Group velocity

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Frequency dispersion in groups of gravity waves on the surface of deep water. The red square moves with the phase velocity, and the 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.
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. Wave group.gif
Frequency dispersion in groups of gravity waves on the surface of deep water. The      red square moves with the phase velocity, and the      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.
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.
Propagation of a wave packet demonstrating a phase velocity greater than the group velocity without dispersion. Wave packet propagation (phase faster than group, nondispersive).gif
Propagation of a wave packet demonstrating a phase velocity greater than the group velocity without dispersion.
This shows a wave with the group velocity and phase velocity going in different directions. 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). Wave opposite-group-phase-velocity.gif
This shows a wave with the group velocity and phase velocity going in different directions. 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 envelope shape of the wave's amplitudes—known as the modulation or envelope of the wave—propagates through space.

Wave oscillation that travels through space and matter

In physics, mathematics, and related fields, a wave is a disturbance of a field in which a physical attribute oscillates repeatedly at each point or propagates from each point to neighboring points, or seems to move through space.

Velocity rate of change of the position of an object as a function of time, and the direction of that change

The velocity of an object is the rate of change of its position with respect to a frame of reference, and is a function of time. Velocity is equivalent to a specification of an object's speed and direction of motion. Velocity is a fundamental concept in kinematics, the branch of classical mechanics that describes the motion of bodies.

Envelope (waves) function describing the extremes of an oscillating signal

In physics and engineering, the envelope of an oscillating signal is a smooth curve outlining its extremes. The envelope thus generalizes the concept of a constant amplitude. The figure illustrates a modulated sine wave varying between an upper and a lower envelope. The envelope function may be a function of time, space, angle, or indeed of any variable.

Contents

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, within which one can discern individual wavelets of differing wavelengths traveling at different speeds. The shorter waves travel faster than the group as a whole, but their amplitudes diminish as they approach the leading edge of the group. The longer waves travel more slowly, and their amplitudes diminish as they emerge from the trailing boundary of the group.

Capillary wave Wave traveling along the phase boundary of a fluid, whose dynamics and phase velocity are dominated by the effects of surface tension

A capillary wave is a wave traveling along the phase boundary of a fluid, whose dynamics and phase velocity are dominated by the effects of surface tension.

Definition and interpretation

Definition

A wave packet.
The envelope of the wave packet. The envelope moves at the group velocity. Wave packet.svg
  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]

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 phase velocity is: vp = ω/k.

Angular frequency physical quantity

In physics, angular frequencyω is a scalar measure of rotation rate. It refers to the angular displacement per unit time or the rate of change of the phase of a sinusoidal waveform, or as the rate of change of the argument of the sine function.

Phase velocity rate at which the phase of the wave propagates in space

The phase velocity of a wave is the rate at which the phase of the wave propagates in space. This is the velocity at which the phase of any one frequency component of the wave travels. For such a component, any given phase of the wave will appear to travel at the phase velocity. The phase velocity is given in terms of the wavelength λ (lambda) and time period T as

The function ω(k), which gives ω as a function of k, is known as the dispersion relation.

Function (mathematics) Mapping that associates a single output value to each input

In mathematics, a function is a relation between sets that associates to every element of a first set exactly one element of the second set. Typical examples are functions from integers to integers or from the real numbers to real numbers.

Dispersion relation Relation of wavelength/wavenumber as a function of a waves frequency

In physical sciences and electrical engineering, dispersion relations describe the effect of dispersion in a medium on the properties of a wave traveling within that medium. A dispersion relation relates the wavelength or wavenumber of a wave to its frequency. From this relation the phase velocity and group velocity of the wave have convenient expressions which then determine the refractive index of the medium. More general than the geometry-dependent and material-dependent dispersion relations, there are the overarching Kramers–Kronig relations that describe the frequency dependence of wave propagation and attenuation.

Proportionality (mathematics) Concept in mathematics

In mathematics, two varying quantities are said to be in a relation of proportionality, if they are multiplicatively connected to a constant, that is, when either their ratio or their product yields a constant. The value of this constant is called the coefficient of proportionality or proportionality constant.

