gruppevelocitydispersion

Group velocity, denoted v_g, is the velocity at which the envelope or overall shape of a modulated wave packet propagates through space. It is distinct from phase velocity, which is the speed of individual Fourier components. In non-dispersive media, both velocities coincide with the propagation speed of the wave.

In mathematical terms, for a wave with angular frequency ω(k) as a function of wave number k, the group velocity is v_g = dω/dk. This relates to how the spectrum of frequencies within a short pulse travels and how the pulse's envelope moves; the energy and information carried by the pulse generally follow this velocity in many cases, but details depend on the system and definitions of signal velocity.

Group velocity varies in dispersive media where the refractive index depends on frequency, such as optical fibers, water waves, plasmas, and photonic crystals. The dispersion relation in such media yields different slopes; near resonances or band edges, v_g can be reduced, increased, or become negative.

The phenomenon of superluminal or negative group velocity does not violate causality. The front velocity or signal velocity, which sets the ultimate limit for information transfer, remains bounded by the speed of light in vacuum in relativity. Anomalous dispersion can produce apparent speed changes for the pulse peak without transmitting information faster than light.

Applications include fiber-optic communication, waveform shaping, and the study of wave propagation in complex media. Understanding group velocity aids in designing systems that control pulse propagation, such as dispersion compensation and slow or fast light experiments.