View source for Group velocity

{{Short description|Physical quantity}}
[[Image:Wave group.gif|frame|[[Dispersion (water waves)|Frequency dispersion]] in groups of [[gravity wave]]s on the surface of deep water. The {{colorbox|red}} red square moves with the [[phase velocity]], and the <span style="border-radius: 25px; background-color:#77ac30; border:1px solid;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</span> green circles propagate with the group velocity. In this deep-water case, ''the phase velocity is twice the group velocity''. The red square overtakes two green circles when moving from the left to the right of the figure.{{paragraph}}
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.{{paragraph}}
For surface gravity waves, the water particle velocities are much smaller than the phase velocity, in most cases.]]
[[File:Wave packet propagation (phase faster than group, nondispersive).gif|thumb|Propagation of a wave packet demonstrating a phase velocity greater than the group velocity without dispersion.]]
[[Image:Wave opposite-group-phase-velocity.gif|thumb|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
 | author2=Rechtsman, Mikael C
 | author3=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
 | bibcode=2012OExpr..20.8907N
 | pmid=22513601
 | doi-access=free
 }}</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).]]

The '''group velocity''' of a [[wave]] is the [[velocity]] with which the overall envelope shape of the wave's amplitudes—known as the ''modulation'' or ''[[envelope (waves)|envelope]]'' of the wave—propagates through space.

For example, if a stone is thrown into the middle of a very still pond, a circular pattern of waves with a quiescent center appears in the water, also known as a [[capillary wave]]. The expanding ring of waves is the '''wave group''' or [[wave packet]], within which one can discern individual waves that travel faster than the group as a whole. The amplitudes of the individual waves grow as they emerge from the trailing edge of the group and diminish as they approach the leading edge of the group.

== History ==
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>

== Definition and interpretation ==
[[File:Wave packet.svg|thumb|{{legend-line|dodgerblue solid|A [[wave packet]].}}{{legend-line|red dashed|The ''envelope'' of the wave packet. The envelope moves at the group velocity.}}]]
The group velocity {{math|''v''<sub>g</sub>}} is defined by the equation:<ref>{{Citation | publisher = Dover | isbn = 978-0-486-49556-9 | last = Brillouin | first = Léon | author-link = Léon Brillouin | title = Wave Propagation in Periodic Structures: Electric Filters and Crystal Lattices | year = 2003 | orig-year = 1946 | page=75 }}</ref><ref>{{Citation | publisher = Cambridge University Press | isbn = 978-0-521-01045-0 | last = Lighthill | first = James | author-link=James Lighthill | title = Waves in fluids | year = 2001 | orig-year=1978 | page=242 }}</ref><ref>{{harvtxt|Lighthill|1965}}</ref><ref>{{harvtxt|Hayes|1973}}</ref>
:<math>v_{\rm g} \ \equiv\  \frac{\partial \omega}{\partial k}\,</math>

where {{math|''ω''}} is the wave's [[angular frequency]] (usually expressed in [[radians per second]]), and {{math|''k''}} is the [[angular wavenumber]] (usually expressed in radians per meter). The [[phase velocity]] is: {{math|''v''<sub>p</sub> {{=}} ''ω''/''k''}}.

The [[function (mathematics)|function]] {{math|''ω''(''k'')}}, which gives {{math|''ω''}} as a function of {{math|''k''}}, is known as the [[dispersion relation]].

* If  {{math|''ω''}} is [[proportionality (mathematics)|directly proportional]] to  {{math|''k''}}, then the group velocity is exactly equal to the phase velocity. A wave of any shape will travel undistorted at this velocity.
* If ''ω'' is a linear function of ''k'', but not directly proportional {{math|(''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.
* If {{math|''ω''}} is not a linear function of {{math|''k''}}, the envelope of a wave packet will become distorted as it travels. Since a wave packet contains a range of different frequencies (and hence different values of {{math|''k''}}), the group velocity {{math|''∂ω/∂k''}} will be different for different values of {{math|''k''}}. Therefore, the envelope does not move at a single velocity, but its wavenumber components ({{math|''k''}}) move at different velocities, distorting the envelope. If the wavepacket has a narrow range of frequencies, and {{math|''ω''(''k'')}} is approximately linear over that narrow range, the pulse distortion will be small, in relation to the small nonlinearity. See further discussion [[#Higher-order terms in dispersion|below]]. For example, for [[Gravity wave#Deep water|deep water]] [[gravity waves]], <math display="inline">\omega = \sqrt{gk}</math>, and hence {{math|''v''<sub>g</sub> {{=}} ''v''<sub>p</sub> /2}}.{{paragraph}} This underlies the ''[[Kelvin wake pattern]]'' for the bow wave of all ships and swimming objects.  Regardless of how fast they are moving, as long as their velocity is constant, on each side the wake forms an angle of 19.47° = arcsin(1/3) with the line of travel.<ref>G.B. Whitham (1974). ''Linear and Nonlinear Waves'' (John Wiley & Sons Inc., 1974) pp 409–410 [https://archive.org/details/LinearAndNonlinearWaves Online scan]</ref>            
{{Further|Airy wave theory#Table of wave quantities}}

