# Dispersion relation

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In the 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.

Electrical engineering is a technical discipline concerned with the study, design and application of equipment, devices and systems which use electricity, electronics, and electromagnetism. It emerged as an identified activity in the latter half of the 19th century after commercialization of the electric telegraph, the telephone, and electrical power generation, distribution and use.

In physics, the wavelength is the spatial period of a periodic wave—the distance over which the wave's shape repeats. It is the distance between consecutive corresponding points of the same phase on the wave, such as two adjacent crests, troughs, or zero crossings, and is a characteristic of both traveling waves and standing waves, as well as other spatial wave patterns. The inverse of the wavelength is called the spatial frequency. Wavelength is commonly designated by the Greek letter lambda (λ). The term wavelength is also sometimes applied to modulated waves, and to the sinusoidal envelopes of modulated waves or waves formed by interference of several sinusoids.

In the physical sciences, the wavenumber is the spatial frequency of a wave, measured in cycles per unit distance or radians per unit distance. Whereas temporal frequency can be thought of as the number of waves per unit time, wavenumber is the number of waves per unit distance.

## Contents

Dispersion may be caused either by geometric boundary conditions (waveguides, shallow water) or by interaction of the waves with the transmitting medium. Elementary particles, considered as matter waves, have a nontrivial dispersion relation even in the absence of geometric constraints and other media.

A waveguide is a structure that guides waves, such as electromagnetic waves or sound, with minimal loss of energy by restricting expansion to one dimension or two. There is a similar effect in water waves constrained within a canal, or guns that have barrels which restrict hot gas expansion to maximize energy transfer to their bullets. Without the physical constraint of a waveguide, wave amplitudes decrease according to the inverse square law as they expand into three dimensional space.

In particle physics, an elementary particle or fundamental particle is a subatomic particle with no sub structure, thus not composed of other particles. Particles currently thought to be elementary include the fundamental fermions, which generally are "matter particles" and "antimatter particles", as well as the fundamental bosons, which generally are "force particles" that mediate interactions among fermions. A particle containing two or more elementary particles is a composite particle.

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 the presence of dispersion, wave velocity is no longer uniquely defined, giving rise to the distinction of phase velocity and group velocity.

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 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.

## Dispersion

Dispersion occurs when pure plane waves of different wavelengths have different propagation velocities, so that a wave packet of mixed wavelengths tends to spread out in space. The speed of a plane wave, v, is a function of the wave's wavelength ${\displaystyle \lambda }$:

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.

${\displaystyle v=v(\lambda ).\,}$

The wave's speed, wavelength, and frequency, f, are related by the identity

${\displaystyle v(\lambda )=\lambda \ f(\lambda ).\,}$

The function ${\displaystyle f(\lambda )}$ expresses the dispersion relation of the given medium. Dispersion relations are more commonly expressed in terms of the angular frequency ${\displaystyle \omega =2\pi f}$ and wavenumber ${\displaystyle k=2\pi /\lambda }$. Rewriting the relation above in these variables gives

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.

${\displaystyle \omega (k)=v(k)\ k.\,}$

where we now view f as a function of k. The use of ω(k) to describe the dispersion relation has become standard because both the phase velocity ω/k and the group velocity dω/dk have convenient representations via this function.

The plane waves being considered can be described by

${\displaystyle A(x,t)=A_{0}e^{2\pi i{\frac {x-vt}{\lambda }}}=A_{0}e^{i(kx-\omega t)},}$

where

A is the amplitude of the wave,
A0 = A(0,0),
x is a position along the wave's direction of travel, and
t is the time at which the wave is described.

## Plane waves in vacuum

Plane waves in vacuum are the simplest case of wave propagation: no geometric constraint, no interaction with a transmitting medium.

### Electromagnetic waves in a vacuum

For electromagnetic waves in vacuum, the angular frequency is proportional to the wavenumber:

${\displaystyle \omega =ck.\,}$

This is a linear dispersion relation. In this case, the phase velocity and the group velocity are the same:

${\displaystyle v={\frac {\omega }{k}}={\frac {d\omega }{dk}}=c;}$

they are given by c, the speed of light in vacuum, a frequency-independent constant.

