Group velocity

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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.

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.

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.

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.

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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.

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

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 phase velocity is: vp = ω/k.

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.

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.

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.

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.

• 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. Since a wave packet contains a range of different frequencies (and hence different values of k), the group velocity ∂ω/∂k will be different for different values of k. Therefore, the envelope does not move at a single velocity, but its wavenumber components (k) move at different velocities, distorting the envelope. If the wavepacket has a narrow range of frequencies, and ω(k) 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, ${\textstyle \omega ={\sqrt {gk}}}$, and hence vg = vp/2.
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. [6]

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.

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.

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,

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

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.

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

Then, linearization gives

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

where

${\displaystyle \omega _{0}=\omega (k_{0})}$ and ${\displaystyle \omega '_{0}=\left.{\frac {\partial \omega (k)}{\partial k}}\right|_{k=k_{0}}}$

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

${\displaystyle \alpha (x,t)=e^{i\left(k_{0}x-\omega _{0}t\right)}\int _{-\infty }^{\infty }dk\,A(k)e^{i(k-k_{0})\left(x-\omega '_{0}t\right)}.}$

There are two factors in this expression. The first factor, ${\displaystyle e^{i\left(k_{0}x-\omega _{0}t\right)}}$, describes a perfect monochromatic wave with wavevector k0, with peaks and troughs moving at the phase velocity ${\displaystyle \omega _{0}/k_{0}}$ within the envelope of the wavepacket.

The other factor,

${\displaystyle \int _{-\infty }^{\infty }dk\,A(k)e^{i(k-k_{0})\left(x-\omega '_{0}t\right)}}$,

gives the envelope of the wavepacket. This envelope function depends on position and time only through the combination ${\displaystyle (x-\omega '_{0}t)}$.

Therefore, the envelope of the wavepacket travels at velocity

${\displaystyle \omega '_{0}=\left.{\frac {d\omega }{dk}}\right|_{k=k_{0}}~,}$

which explains the group velocity formula.

Higher-order terms in dispersion

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

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

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

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

with vp = ω/k the phase velocity.

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

{\displaystyle {\begin{aligned}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}}.\end{aligned}}}

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

where

${\displaystyle {\vec {\nabla }}_{\mathbf {k} }\,\omega }$

means the gradient of the angular frequency ω as a function of the wave vector ${\displaystyle \mathbf {k} }$, and ${\displaystyle {\hat {\mathbf {k} }}}$ 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.

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.,

${\displaystyle v_{g}=\left({\frac {\partial (\operatorname {Re} k)}{\partial \omega }}\right)^{-1}.}$

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

${\displaystyle {\frac {c}{v_{g}}}=n+\omega {\frac {\partial n}{\partial \omega }}.}$

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, ${\displaystyle v_{g}}$ becomes infinite (surpassing even the speed of light in vacuum), and ${\displaystyle v_{g}}$ 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]

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References

Notes

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