===Derivation===
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
 | url=https://archive.org/details/introductiontoqu00grif_200
 | url-access=limited
 | publisher=[[Prentice Hall]]
 | year=1995
 | page=[https://archive.org/details/introductiontoqu00grif_200/page/n61 48]
| isbn=9780131244054
 }}</ref><ref>
{{cite book
 | title = Quantum Mechanics: An Introduction for Device Physicists and Electrical Engineers
 | edition = 2nd
 | author = David K. Ferry
 | publisher = CRC Press
 | year = 2001
 | isbn = 978-0-7503-0725-3
 | pages = 18–19
 | url = https://books.google.com/books?id=imvYBULWPMQC&pg=PA18
| bibcode = 2001qmid.book.....F
 }}</ref>

Consider a [[wave packet]] as a function of position {{math|''x''}} and time {{math|''t'': ''α''(''x'',''t'')}}.

Let {{math|''A''(''k'')}} be its Fourier transform at time {{nowrap|{{math|''t'' {{=}} 0}}}},
:<math> \alpha(x, 0) = \int_{-\infty}^\infty dk \, A(k) e^{ikx}.</math>

By the [[superposition principle]], the wavepacket at any time  {{math|''t''}} is
:<math> \alpha(x, t) = \int_{-\infty}^\infty dk \, A(k) e^{i(kx - \omega t)},</math>
where  {{math|''ω''}} is implicitly a function of  {{math|''k''}}.

Assume that the wave packet  {{math|''α''}} is almost [[monochromatic]], so that  {{math|''A''(''k'')}} is sharply peaked around a central [[wavenumber]]  {{math|''k''<sub>0</sub>}}.

Then, [[linearization]] gives
:<math>\omega(k) \approx \omega_0 + \left(k - k_0\right)\omega'_0</math>
where 
:<math>\omega_0 = \omega(k_0)</math> and <math>\omega'_0 = \left.\frac{\partial \omega(k)}{\partial k}\right|_{k=k_0}</math> 
(see next section for discussion of this step). Then, after some algebra,
:<math> \alpha(x,t) = e^{i\left(k_0 x - \omega_0 t\right)}\int_{-\infty}^\infty dk \, A(k) e^{i(k - k_0)\left(x - \omega'_0 t\right)}.</math>

There are two factors in this expression. The first factor, <math>e^{i\left(k_0 x - \omega_0 t\right)}</math>, describes a perfect monochromatic wave with wavevector  {{math|''k''<sub>0</sub>}}, with peaks and troughs moving at the [[phase velocity]] <math>\omega_0/k_0</math>  within the envelope of the wavepacket.

The other factor, 
:<math>\int_{-\infty}^\infty dk \, A(k) e^{i(k - k_0)\left(x - \omega'_0 t\right)}</math>, 
gives the envelope of the wavepacket. This envelope function depends on position and time ''only'' through the combination <math>(x - \omega'_0 t)</math>.

Therefore, the envelope of the wavepacket travels at velocity 
:<math>\omega'_0 = \left.\frac{d\omega}{dk}\right|_{k=k_0}~,</math> 
which explains the group velocity formula.

=== Other expressions ===
For light, the refractive index {{math|''n''}}, vacuum wavelength  {{math|''λ<sub>0</sub>''}}, and wavelength in the medium  {{math|''λ''}}, are related by
:<math>\lambda_0 = \frac{2\pi c}{\omega}, \;\; \lambda = \frac{2\pi}{k} = \frac{2\pi v_{\rm p}}{\omega}, \;\; n = \frac{c}{v_{\rm p}} = \frac{\lambda_0}{\lambda},</math>
with  {{math|''v''<sub>p</sub>&nbsp;{{=}}&nbsp;''ω''/''k''}} the [[phase velocity]].

