infty dk \, A(k) e^{i(kx-\omega t)},</math>
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| where ω is implicitly a function of ''k''. We assume that the wave packet α is almost [[monochromatic]], so that ''A''(''k'') is nonzero only in the vicinity of a central [[wavenumber]] ''k''<sub>0</sub>. Then, [[linearization]] gives:
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| :<math>\omega(k) \approx \omega_0 + (k-k_0)\omega'_0</math>
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| where <math>\omega_0=\omega(k_0)</math> and <math>\omega'_0=\frac{\partial \omega(k)}{\partial k} |_{k=k_0}</math>. Then, after some algebra,
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| :<math> \alpha(x,t)= e^{it(\omega'_0 k_0-\omega_0)}\int_{-\infty}^\infty dk \, A(k) e^{ik(x-\omega'_0 t)}.</math>
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| The factor in front of the integral has absolute value 1. Therefore,
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| :<math> |\alpha(x,t)| = |\alpha(x-\omega'_0 t, 0)|, \,</math>
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| i.e. the envelope of the wavepacket travels at velocity <math>\omega'_0=(d\omega/dk)_{k=k_0}</math>. This explains the group velocity formula.
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| ====Higher order terms in dispersion====
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| [[File:Wave disp.gif|thumb|388px|right|Distortion of wave groups by higher-order dispersion effects, for [[surface gravity wave]]s on deep water (with ''v''<sub>g</sub> = ½''v''<sub>p</sub>). The superposition of three wave components – with respectively 22, 25 and 29 wavelengths, fitting in a [[periodic function|periodic]] horizontal domain of 2 km length – is shown. The wave [[amplitude]]s of the components are respectively 1, 2 and 1 metre.]]
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| Part of the previous derivation is the assumption:
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| :<math>\omega(k) \approx \omega_0 + (k-k_0)\omega'_0</math>
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| If the wavepacket has a relatively large frequency spread, or if the dispersion <math>\omega(k)</math> 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.
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| The next-higher term in the Taylor series (related to the second derivative of <math>\omega(k)</math>) is called [[group velocity dispersion]]. This is an important effect in the propagation of signals through [[optical fiber]]s and in the design of high-power, short-pulse lasers.
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| === Physical interpretation ===
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| The group velocity is often thought of as the velocity at which [[energy]] or [[Physical information|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 [[wave]]form. 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.<ref name=nemirovsky2012negative /> However, in all these cases, photons continue to propagate at the expected speed of light in the medium.<ref name="GSBKB06">{{Citation |first1=George M. |last1=Gehring |first2=Aaron |last2=Schweinsberg |first3=Christopher |last3=Barsi |first4=Natalie |last4=Kostinski |first5=Robert W. |last5=Boyd |title=Observation of a Backward Pulse Propagation Through a Medium with a Negative Group Velocity |journal=Science |volume=312 |pages=895–897 |year=2006 |doi=10.1126/science.1124524 |pmid=16690861 |issue=5775|bibcode = 2006Sci...312..895G }}</ref><ref name="DEWSL06">{{Citation |first1=Gunnar |last1=Dolling |first2=Christian |last2=Enkrich |first3=Martin |last3=Wegener |first4=Costas M. |last4=Soukoulis |first5=Stefan |last5=Linden |title=Simultaneous Negative Phase and Group Velocity of Light in a Metamaterial |journal=Science |volume=312 |pages=892–894 |year=2006 |doi=10.1126/science.1126021 |pmid=16690860 |issue=5775 |bibcode = 2006Sci...312..892D }}</ref><ref name="SLBBJ05">{{Citation |first1=A. |last1=Schweinsberg |first2=N. N. |last2=Lepeshkin |first3=M.S. |last3=Bigelow |first4=R. W. |last4=Boyd |first5=S. |last5=Jarabo |title=Observation of superluminal and slow light propagation in erbium-doped optical fiber |journal=Europhysics Letters |volume=73 |issue=2 |pages=218–224 |year=2005 |doi=10.1209/epl/i2005-10371-0|bibcode = 2006EL.....73..218S }}</ref><ref name="BLSB06">{{Citation |first1=Matthew S. |last1=Bigelow |first2=Nick N. |last2=Lepeshkin |first3=Heedeuk |last3=Shin |first4=Robert W. |last4=Boyd |title=Propagation of a smooth and discontinuous pulses through materials with very large or very small group velocities |journal=Journal of Physics: Condensed Matter |volume=18 |pages=3117–3126 |year=2006 |doi=10.1088/0953-8984/18/11/017 |issue=11|bibcode = 2006JPCM...18.3117B }}</ref> | |
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| [[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.<ref name="DEWSL06"/> Materials that exhibit large anomalous dispersion allow the group velocity of the light to exceed ''c'' and/or become negative.<ref name="BLSB06"/><ref>{{Citation |doi=10.1109/JPROC.2010.2052910 |volume=98 |issue=10 |pages=1775–1786 |last1=Withayachumnankul |first1=W. |first2=B. M. |last2=Fischer |first3=B. |last3=Ferguson |first4=B. R. |last4=Davis |first5=D. |last5=Abbott |title=A Systemized View of Superluminal Wave Propagation |journal=Proceedings of the IEEE | year=2010}}</ref>
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| === History ===
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| The idea of a group velocity distinct from a wave's [[phase velocity]] was first proposed by [[William Rowan Hamilton|W.R. Hamilton]] in 1839, and the first full treatment was by [[John Strutt, 3rd Baron Rayleigh|Rayleigh]] in his "Theory of Sound" in 1877.<ref>{{Citation |last=Brillouin |first=Léon |title=Wave Propagation and Group Velocity |publisher=Academic Press Inc. |location=New York |year=1960 |oclc=537250}}</ref> | |
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| === Other expressions ===
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| For light, the refractive index ''n'', vacuum wavelength λ<sub>0</sub>, and wavelength in the medium λ, are related by
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| :<math>\lambda_0=\frac{2\pi c}{\omega}, \;\; \lambda = \frac{2\pi}{k} = \frac{2\pi v_p}{\omega}, \;\; n=\frac{c}{v_p}=\frac{\lambda_0}{\lambda},</math>
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| with ''v''<sub>p</sub> = ''ω''/''k'' the phase velocity.
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| The group velocity, therefore, satisfies:
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| :<math>v_g = \frac{c}{n + \omega \frac{\partial n}{\partial \omega}} = \frac{c}{n - \lambda_0 \frac{\partial n}{\partial \lambda_0}} = v_p \left( 1+\frac{\lambda}{n} \frac{\partial n}{\partial \lambda} \right) = v_p - \lambda \frac{\partial v_p}{\partial \lambda} = v_p + k \frac{\partial v_p}{\partial k}.</math>
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| ==In three dimensions==
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| {{see also|Plane wave}}
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| 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:<ref>[http://books.google.com/books?id=cC0Kye7nHEEC&pg=PA239 Atmospheric and oceanic fluid dynamics: fundamentals and large-scale circulation, by Geoffrey K. Vallis, p239]</ref>
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| :One dimension: <math>v_p = \omega/k, \quad v_g = \frac{\partial \omega}{\partial k}, \,</math>
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| :Three dimensions: <math>\mathbf{v}_p = \hat{\mathbf{k}} \frac{\omega}{|\mathbf{k}|}, \quad \mathbf{v}_g = \vec{\nabla}_{\mathbf{k}} \, \omega \,</math>
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| where <math>\vec{\nabla}_{\mathbf{k}} \, \omega</math> means the [[gradient]] of the [[angular frequency]] <math>\omega</math> as a function of the wave vector <math>\mathbf{k}</math>, and <math>\hat{\mathbf{k}}</math> is the [[unit vector]] in direction '''k'''.
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| 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.
