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 phase velocity and group velocity of waves 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, wave velocity is no longer uniquely defined, giving rise to the distinction of phase velocity and group velocity. Dispersion Dispersion occurs when sinusoidal 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, , is a function of the wave's wavelength : The wave's speed, wavelength, and frequency, f, are related by the identity The function expresses the dispersion relation of the given medium. Dispersion relations are more commonly expressed in terms of the angular frequency and wavenumber . Rewriting the relation above in these variables gives 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 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: This is a linear dispersion relation. In this case, the phase velocity and the group velocity are the same: 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