The group velocity, therefore, can be calculated by any of the following formulas,
:<math> \begin{align}
  v_{\rm g} &= \frac{c}{n + \omega \frac{\partial n}{\partial \omega}}
       = \frac{c}{n - \lambda_0 \frac{\partial n}{\partial \lambda_0}}\\
      &= v_{\rm p} \left(1 + \frac{\lambda}{n} \frac{\partial n}{\partial \lambda}\right)
       = v_{\rm p} - \lambda \frac{\partial v_{\rm p}}{\partial \lambda} = v_{\rm p} + k \frac{\partial v_{\rm p}}{\partial k}.
\end{align}</math>

==Dispersion==
{{main|Group velocity dispersion}}
[[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 {{math|''v''<sub>g</sub> {{=}} ½''v''<sub>p</sub>}}).{{paragraph}}
This shows the superposition of three wave components—with respectively 22, 25 and 29 wavelengths fitting in a [[periodic function|periodic]] horizontal domain of 2&nbsp;km length. The wave [[amplitude]]s of the components are respectively 1, 2 and 1 meter.]]
Part of the previous derivation is the [[Taylor series|Taylor series approximation]] that:
:<math>\omega(k) \approx \omega_0 + (k - k_0)\omega'_0(k_0)</math>

If the wavepacket has a relatively large frequency spread, or if the dispersion {{math|''ω(k)''}} has sharp variations (such as due to a [[resonance]]), or if the packet travels over very long distances, this assumption is not valid, and higher-order terms in the Taylor expansion become important.

As a result, the envelope of the wave packet not only moves, but also ''distorts,'' in a manner that can be described by the material's [[group velocity dispersion]]. 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. This is an important effect in the propagation of signals through [[optical fiber]]s and in the design of high-power, short-pulse lasers.

== Relation to phase velocity, refractive index and transmission speed ==
{{excerpt|Phase velocity|Group velocity}}
{{excerpt|Phase velocity|Refractive index}}

==In three dimensions==
{{See also|Plane wave}}
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>[https://books.google.com/books?id=cC0Kye7nHEEC&pg=PA239 Atmospheric and oceanic fluid dynamics: fundamentals and large-scale circulation, by Geoffrey K. Vallis, p239]</ref>
*One dimension: <math>v_{\rm p} = \omega/k, \quad v_{\rm g} = \frac{\partial \omega}{\partial k}, \,</math>
*Three dimensions: <math>(v_{\rm p})_i = \frac{\omega}{{k}_i}, \quad \mathbf{v}_{\rm g} = \vec{\nabla}_{\mathbf{k}} \, \omega \,</math>
where <math display="block">\vec{\nabla}_{\mathbf{k}} \, \omega</math> means the [[gradient]] of the [[angular frequency]] {{mvar|ω}} as a function of the wave vector <math>\mathbf{k}</math>, and <math>\hat{\mathbf{k}}</math> is the [[unit vector]] in direction '''k'''.

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.

== In lossy or gainful media ==
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 [[wave]]form. However, if the wave is travelling through an absorptive or gainful medium, this does not always hold. In these cases the group velocity may not be a well-defined quantity, or may not be a meaningful quantity.

In his text “Wave Propagation in Periodic Structures”,<ref>{{cite book |author=Brillouin, L. |title=Wave Propagation in Periodic Structures |url=https://archive.org/details/in.ernet.dli.2015.166889 |page=75 |publisher=McGraw Hill |place=New York |year=1946}}</ref> [[Léon Brillouin|Brillouin]] argued that in a lossy medium the group velocity ceases to have a clear physical meaning. An example concerning the transmission of electromagnetic waves through an atomic gas is given by Loudon.<ref>{{cite book |author=Loudon, R. |title=The Quantum Theory of Light |publisher=Oxford |year=1973}}</ref> Another example is mechanical waves in the [[solar photosphere]]: The waves are damped (by radiative heat flow from the peaks to the troughs), and related to that, the energy velocity is often substantially lower than the waves' group velocity.<ref>{{cite journal |author=Worrall, G. |journal=Solar Physics |title=On the Effect of Radiative Relaxation on the Flux of Mechanical-Wave Energy in the Solar Atmosphere |volume=279 |issue=1 |pages=43–52 |year=2012 |doi=10.1007/s11207-012-9982-z|bibcode = 2012SoPh..279...43W |s2cid=119595058 }}</ref>

