In thephysical sciences andelectrical engineering,dispersion relations describe the effect ofdispersion on the properties of waves in a medium. A dispersion relation relates thewavelength orwavenumber of a wave to itsfrequency. Given the dispersion relation, one can calculate the frequency-dependentphase velocity andgroup velocity of each sinusoidal component of a wave in the medium, as a function of frequency. In addition to the geometry-dependent and material-dependent dispersion relations, the overarchingKramers–Kronig relations describe the frequency-dependence ofwave propagation andattenuation.

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 asmatter waves, have a nontrivial dispersion relation, even in the absence of geometric constraints and other media.

In the presence of dispersion, a wave does not propagate with an unchanging waveform, giving rise to the distinct frequency-dependentphase velocity andgroup velocity.

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

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

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 theangular frequency${\displaystyle \omega =2\pi f}$ andwavenumber${\displaystyle k=2\pi /\lambda }$. Rewriting the relation above in these variables gives

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

where we now viewf as a function ofk. The use ofω(k) to describe the dispersion relation has become standard because both thephase velocityω/k and thegroup velocitydω/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 are the simplest case of wave propagation: no geometric constraint, no interaction with a transmitting medium.

Forelectromagnetic waves in vacuum, the angular frequency is proportional to the wavenumber:$${\displaystyle \omega =ck.}$$

This is alinear dispersion relation, in which case the waves are said to benon-dispersive.^[1] That is, the phase velocity and the group velocity are the same:$${\displaystyle v=\frac{\omega }{k}=\frac{d\omega }{dk}=c,}$$and thus both are equal to thespeed of light in vacuum, which is frequency-independent.

Forde Broglie matter waves the frequency dispersion relation is non-linear:$${\displaystyle \omega (k)\approx \frac{m_0{c}^2}{\hslash }+\frac{\hslash k^2}{2m_0}.}$$The equation says the matter wave frequency${\displaystyle \omega }$ in vacuum varies with wavenumber (${\displaystyle k=2\pi /\lambda }$) in the non-relativistic approximation. The variation has two parts: a constant part due to the de Broglie frequency of the rest mass (${\displaystyle \hslash {\omega }_0=m_0{c}^2}$) and a quadratic part due to kinetic energy.

While applications of matter waves occur at non-relativistic velocity,de Broglie appliedspecial relativity to derive his waves. Starting from the relativisticenergy–momentum relation:$${\displaystyle E^2=(p\text{c}{)}^2+{\left(m_0{\text{c}}^2\right)}^2}$$use thede Broglie relations for energy and momentum formatter waves,$${\displaystyle E=\hslash \omega ,\mathbf{p}=\hslash \mathbf{k},}$$whereω is theangular frequency andk is thewavevector with magnitude|k| =k, equal to thewave number. Divide by${\displaystyle \hslash }$ and take the square root. This gives therelativistic frequency dispersion relation:$${\displaystyle \omega (k)=\sqrt{k^2{c}^2+{\left(\frac{m_0{c}^2}{\hslash }\right)}^2}.}$$

Practical work with matter waves occurs at non-relativistic velocity. To approximate, we pull out the rest-mass dependent frequency:$${\displaystyle \omega =\frac{m_0{c}^2}{\hslash }\sqrt{1+{\left(\frac{\hslash k}{m_{0}c}\right)}^2}.}$$

Then we see that the${\displaystyle \hslash /c}$ factor is very small so for${\displaystyle k}$ not too large, we expand${\displaystyle \sqrt{1+x^2}\approx 1+x^2/2,}$ and multiply:$${\displaystyle \omega (k)\approx \frac{m_0{c}^2}{\hslash }+\frac{\hslash k^2}{2m_0}.}$$This gives the non-relativistic approximation discussed above. If we start with the non-relativisticSchrödinger equation we will end up without the first, rest mass, term.

Animation: phase and group velocity of electrons

This animation portrays the de Broglie phase and group velocities (in slow motion) of three free electrons traveling over a field 0.4ångströms in width. The momentum per unit mass (proper velocity) of the middle electron is lightspeed, so that its group velocity is 0.707c. The top electron has twice the momentum, while the bottom electron has half. Note that as the momentum increases, the phase velocity decreases down toc, whereas the group velocity increases up toc, until the wave packet and its phase maxima move together near the speed of light, whereas the wavelength continues to decrease without bound. Both transverse and longitudinal coherence widths (packet sizes) of such high energy electrons in the lab may be orders of magnitude larger than the ones shown here.

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

The name "dispersion relation" originally comes fromoptics. It is possible to make the effective speed of light dependent on wavelength by making light pass through a material which has a non-constantindex of refraction, or by using light in a non-uniform medium such as awaveguide. 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,${\displaystyle \frac{\mathrm{\partial }\omega }{\mathrm{\partial }k}}$ is known as thegroup velocity^[2] and corresponds to the speed at which the peak of the pulse propagates, a value different from thephase velocity.^[3]

The dispersion relation for deepwater waves is often written as

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

whereg 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

$${\displaystyle v_g=\frac{d\omega }{dk}=\frac{1}{2}{v}_p.}$$

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

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

whereT is the tension force in the string, andμ 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.

In the study of solids, the study of the dispersion relation of electrons is of paramount importance. The periodicity of crystals means that manylevels 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 theband structure of a material. Properties of the band structure define whether the material is aninsulator,semiconductor orconductor.

Phonons are to sound waves in a solid what photons are to light: they are the quanta that carry it. The dispersion relation ofphonons 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 theBrillouin zone are calledacoustic phonons, since they correspond to classical sound in the limit of long wavelengths. The others areoptical phonons, since they can be excited by electromagnetic radiation.

With high-energy (e.g., 200 keV, 32 fJ) electrons in atransmission electron microscope, the energy dependence of higher-orderLaue zone (HOLZ) lines in convergent beamelectron diffraction (CBED) patterns allows one, in effect, todirectly image cross-sections of a crystal's three-dimensionaldispersion surface.^[5] Thisdynamical effect has found application in the precise measurement of lattice parameters, beam energy, and more recently for the electronics industry: lattice strain.

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 byPierre-Simon Laplace in 1776.^[7]

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

• Ellipsometry
• Ultrashort pulse
• Waves in plasmas

• Ablowitz, Mark J. (2011-09-08).Nonlinear Dispersive Waves. Cambridge, UK; New York: Cambridge University Press.ISBN 978-1-107-01254-7.OCLC 714729246.

• Poster on CBED simulations to help visualize dispersion surfaces, by Andrey Chuvilin and Ute Kaiser
• Angular frequency calculator

• Equations of physics

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