Dispersive

Dispersive is an adjective used to describe systems in which waves of different frequencies propagate at different speeds, causing a wave packet to spread as it travels. This behavior arises when a medium or a governing equation imposes a frequency-dependent relationship between energy, momentum, and velocity.

In physics, dispersion is commonly discussed in optics and acoustics. A medium is dispersive if its refractive index n depends on frequency, leading to a nontrivial dispersion relation ω(k). The phase velocity v_p = ω/k and the group velocity v_g = dω/dk describe the propagation of a single-frequency wave and of a wave packet, respectively. Depending on how n(ω) changes with frequency, dispersion can be categorized as normal or anomalous. Normal dispersion typically causes higher-frequency components to travel more slowly than lower-frequency ones, while anomalous dispersion has the opposite tendency. Dispersion explains why prisms separate white light into colors and why short pulses broaden as they travel through optical fibers, affecting communication bandwidth.

Dispersive effects also appear in water waves, acoustics, and other wave systems, where the speed of wave components varies with wavelength, leading to spreading or the development of complex wave patterns.

In mathematics, dispersive partial differential equations model waves whose amplitudes spread over time. Examples include the linear Schrödinger equation and the Korteweg–de Vries equation. Dispersive terms often compete with nonlinear effects, giving rise to phenomena such as wave packet spreading and solitons.