### De Broglie dispersion relations

Total energy, momentum, and mass of particles are connected through the relativistic dispersion relation: [1]

${\displaystyle E^{2}=(mc^{2})^{2}+(pc)^{2},}$

which in the ultrarelativistic limit is

${\displaystyle E=pc}$

and in the nonrelativistic limit is

${\displaystyle E=mc^{2}+{\frac {p^{2}}{2m}},}$

where ${\displaystyle m}$ is the invariant mass. In the nonrelativistic limit, ${\displaystyle mc^{2}}$ is a constant, and ${\displaystyle p^{2}/(2m)}$ is the familiar kinetic energy expressed in terms of the momentum ${\displaystyle p=mv}$.

The transition from ultrarelativistic to nonrelativistic behaviour shows up as a slope change from p to p2 as shown in the log–log dispersion plot of E vs. p.

Elementary particles, atomic nuclei, atoms, and even molecules behave in some contexts as matter waves. According to the de Broglie relations, their kinetic energy E can be expressed as a frequency ω, and their momentum p as a wavenumber k, using the reduced Planck constant ħ:

${\displaystyle E=\hbar \omega ,\quad p=\hbar k.}$

Accordingly, angular frequency and wavenumber are connected through a dispersion relation, which in the nonrelativistic limit reads

${\displaystyle \omega ={\frac {\hbar k^{2}}{2m}}.}$

## Frequency versus wavenumber

As mentioned above, when the focus in a medium is on refraction rather than absorption—that is, on the real part of the refractive index—it is common to refer to the functional dependence of angular frequency on wavenumber as the dispersion relation. For particles, this translates to a knowledge of energy as a function of momentum.

### Waves and optics

The name "dispersion relation" originally comes from optics. It is possible to make the effective speed of light dependent on wavelength by making light pass through a material which has a non-constant index of refraction, or by using light in a non-uniform medium such as a waveguide. In this case, the waveform will spread over time, such that a narrow pulse will become an extended pulse, i.e., be dispersed. In these materials, ${\displaystyle {\frac {\partial \omega }{\partial k}}}$ is known as the group velocity [2] and corresponds to the speed at which the peak of the pulse propagates, a value different from the phase velocity. [3]

### Deep water waves

The dispersion relation for deep water waves is often written as

${\displaystyle \omega ={\sqrt {gk}},}$

where g is the acceleration due to gravity. Deep water, in this respect, is commonly denoted as the case where the water depth is larger than half the wavelength. [4] In this case the phase velocity is

${\displaystyle v_{p}={\frac {\omega }{k}}={\sqrt {\frac {g}{k}}},}$

and the group velocity is

${\displaystyle v_{g}={\frac {d\omega }{dk}}={\frac {1}{2}}v_{p}.}$

### Waves on a string

For an ideal string, the dispersion relation can be written as

${\displaystyle \omega =k{\sqrt {\frac {T}{\mu }}},}$

where T is the tension force in the string, and μ is the string's mass per unit length. As for the case of electromagnetic waves in vacuum, ideal strings are thus a non-dispersive medium, i.e. the phase and group velocities are equal and independent (to first order) of vibration frequency.

For a nonideal string, where stiffness is taken into account, the dispersion relation is written as

${\displaystyle \omega ^{2}={\frac {T}{\mu }}k^{2}+\alpha k^{4},}$

where ${\displaystyle \alpha }$ is a constant that depends on the string.

### Solid state

In the study of solids, the study of the dispersion relation of electrons is of paramount importance. The periodicity of crystals means that many levels of energy are possible for a given momentum and that some energies might not be available at any momentum. The collection of all possible energies and momenta is known as the band structure of a material. Properties of the band structure define whether the material is an insulator, semiconductor or conductor.

### Phonons

Phonons are to sound waves in a solid what photons are to light: they are the quanta that carry it. The dispersion relation of phonons is also non-trivial and important, being directly related to the acoustic and thermal properties of a material. For most systems, the phonons can be categorized into two main types: those whose bands become zero at the center of the Brillouin zone are called acoustic phonons, since they correspond to classical sound in the limit of long wavelengths. The others are optical phonons, since they can be excited by electromagnetic radiation.

### Electron optics

With high-energy (e.g., 200 keV, 32 fJ) electrons in a transmission electron microscope, the energy dependence of higher-order Laue zone (HOLZ) lines in convergent beam electron diffraction (CBED) patterns allows one, in effect, to directly image cross-sections of a crystal's three-dimensional dispersion surface. [5] This dynamical effect has found application in the precise measurement of lattice parameters, beam energy, and more recently for the electronics industry: lattice strain.