Despite this ambiguity, a common way to extend the concept of group velocity to complex media is to consider spatially damped plane wave solutions inside the medium, which are characterized by a ''complex-valued'' wavevector. Then, the imaginary part of the wavevector is arbitrarily discarded and the usual formula for group velocity is applied to the real part of wavevector, i.e.,
:<math>v_{\rm g} = \left(\frac{\partial \left(\operatorname{Re} k\right)}{\partial \omega}\right)^{-1} .</math>

Or, equivalently, in terms of the real part of complex [[refractive index]], {{math|1=<u>''n''</u> = ''n'' + ''iκ''}}, one has<ref name=Boyd1170885>{{Cite journal |pmid = 19965419| year = 2009 |last1 = Boyd| first1 = R. W. |title = Controlling the velocity of light pulses |journal = Science |volume = 326| issue = 5956 |pages = 1074–7 |last2 = Gauthier |first2 = D. J. |doi = 10.1126/science.1170885 |bibcode = 2009Sci...326.1074B |url = http://www.phy.duke.edu/~qelectron/pubs/Science326_1074_2009.pdf |citeseerx = 10.1.1.630.2223 |s2cid = 2370109}}</ref>
:<math>\frac{c}{v_{\rm g}} = n + \omega \frac{\partial n}{\partial \omega} .</math>

It can be shown that this generalization of group velocity continues to be related to the apparent speed of the peak of a wavepacket.<ref>{{cite web |title=Dispersion |url=https://scholar.harvard.edu/files/david-morin/files/waves_dispersion.pdf |archive-url=https://web.archive.org/web/20120521232240/http://www.people.fas.harvard.edu/~djmorin/waves/dispersion.pdf |archive-date=2012-05-21 |url-status=live |website=people.fas.harvard.edu |first=David |last=Morin |date=2009 |access-date=2019-07-11}}</ref> The above definition is not universal, however: alternatively one may consider the time damping of standing waves (real {{mvar|k}}, complex {{mvar|ω}}), or, allow group velocity to be a complex-valued quantity.<ref>{{cite journal |doi=10.1063/1.860877 |title=Real group velocity in a medium with dissipation |journal=Physics of Fluids B: Plasma Physics |volume=5 |issue=5 |pages=1383 |year=1993 |last1=Muschietti |first1=L. |last2=Dum |first2=C. T. |bibcode = 1993PhFlB...5.1383M }}</ref><ref>{{cite journal |doi=10.1103/PhysRevE.81.056602 |pmid=20866345 |title=Complex group velocity and energy transport in absorbing media |journal=Physical Review E |volume=81 |issue=5 |pages=056602 |year=2010 |last1=Gerasik |first1=Vladimir |last2=Stastna |first2=Marek |bibcode = 2010PhRvE..81e6602G }}</ref> Different considerations yield distinct velocities, yet all definitions agree for the case of a lossless, gainless medium.

The above generalization of group velocity for complex media can behave strangely, and the example of [[anomalous dispersion]] serves as a good illustration.
At the edges of a region of anomalous dispersion, <math>v_{\rm g}</math> becomes infinite (surpassing even the [[speed of light]] in vacuum), and <math>v_{\rm g}</math> may easily become negative 
(its sign opposes Re{{mvar|k}}) inside the band of anomalous dispersion.<ref name="DEWSL06"/><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|s2cid=15100571 }}</ref>

=== Superluminal group velocities ===
Since the 1980s, various experiments have verified that it is possible for the group velocity (as defined above) of [[laser]] light pulses sent through lossy materials, or gainful materials, to significantly exceed the [[speed of light in vacuum]] {{mvar|c}}. The peaks of wavepackets were also seen to move faster than {{mvar|c}}.

In all these cases, however, there is no possibility that signals could be carried [[faster than light|faster than the speed of light in vacuum]], since the high value of {{mvar|v}}<sub>{{mvar|g}}</sub> does not help to speed up the true motion of the sharp wavefront that would occur at the start of any real signal. Essentially the seemingly superluminal transmission is an artifact of the narrow band approximation used above to define group velocity and happens because of resonance phenomena in the intervening medium. In a wide band analysis it is seen that the apparently paradoxical speed of propagation of the signal envelope is actually the result of local interference of a wider band of frequencies over many cycles, all of which propagate perfectly causally and at phase velocity. The result is akin to the fact that shadows can travel faster than light, even if the light causing them always propagates at light speed; since the phenomenon being measured is only loosely connected with causality, it does not necessarily respect the rules of causal propagation, even if it under normal circumstances does so and leads to a common intuition.<ref name=Boyd1170885/><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 |s2cid=29012046 |url=https://publikationen.bibliothek.kit.edu/1000010972/901157 }}</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 |s2cid=38556364 }}</ref><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 |s2cid=28800603 }}</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 |url=http://www.hep.princeton.edu/%7Emcdonald/examples/optics/schweinsberg_epl_73_218_06.pdf |citeseerx=10.1.1.205.5564 |s2cid=250852270 }}</ref>

