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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.

In physics, a **wave** is a disturbance that transfers energy through matter or space, with little or no associated mass transport. Waves consist of oscillations or vibrations of a physical medium or a field, around relatively fixed locations. From the perspective of mathematics, waves, as functions of time and space, are a class of signals.

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.

- Definition and interpretation
- Definition
- Derivation
- History
- Other expressions
- In three dimensions
- In lossy or gainful media
- Superluminal group velocities
- See also
- References
- Notes
- Further reading
- External links

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.

The group velocity *v _{g}* is defined by the equation:

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: *v _{p}* =

In physics, **angular frequency***ω* 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** was originally the idealization of how a varying quantity depends on another quantity. For example, the position of a planet is a *function* of time. Historically, the concept was elaborated with the infinitesimal calculus at the end of the 17th century, and, until the 19th century, the functions that were considered were differentiable. The concept of function was formalized at the end of the 19th century in terms of set theory, and this greatly enlarged the domains of application of the concept.

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,*ω*= √*gk*, and hence*v*=_{g}*v*/2._{p}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:

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 *k*_{0}.

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 *k*_{0}, 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.

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.

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] }

For light, the refractive index *n*, vacuum wavelength *λ _{0}*, and wavelength in the medium

with *v _{p}* =

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

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.

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] }

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 v_{g} 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] }

The **wave equation** is an important second-order linear partial differential equation for the description of waves—as they occur in classical physics—such as mechanical waves or light waves. It arises in fields like acoustics, electromagnetics, and fluid dynamics.

In the physics of wave propagation, a **plane wave** is a wave whose wavefronts are infinite parallel planes. Mathematically a plane wave takes the form

**Matter waves** are a central part of the theory of quantum mechanics, being an example of wave–particle duality. All matter can exhibit wave-like behavior. For example, a beam of electrons can be diffracted just like a beam of light or a water wave. The concept that matter behaves like a wave was proposed by Louis de Broglie in 1924. It is also referred to as the *de Broglie hypothesis*. Matter waves are referred to as *de Broglie waves*.

In physics, a **wave vector** is a vector which helps describe a wave. Like any vector, it has a magnitude and direction, both of which are important: Its magnitude is either the wavenumber or angular wavenumber of the wave, and its direction is ordinarily the direction of wave propagation.

In physics, a **free particle** is a particle that, in some sense, is not bound by an external force, or equivalently not in a region where its potential energy varies. In classical physics, this means the particle is present in a "field-free" space. In quantum mechanics, it means a region of uniform potential, usually set to zero in the region of interest since potential can be arbitrarily set to zero at any point in space.

In theoretical physics, the (one-dimensional) **nonlinear Schrödinger equation** (**NLSE**) is a nonlinear variation of the Schrödinger equation. It is a classical field equation whose principal applications are to the propagation of light in nonlinear optical fibers and planar waveguides and to Bose-Einstein condensates confined to highly anisotropic cigar-shaped traps, in the mean-field regime. Additionally, the equation appears in the studies of small-amplitude gravity waves on the surface of deep inviscid (zero-viscosity) water; the Langmuir waves in hot plasmas; the propagation of plane-diffracted wave beams in the focusing regions of the ionosphere; the propagation of Davydov's alpha-helix solitons, which are responsible for energy transport along molecular chains; and many others. More generally, the NLSE appears as one of universal equations that describe the evolution of slowly varying packets of quasi-monochromatic waves in weakly nonlinear media that have dispersion. Unlike the linear Schrödinger equation, the NLSE never describes the time evolution of a quantum state. The 1D NLSE is an example of an integrable model.

In fluid dynamics, **dispersion** of water waves generally refers to frequency dispersion, which means that waves of different wavelengths travel at different phase speeds. Water waves, in this context, are waves propagating on the water surface, with gravity and surface tension as the restoring forces. As a result, water with a free surface is generally considered to be a dispersive medium.

The **electromagnetic wave equation** is a second-order partial differential equation that describes the propagation of electromagnetic waves through a medium or in a vacuum. It is a three-dimensional form of the wave equation. The homogeneous form of the equation, written in terms of either the electric field **E** or the magnetic field **B**, takes the form:

The **theoretical and experimental justification for the Schrödinger equation** motivates the discovery of the Schrödinger equation, the equation that describes the dynamics of nonrelativistic particles. The motivation uses photons, which are relativistic particles with dynamics determined by Maxwell's equations, as an analogue for all types of particles.

The **Frank–Tamm formula** yields the amount of Cherenkov radiation emitted on a given frequency as a charged particle moves through a medium at superluminal velocity. It is named for Russian physicists Ilya Frank and Igor Tamm who developed the theory of the Cherenkov effect in 1937, for which they were awarded a Nobel Prize in Physics in 1958.

In optics, the term **soliton** is used to refer to any optical field that does not change during propagation because of a delicate balance between nonlinear and linear effects in the medium. There are two main kinds of solitons:

In fluid dynamics, a **Stokes wave** is a non-linear and periodic surface wave on an inviscid fluid layer of constant mean depth. This type of modelling has its origins in the mid 19th century when Sir George Stokes – using a perturbation series approach, now known as the **Stokes expansion** – obtained approximate solutions for non-linear wave motion.

In fluid dynamics, **wave shoaling** is the effect by which surface waves entering shallower water change in wave height. It is caused by the fact that the group velocity, which is also the wave-energy transport velocity, changes with water depth. Under stationary conditions, a decrease in transport speed must be compensated by an increase in energy density in order to maintain a constant energy flux. Shoaling waves will also exhibit a reduction in wavelength while the frequency remains constant.

In fluid dynamics, a flow with periodic variations is known as **pulsatile flow**, or as **Womersley flow**. The flow profiles was first derived by John R. Womersley (1907–1958) in his work with blood flow in arteries. The cardiovascular system of chordate animals is a very good example where pulsatile flow is found, but pulsatile flow is also observed in engines and hydraulic systems, as a result of rotating mechanisms pumping the fluid.

**Multidimensional seismic data processing** forms a major component of seismic profiling, a technique used in geophysical exploration. The technique itself has various applications, including mapping ocean floors, determining the structure of sediments, mapping subsurface currents and hydrocarbon exploration. Since geophysical data obtained in such techniques is a function of both space and time, multidimensional signal processing techniques may be better suited for processing such data.

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*Wave Propagation and Group Velocity*, New York: Academic Press Inc., OCLC 537250 - ↑ Atmospheric and oceanic fluid dynamics: fundamentals and large-scale circulation, by Geoffrey K. Vallis, p239
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- Greg Egan has an excellent Java applet on his web site that illustrates the apparent difference in group velocity from phase velocity.
- Maarten Ambaum has a webpage with movie demonstrating the importance of group velocity to downstream development of weather systems.
- Phase vs. Group Velocity – Various Phase- and Group-velocity relations (animation)

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