dispersionsrelation

A dispersion relation, also known as a dispersion law, is a functional relationship between frequency ω (or f) and wavenumber k (or wavevector) for waves in a given medium or system. It describes how waves of different wavelengths propagate and determines the phase velocity v_p = ω/k and the group velocity v_g = dω/dk, which govern the motion of individual wave crests and of wave packets, respectively.

In continuous media, the dispersion relation expresses how ω depends on k for that medium. Examples include electromagnetic waves in vacuum, where ω = c k giving v_p and v_g both equal to the speed of light c; electromagnetic waves in a medium, where n(ω) makes v_p = c/n(ω) and v_g = dω/dk generally different; and surface gravity waves on fluids, where ω^2 = g k for deep water and ω^2 ≈ g h k^2 in shallow water, with additional terms from surface tension.

In solids and lattices, dispersion relations relate energy and momentum. Phonons have ω(k) determined by interatomic forces, featuring acoustic branches (ω ∝ k for small k) and optical branches (finite ω at k = 0). In quantum mechanics, the energy–momentum relation translates to E = ħ ω and p = ħ k; for a free particle, ω(k) ∝ k^2.

Dispersion relations explain phenomena such as the spreading of wave packets, frequency-dependent refraction, and the existence of band gaps in periodic media like photonic and electronic crystals. They can be anisotropic, varying with direction, and are obtained by solving the governing equations under appropriate boundary conditions.