==See also==
{{div col|colwidth=22em}}
*[[Wave propagation]]
*[[Dispersion (water waves)]]
*[[Dispersion (optics)]]
*[[Wave propagation speed]]
*[[Group delay]] 
*[[Group velocity dispersion]]
*[[Group delay dispersion]]
*[[Phase delay]]
*[[Phase velocity]]
*[[Signal velocity]]
*[[Slow light]]
*[[Front velocity]]
*[[Matter wave#Group velocity]]
*[[Soliton]]
{{div col end}}

==References==

===Notes===

{{Reflist|30em}}

===Further reading===

{{refbegin}}
*Crawford jr.,  Frank S. (1968).  ''Waves (Berkeley Physics Course, Vol. 3)'', McGraw-Hill,  {{ISBN|978-0070048607}} [https://archive.org/details/Waves_371 Free online version]
* {{Citation
| last1 = Tipler
| first1 = Paul A.
| first2 = Ralph A.
| last2 = Llewellyn
| year = 2003
| title = Modern Physics
| edition = 4th
| location = New York
| publisher = W. H. Freeman and Company
| isbn = 978-0-7167-4345-3
| postscript = .
| page=223
}}
* {{Citation
| doi = 10.1103/PhysRev.105.1129
| volume = 105
| issue = 4
| pages = 1129–1137
| last = Biot
| first = M. A.
| title = General theorems on the equivalence of group velocity and energy transport
| journal = Physical Review
| year = 1957
|bibcode = 1957PhRv..105.1129B }}
* {{Citation
| doi = 10.1002/cpa.3160140337
| volume = 14
| issue = 3
| pages = 675–691
| last = Whitham
| first = G. B.
| title = Group velocity and energy propagation for three-dimensional waves
| journal = Communications on Pure and Applied Mathematics
| year = 1961
| citeseerx = 10.1.1.205.7999
}}
* {{Citation
| doi = 10.1093/imamat/1.1.1
| volume = 1
| issue = 1
| pages = 1–28
| last = Lighthill
| first = M. J.
| title = Group velocity
| journal = IMA Journal of Applied Mathematics
| year = 1965
}}
* {{Citation
| doi = 10.1098/rspa.1968.0034
| volume = 302
| issue = 1471
| pages = 529–554
| last1 = Bretherton
| first1 = F. P.
| author1-link = Francis Patton Bretherton
| first2 = C. J. R.
| last2 = Garrett
| title = Wavetrains in inhomogeneous moving media
| journal = Proceedings of the Royal Society of London
| series =Series A, Mathematical and Physical Sciences
| year = 1968
|bibcode = 1968RSPSA.302..529B | s2cid = 202575349
}}
* {{Citation
| doi = 10.1098/rspa.1973.0021
| volume = 332
| issue = 1589
| pages = 199–221
| last = Hayes
| first = W. D.
| title = Group velocity and nonlinear dispersive wave propagation
| journal = Proceedings of the Royal Society of London
| series = Series A, Mathematical and Physical Sciences
| year = 1973
|bibcode = 1973RSPSA.332..199H | s2cid = 121521673
}}
* {{Citation
| publisher = Wiley
| isbn = 978-0471940906
| last = Whitham
| first = G. B.
| title = Linear and nonlinear waves
| year = 1974
}}
{{refend}}

==External links==
* [[Greg Egan]] has an excellent Java applet on [https://web.archive.org/web/20050615174625/http://gregegan.customer.netspace.net.au/APPLETS/20/20.html his web site] that illustrates the apparent difference in group velocity from [[phase velocity]].
* 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.
* [https://web.archive.org/web/20200418070125/http://www.engineeredmiracle.com/fady/GroupAndPhaseVelocities/ Phase vs. Group Velocity ] – Various Phase- and Group-velocity relations (animation)

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