## History

Isaac Newton studied refraction in prisms but failed to recognize the material dependence of the dispersion relation, dismissing the work of another researcher whose measurement of a prism's dispersion did not match Newton's own. [6]

Dispersion of waves on water was studied by Pierre-Simon Laplace in 1776. [7]

The universality of the Kramers–Kronig relations (1926–27) became apparent with subsequent papers on the dispersion relation's connection to causality in the scattering theory of all types of waves and particles. [8]

## Related Research Articles

In physics, a phonon is a collective excitation in a periodic, elastic arrangement of atoms or molecules in condensed matter, specifically in solids and some liquids. Often designated a quasiparticle, it represents an excited state in the quantum mechanical quantization of the modes of vibrations of elastic structures of interacting particles.

A wavenumber–frequency diagram is a plot displaying the relationship between the wavenumber and the frequency of certain phenomena. Usually frequencies are placed on the vertical axis, while wavenumbers are placed on the horizontal axis.

In optics, dispersion is the phenomenon in which the phase velocity of a wave depends on its frequency.

In thermodynamics and solid state physics, the Debye model is a method developed by Peter Debye in 1912 for estimating the phonon contribution to the specific heat (heat capacity) in a solid. It treats the vibrations of the atomic lattice (heat) as phonons in a box, in contrast to the Einstein model, which treats the solid as many individual, non-interacting quantum harmonic oscillators. The Debye model correctly predicts the low temperature dependence of the heat capacity, which is proportional to – the Debye T3 law. Just like the Einstein model, it also recovers the Dulong–Petit law at high temperatures. But due to simplifying assumptions, its accuracy suffers at intermediate temperatures.

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.

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 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.

Wave propagation is any of the ways in which waves travel.

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 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, Airy wave theory gives a linearised description of the propagation of gravity waves on the surface of a homogeneous fluid layer. The theory assumes that the fluid layer has a uniform mean depth, and that the fluid flow is inviscid, incompressible and irrotational. This theory was first published, in correct form, by George Biddell Airy in the 19th century.

Rossby-gravity waves are equatorially trapped waves, meaning that they rapidly decay as their distance increases away from the equator. These waves have the same trapping scale as Kelvin waves, more commonly known as the equatorial Rossby deformation radius. They always carry energy eastward, but their 'crests' and 'troughs' may propagate westward if their periods are long enough.

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.

Brillouin spectroscopy is an empirical spectroscopy technique which allows the determination of elastic moduli of materials. The technique uses inelastic scattering of light when it encounters acoustic phonons in a crystal, a process known as Brillouin scattering, to determine phonon energies and therefore interatomic potentials of a material. The scattering occurs when an electromagnetic wave interacts with a density wave, photon-phonon scattering.

The Planck–Einstein relation is also referred to as the Einstein relation, Planck's energy–frequency relation, the Planck relation, and the Planck equation. Also the eponym Planck formula belongs on this list, but also often refers to Planck's law instead. These various eponyms are far from standard; they are used only sporadically, neither regularly nor very widely. They refer to a formula integral to quantum mechanics, which states that the energy of a photon, E, known as photon energy, is proportional to its frequency, ν:

## References

1. Taylor. Classical Mechanics. University Science Books. p. 652. ISBN   1-891389-22-X.
2. F. A. Jenkins and H. E. White (1957). Fundamentals of optics. New York: McGraw-Hill. p. 223. ISBN   0-07-032330-5.
3. R. A. Serway, C. J. Moses and C. A. Moyer (1989). Modern Physics. Philadelphia: Saunders. p. 118. ISBN   0-534-49340-8.
4. R. G. Dean and R. A. Dalrymple (1991). Water wave mechanics for engineers and scientists. Advanced Series on Ocean Engineering. 2. World Scientific, Singapore. ISBN   978-981-02-0420-4. See page 64–66.
5. P. M. Jones, G. M. Rackham and J. W. Steeds (1977). "Higher order Laue zone effects in electron diffraction and their use in lattice parameter determination". Proceedings of the Royal Society. A 354: 197.
6. Westfall, Richard S. (1983). (illustrated, revised ed.). Cambridge University. p. 276. ISBN   9780521274357.
7. A. D. D. Craik (2004). "The origins of water wave theory". Annual Review of Fluid Mechanics. 36: 1–28. Bibcode:2004AnRFM..36....1C. doi:10.1146/annurev.fluid.36.050802.122118.
8. John S. Toll (1956). "Causality and the dispersion relation: Logical foundations". Phys. Rev. 104 (6): 1760–1770. Bibcode:1956PhRv..104.1760T. doi:10.1103/PhysRev.104.1760.