Толқын жиілігіне байланысты толқын ұзындығы мен толқын санының қатынасы

Relation of wavelength/wavenumber as a function of a wave's frequency

In the physical sciences and electrical engineering, dispersion relations describe the effect of dispersion on the properties of waves in a medium. A dispersion relation relates the wavelength or wavenumber of a wave to its frequency. Given the dispersion relation, one can calculate the frequency dependent phase velocity and group 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 overarching Kramers–Kronig relations describe the frequency dependence of wave propagation and attenuation. 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. In the presence of dispersion, a wave does not propagate with an unchanging waveform, giving rise to the distinct frequency dependent phase velocity and group velocity.

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

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.

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 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.

With high energy (e. g., ) 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. This dynamical 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. Dispersion of waves on water was studied by Pierre Simon Laplace in 1776. 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.