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| ==See also==
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| *[[Wave propagation]]
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| *[[Dispersion (optics)]] for a full discussion of wave velocities
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| *[[Phase velocity]]
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| *[[Front velocity]]
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| *[[Group delay]] -- "The group velocity of light in a medium is the inverse of the group delay per unit length."<ref>http://www.rp-photonics.com/group_delay.html</ref>
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| *[[Phase delay]]
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| *[[Signal velocity]]
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| *[[Slow light]]
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| *[[Wave propagation speed]]
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| *[[Defining equation (physics)]]
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| *[[Matter wave#Group velocity]]
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| ==References==
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| ===Notes===
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| {{reflist}}
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| ===Further reading===
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| <div class="references-small"> | |
| * {{Citation
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| | last1 = Tipler
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| | first1 = Paul A.
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| | first2 = Ralph A.
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| | year = 2003
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| | title = Modern Physics
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| | edition = 4th
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| | location = New York
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| | publisher = W. H. Freeman and Company
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| | isbn = 0-7167-4345-0
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| | postscript = .
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| | unused_data = last 2 = Llewellyn
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| }} 223 p.
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| * {{Citation
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| | doi = 10.1103/PhysRev.105.1129
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| | volume = 105
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| | issue = 4
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| | pages = 1129–1137
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| | last = Biot
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| | first = M. A.
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| | title = General theorems on the equivalence of group velocity and energy transport
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| | journal = Physical Review
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| | year = 1957
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| |bibcode = 1957PhRv..105.1129B }}
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| * {{Citation
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| | doi = 10.1002/cpa.3160140337
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| | volume = 14
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| | issue = 3
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| | pages = 675–691
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| | last = Whitham
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| | first = G. B.
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| | title = Group velocity and energy propagation for three-dimensional waves
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| | journal = Communications on Pure and Applied Mathematics
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| | year = 1961
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| }}
| |
| * {{Citation
| |
| | doi = 10.1093/imamat/1.1.1
| |
| | volume = 1
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| | issue = 1
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| | pages = 1–28
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| | last = Lighthill
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| | first = M. J.
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| | title = Group velocity
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| | journal = IMA Journal of Applied Mathematics
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| | year = 1965
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| }}
| |
| * {{Citation
| |
| | doi = 10.1098/rspa.1968.0034
| |
| | volume = 302
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| | issue = 1471
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| | pages = 529–554
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| | last1 = Bretherton
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| | first1 = F. P.
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| | first2 = C. J. R.
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| | last2 = Garrett
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| | title = Wavetrains in inhomogeneous moving media
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| | journal = Proceedings of the Royal Society of London
| |
| | series =Series A, Mathematical and Physical Sciences
| |
| | year = 1968
| |
| |bibcode = 1968RSPSA.302..529B }}
| |
| * {{Citation
| |
| | doi = 10.1098/rspa.1973.0021
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| | volume = 332
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| | issue = 1589
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| | pages = 199–221
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| | last = Hayes
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| | first = W. D.
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| | title = Group velocity and nonlinear dispersive wave propagation
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| | journal = Proceedings of the Royal Society of London
| |
| | series = Series A, Mathematical and Physical Sciences
| |
| | year = 1973
| |
| |bibcode = 1973RSPSA.332..199H }}
| |
| * {{Citation
| |
| | publisher = Wiley
| |
| | isbn = 0471940909
| |
| | last = Whitham
| |
| | first = G. B.
| |
| | title = Linear and nonlinear waves
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| | year = 1974
| |
| }}
| |
| </div> | |
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| ==External links==
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| * [[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]].
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| * [http://publicliterature.org/tools/group_and_phase_velocity/ Group and Phase Velocity] - Java applet with configurable group velocity and frequency.
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| * 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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| {{Velocities of Waves}}
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| {{DEFAULTSORT:Group Velocity}}
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| [[Category:Radio frequency propagation]]
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| [[Category:Optics]]
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| [[Category:Wave mechanics]]
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| [[Category:Physical quantities]]
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| [[fr:Vitesse d'une onde#Vitesse de groupe]]
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| [[nl:Voortplantingssnelheid#Fase- en groepssnelheid]]
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