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. (The group velocity is positive and the phase velocity is negative.)

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, 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. The shorter waves travel more slowly and they die out 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 v_g is defined by the equation:

$v_g \ \equiv\ \frac{\partial \omega}{\partial k}\,$

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. 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.

Derivation

One derivation of the formula for group velocity is as follows.

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:

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

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

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

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:

$\omega(k) \approx \omega_0 + (k-k_0)\omega'_0$

where $\omega_0=\omega(k_0)$ and $\omega'_0=\frac{\partial \omega(k)}{\partial k} |_{k=k_0}$. Then, after some algebra,

$\alpha(x,t)= e^{it(\omega'_0 k_0-\omega_0)}\int_{-\infty}^\infty dk \, A(k) e^{ik(x-\omega'_0 t)}.$

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

$|\alpha(x,t)| = |\alpha(x-\omega'_0 t, 0)|, \,$

i.e. the envelope of the wavepacket travels at velocity $\omega'_0=(d\omega/dk)_{k=k_0}$. 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 v_g = ½v_p). The superposition of three wave components – with respectively 22, 25 and 29 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 metre.

Part of the previous derivation is the assumption:

$\omega(k) \approx \omega_0 + (k-k_0)\omega'_0$

If the wavepacket has a relatively large frequency spread, or if the dispersion $\omega(k)$ 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 $\omega(k)$) is called group velocity dispersion.

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. However, in all these cases, photons continue to propagate at the expected speed of light in the medium.

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

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.

Other expressions

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

$\lambda_0=\frac{2\pi c}{\omega}, \;\; \lambda = \frac{2\pi}{k} = \frac{2\pi v_p}{\omega}, \;\; n=\frac{c}{v_p}=\frac{\lambda_0}{\lambda},$

with v_p = ω/k the phase velocity.

The group velocity, therefore, satisfies:

$v_g = \frac{c}{n + \omega \frac{\partial n}{\partial \omega}} = \frac{c}{n - \lambda_0 \frac{\partial n}{\partial \lambda_0}} = v_p \left( 1+\frac{\lambda}{n} \frac{\partial n}{\partial \lambda} \right) = v_p - \lambda \frac{\partial v_p}{\partial \lambda} = v_p + k \frac{\partial v_p}{\partial k}.$

In three dimensions

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:

One dimension: $v_p = \omega/k, \quad v_g = \frac{\partial \omega}{\partial k}, \,$

Three dimensions: $\mathbf{v}_p = \hat{\mathbf{k}} \frac{\omega}{|\mathbf{k}|}, \quad \mathbf{v}_g = \vec{\nabla}_{\mathbf{k}} \, \omega \,$

where $\vec{\nabla}_{\mathbf{k}} \, \omega$ means the gradient of the angular frequency $\omega$ as a function of the wave vector $\mathbf{k}$, and $\hat{\mathbf{k}}$ 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.

Matter-wave group velocity

See also: Matter wave

Albert 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 that

$v_g = \frac{\partial \omega}{\partial k} = \frac{\partial (E/\hbar)}{\partial (p/\hbar)} = \frac{\partial E}{\partial p}$

where E is the total energy of the particle, p is its momentum, ħ is the reduced Planck constant. For a free non-relativistic particle it follows that

$\begin{align} v_g &= \frac{\partial E}{\partial p} = \frac{\partial}{\partial p} \left( \frac{1}{2}\frac{p^2}{m} \right),\\ &= \frac{p}{m},\\ &= v. \end{align}$

where $m$ is the mass of the particle and v its velocity.

Also in special relativity we find that

$\begin{align} v_g &= \frac{\partial E}{\partial p} = \frac{\partial}{\partial p} \left( \sqrt{p^2c^2+m^2c^4} \right),\\ &= \frac{pc^2}{\sqrt{p^2c^2 + m^2c^4}},\\ &= \frac{p}{m\sqrt{\left(\frac{p}{mc}\right)^2+1}},\\ &= \frac{p}{m\gamma},\\ &= \frac{mv\gamma}{m\gamma},\\ &= v. \end{align}$

where m is the mass of the particle, c is the speed of light in a vacuum, $\gamma$ 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.^{[citation needed]}

See also

• Wave propagation
• Dispersion (optics) for a full discussion of wave velocities
• Phase velocity
• Front velocity
• Group delay and phase delay
• Signal velocity
• Slow light
• Wave propagation speed
• Defining equation (physics)

References

Further reading

• Tipler, Paul A. (2003), Modern Physics (4th 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 Review 105 (4): 1129–1137, Bibcode:1957PhRv..105.1129B, doi:10.1103/PhysRev.105.1129 
• Whitham, G. B. (1961), "Group velocity and energy propagation for three-dimensional waves", Communications on Pure and Applied Mathematics 14 (3): 675–691, doi:10.1002/cpa.3160140337 
• Lighthill, M. J. (1965), "Group velocity", IMA Journal of Applied Mathematics 1 (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, Bibcode:1968RSPSA.302..529B, 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, Bibcode:1973RSPSA.332..199H, doi:10.1098/rspa.1973.0021 
• Whitham, G. B. (1974), Linear and nonlinear waves, Wiley, ISBN 0471940909 

External links

• Greg Egan has an excellent Java applet on his web site that illustrates the apparent difference in group velocity from phase velocity.
• Group and Phase Velocity - Java applet with configurable group velocity and frequency.
• Maarten Ambaum has a webpage with movie demonstrating the importance of group velocity to downstream development of weather systems.