Wave packet short "burst" or "envelope" of localized wave action that travels as a unit

In physics, a wave packet is a short "burst" or "envelope" of localized wave action that travels as a unit. A wave packet can be analyzed into, or can be synthesized from, an infinite set of component sinusoidal waves of different wavenumbers, with phases and amplitudes such that they interfere constructively only over a small region of space, and destructively elsewhere. Each component wave function, and hence the wave packet, are solutions of a wave equation. Depending on the wave equation, the wave packet's profile may remain constant or it may change (dispersion) while propagating.

Wake Region of recirculating flow immediately behind or downstream of a moving or stationary solid body


In fluid dynamics, a wake may either be:

Derivation

One derivation of the formula for group velocity is as follows. [7] [8]

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,

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

where ω is implicitly a function of k.

Assume that the wave packet α is almost monochromatic, so that A(k) is sharply peaked around a central wavenumber k0.

Then, linearization gives

where

and

(see next section for discussion of this step). Then, after some algebra,

There are two factors in this expression. The first factor, , describes a perfect monochromatic wave with wavevector k0, with peaks and troughs moving at the phase velocity within the envelope of the wavepacket.

The other factor,

,

gives the envelope of the wavepacket. This envelope function depends on position and time only through the combination .

Therefore, the envelope of the wavepacket travels at velocity

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

Part of the previous derivation is the Taylor series approximation that:

If the wavepacket has a relatively large frequency spread, or if the dispersion 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 fibers and in the design of high-power, short-pulse lasers.

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. [9]

Other expressions

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

with vp = ω/k the phase velocity.

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

In three dimensions

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: [10]

One dimension:
Three dimensions:

where

means the gradient of the angular frequency ω as a function of the wave vector , and 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 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”, [11] Brillouin argued that in a dissipative 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. [12] 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. [13]

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.,

Or, equivalently, in terms of the real part of complex refractive index, n = n+iκ, one has [14]

It can be shown that this generalization of group velocity continues to be related to the apparent speed of the peak of a wavepacket.[ citation needed ] The above definition is not universal, however: alternatively one may consider the time damping of standing waves (real k, complex ω), or, allow group velocity to be a complex-valued quantity. [15] [16] 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, becomes infinite (surpassing even the speed of light in vacuum), and may easily become negative (its sign opposes Rek) inside the band of anomalous dispersion. [17] [18] [19]

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 c. The peaks of wavepackets were also seen to move faster than c.

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 vg 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. [14] [17] [18] [20] [21]

See also

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References

Notes

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  4. Lighthill (1965)
  5. Hayes (1973)
  6. G.B. Whitham (1974). Linear and Nonlinear Waves (John Wiley & Sons Inc., 1974) pp 409–410 Online scan
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  8. David K. Ferry (2001). Quantum Mechanics: An Introduction for Device Physicists and Electrical Engineers (2nd ed.). CRC Press. pp. 18–19. Bibcode:2001qmid.book.....F. ISBN   978-0-7503-0725-3.
  9. Brillouin, Léon (1960), Wave Propagation and Group Velocity, New York: Academic Press Inc., OCLC   537250
  10. Atmospheric and oceanic fluid dynamics: fundamentals and large-scale circulation, by Geoffrey K. Vallis, p239
  11. Brillouin, L. (1946). Wave Propagation in Periodic Structures. New York: McGraw Hill.
  12. Loudon, R. (1973). The Quantum Theory of Light. Oxford.
  13. Worrall, G. (2012). "On the Effect of Radiative Relaxation on the Flux of Mechanical-Wave Energy in the Solar Atmosphere". Solar Physics. 279 (1): 43–52. Bibcode:2012SoPh..279...43W. doi:10.1007/s11207-012-9982-z.
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  21. Schweinsberg, A.; Lepeshkin, N. N.; Bigelow, M.S.; Boyd, R. W.; Jarabo, S. (2005), "Observation of superluminal and slow light propagation in erbium-doped optical fiber" (PDF), Europhysics Letters, 73 (2): 218–224, Bibcode:2006EL.....73..218S, CiteSeerX   10.1.1.205.5564 , doi:10.1209/epl/i2005-10371-0

Further reading