# Group velocity

Frequency 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 the phase velocity, in most cases.
This shows a wave with the group velocity and phase velocity going in different directions.[1] The group velocity is positive (i.e., the 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 shape of the waves' 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. The 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 their amplitudes diminish as they approach the leading edge. The shorter waves travel more slowly, and their amplitudes diminish as they emerge from the trailing boundary of the group.

## Definition and interpretation

### Definition

Solid line: A wave packet. Dashed line: The envelope of the wave packet. The envelope moves at the group velocity.

The group velocity vg is defined by the equation:[2][3][4][5]

${\displaystyle v_{g}\ \equiv \ {\frac {\partial \omega }{\partial k}}\,}$

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).

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. A wave of any shape will travel undistorted at this velocity.
• 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.
• 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 below.

### Derivation

One derivation of the formula for group velocity is as follows.[6][7]

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:

${\displaystyle \alpha (x,0)=\int _{-\infty }^{\infty }dk\,A(k)e^{ikx},}$

By the superposition principle, the wavepacket at any time t is:

${\displaystyle \alpha (x,t)=\int _{-\infty }^{\infty }dk\,A(k)e^{i(kx-\omega t)},}$

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 k0. Then, linearization gives:

${\displaystyle \omega (k)\approx \omega _{0}+(k-k_{0})\omega '_{0}}$
${\displaystyle \alpha (x,t)=e^{it(\omega '_{0}k_{0}-\omega _{0})}\int _{-\infty }^{\infty }dk\,A(k)e^{ik(x-\omega '_{0}t)}.}$

The factor in front of the integral has absolute value 1. Therefore,

${\displaystyle |\alpha (x,t)|=|\alpha (x-\omega '_{0}t,0)|,\,}$

i.e. the envelope of the wavepacket travels at velocity ${\displaystyle \omega '_{0}=(d\omega /dk)_{k=k_{0}}}$. This explains the group velocity formula.

#### Higher order terms in dispersion

Distortion of wave groups by higher-order dispersion effects, for surface gravity waves on deep water (with vg = ½vp). The superposition of three wave components – with respectively 22, 25 and 29 meter wavelengths, fitting in a periodic horizontal domain of 2 km length – is shown. The wave amplitudes of the components are respectively 1, 2 and 1 meter.

Part of the previous derivation is the assumption:

${\displaystyle \omega (k)\approx \omega _{0}+(k-k_{0})\omega '_{0}}$

If the wavepacket has a relatively large frequency spread, or if the dispersion ${\displaystyle \omega (k)}$ 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 ${\displaystyle \omega (k)}$) is called group velocity dispersion. This is an important effect in the propagation of signals through optical fibers and in the design of high-power, short-pulse lasers.

### 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.[1] However, in all these cases, photons continue to propagate at the expected speed of light in the medium.[8][9][10][11]

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.[9] Materials that exhibit large anomalous dispersion allow the group velocity of the light to exceed c and/or become negative.[11][12]

### History

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.[13]

### Other expressions

For light, the refractive index n, vacuum wavelength λ0, and wavelength in the medium λ, are related by

${\displaystyle \lambda _{0}={\frac {2\pi c}{\omega }},\;\;\lambda ={\frac {2\pi }{k}}={\frac {2\pi v_{p}}{\omega }},\;\;n={\frac {c}{v_{p}}}={\frac {\lambda _{0}}{\lambda }},}$

with vp = ω/k the phase velocity.

The group velocity, therefore, can be calculated by any of the following formulas:

${\displaystyle v_{g}={\frac {c}{n+\omega {\frac {\partial n}{\partial \omega }}}}={\frac {c}{n-\lambda _{0}{\frac {\partial n}{\partial \lambda _{0}}}}}=v_{p}\left(1+{\frac {\lambda }{n}}{\frac {\partial n}{\partial \lambda }}\right)=v_{p}-\lambda {\frac {\partial v_{p}}{\partial \lambda }}=v_{p}+k{\frac {\partial v_{p}}{\partial k}}.}$

## In three dimensions

{{#invoke:see also|seealso}} 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:[14]

One dimension: ${\displaystyle v_{p}=\omega /k,\quad v_{g}={\frac {\partial \omega }{\partial k}},\,}$
Three dimensions: ${\displaystyle \mathbf {v} _{p}={\hat {\mathbf {k} }}{\frac {\omega }{|\mathbf {k} |}},\quad \mathbf {v} _{g}={\vec {\nabla }}_{\mathbf {k} }\,\omega \,}$

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.

## References

### Notes

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14. Atmospheric and oceanic fluid dynamics: fundamentals and large-scale circulation, by Geoffrey K. Vallis, p239
15. http://www.rp-photonics.com/group_delay.html

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