Dispersion relation

A dispersion relation connects a wave’s angular frequency to its wave number, specifying how its frequency depends on spatial scale in a medium or system.

Angular frequency: The rate of change of a wave’s phase, measured in radians per unit time. It is the temporal quantity paired with wave number in the relation.
Wave equation: A differential equation describing how a wave-like quantity varies across space and time. Solving a wave equation for sinusoidal modes yields a dispersion relation.
Seismology: The study of earthquakes and the propagation of seismic waves through Earth. Seismic dispersion reveals how wave speed varies with frequency and subsurface structure.
Nondispersive wave: A wave whose phase velocity does not vary with frequency or wave number. Its linear frequency–wave-number relation makes all spectral components travel at the same speed.
Damping: The loss of wave amplitude over time or distance through energy dissipation or leakage. Damping makes frequency or wave number complex and broadens ideal modes.
Wave number: A measure of spatial oscillation, equal to the phase change per unit distance. It is the spatial quantity whose dependence on frequency defines dispersion.
Plane wave: A wave whose phase varies linearly with position and time across space. Plane-wave solutions make frequency and wave number explicit.
Plasma waves: Collective oscillations that propagate through a plasma of charged particles. Their dispersion relations distinguish modes such as Langmuir and ion-acoustic waves.
Klein–Gordon equation: A relativistic wave equation for a scalar field with nonzero mass. Its massive modes have a nonlinear frequency–wave-number relation, unlike massless waves.
Nonlinear wave: A wave whose evolution depends on its amplitude because its governing dynamics are nonlinear. Nonlinear interactions can make a single fixed frequency–wave-number relation insufficient.

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