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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 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.
HistoryEdit
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.<ref>Template:Citation</ref>
Definition and interpretationEdit
The group velocity Template:Math is defined by the equation:<ref>Template:Citation</ref><ref>Template:Citation</ref><ref>Template:Harvtxt</ref><ref>Template:Harvtxt</ref>
- <math>v_{\rm g} \ \equiv\ \frac{\partial \omega}{\partial k}\,</math>
where Template:Math is the wave's angular frequency (usually expressed in radians per second), and Template:Math is the angular wavenumber (usually expressed in radians per meter). The phase velocity is: Template:Math.
The function Template:Math, which gives Template:Math as a function of Template:Math, is known as the dispersion relation.
- If Template:Math is directly proportional to Template:Math, 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 Template:Math, 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 Template:Math is not a linear function of Template:Math, 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 Template:Math), the group velocity Template:Math will be different for different values of Template:Math. Therefore, the envelope does not move at a single velocity, but its wavenumber components (Template:Math) move at different velocities, distorting the envelope. If the wavepacket has a narrow range of frequencies, and Template:Math is approximately linear over that narrow range, the pulse distortion will be small, in relation to the small nonlinearity. See further discussion below. For example, for deep water gravity waves, <math display="inline">\omega = \sqrt{gk}</math>, and hence Template:Math.Template: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 Online scan</ref>
DerivationEdit
One derivation of the formula for group velocity is as follows.<ref name=Griffiths> Template:Cite book</ref><ref> Template:Cite book</ref>
Consider a wave packet as a function of position Template:Math and time Template:Math.
Let Template:Math be its Fourier transform at time Template:Nowrap,
- <math> \alpha(x, 0) = \int_{-\infty}^\infty dk \, A(k) e^{ikx}.</math>
By the superposition principle, the wavepacket at any time Template:Math is
- <math> \alpha(x, t) = \int_{-\infty}^\infty dk \, A(k) e^{i(kx - \omega t)},</math>
where Template:Math is implicitly a function of Template:Math.
Assume that the wave packet Template:Math is almost monochromatic, so that Template:Math is sharply peaked around a central wavenumber Template:Math.
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 Template:Math, 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 expressionsEdit
For light, the refractive index Template:Math, vacuum wavelength Template:Math, and wavelength in the medium Template:Math, 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 Template:Math 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>
DispersionEdit
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Part of the previous derivation is the 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 Template:Math 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.<ref name="Jackson_1999">Template:Citation</ref>Template:Rp
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 fibers and in the design of high-power, short-pulse lasers.
Relation to phase velocity, refractive index and transmission speedEdit
Template:Excerpt Template:Excerpt
In three dimensionsEdit
Template:See also 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>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>\mathbf{v}_{\rm p} = \frac{\omega}{k} \hat{\mathbf{k}}, \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 Template:Mvar 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 mediaEdit
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 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>Template:Cite book</ref> 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>Template:Cite book</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>Template:Cite journal</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, Template:Math, one has<ref name=Boyd1170885>Template:Cite journal</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>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> The above definition is not universal, however: alternatively one may consider the time damping of standing waves (real Template:Mvar, complex Template:Mvar), or, allow group velocity to be a complex-valued quantity.<ref>Template:Cite journal</ref><ref>Template:Cite journal</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 ReTemplate:Mvar) inside the band of anomalous dispersion.<ref name="DEWSL06"/><ref name="BLSB06"/><ref>Template:Citation</ref>
Superluminal group velocitiesEdit
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 Template:Mvar. The peaks of wavepackets were also seen to move faster than Template:Mvar.
In all these cases, however, there is no possibility that signals could be carried faster than the speed of light in vacuum, since the high value of Template:MvarTemplate:Mvar 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">Template:Citation</ref><ref name="BLSB06">Template:Citation</ref><ref name="GSBKB06">Template:Citation</ref><ref name="SLBBJ05">Template:CitationTemplate:Dead link</ref>
See alsoEdit
- 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
ReferencesEdit
NotesEdit
Further readingEdit
- Crawford jr., Frank S. (1968). Waves (Berkeley Physics Course, Vol. 3), McGraw-Hill, Template:ISBN Free online version
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External linksEdit
- Greg Egan has an excellent Java applet on his web site that illustrates the apparent difference in group velocity from phase velocity.
- Maarten Ambaum has a webpage with movie Template:Webarchive demonstrating the importance of group velocity to downstream development of weather systems.
- Phase vs. Group Velocity – Various Phase- and Group-velocity relations (animation)