2024年5月15日发(作者：ktv点歌系统下载)

Group velocity

1

Group velocity

The group velocity of a wave is the velocity with which the overall

shape of the wave's 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 slower and

they die out as they emerge from the trailing boundary of the group.

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 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 surface gravity waves, the water

particle velocities are much smaller than the

phase velocity, in most cases.

Definition and interpretation

Definition

The group velocity v

g

is defined by the equation

This shows a wave with the group velocity and

phase velocity going in different directions. As

we can see, the group velocity is positive and the

phase velocity is negative.

where:

ω is the wave's angular frequency (usually expressed in radians per second);

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.

Group velocity

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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.

[1]

[2]

[3]

[4]

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.

[2]

Materials that exhibit large anomalous dispersion allow the

group velocity of the light to exceed c and/or become negative.

[4]

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.

[5]

Other expressions

For light, the refractive index n, vacuum wavelength λ

0

, and wavelength in the medium λ, are related by

The group velocity, therefore, satisfies:

Matter-wave group velocity

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

where

E is the total energy of the particle,

p is its momentum,

is the reduced Planck constant.

For a free non-relativistic particle it follows that