Gruppenvelocity

Gruppenvelocity, or group velocity, is the velocity at which the envelope of a wave packet travels through a medium. It characterizes how the overall shape or modulation of a wave packet moves, rather than a single harmonic component. The group velocity is defined as the derivative of the angular frequency with respect to the wave vector: vg = dω/dk. In many contexts, such as optics and quantum mechanics, ω(k) is the dispersion relation that relates frequency to wave number.

In nondispersive media, where ω is proportional to k, the group velocity equals the phase velocity (vg = vp). In dispersive media, vg generally differs from the phase velocity vp = ω/k, and can vary with frequency. For an isotropic dielectric with refractive index n(ω) and wave number k = n(ω) ω / c, the group velocity is vg = c / [n(ω) + ω dn/dω]. More generally, in anisotropic or structured media, the group velocity is the gradient in k-space: vg = ∇k ω(k).

Group velocity is closely tied to energy transport in many linear, lossless media, where it often approximates the velocity at which energy and information propagate. In lossy or highly dispersive media, the relationship between vg and energy or information transport becomes more complex.

Applications and phenomena include slow light, fast or even negative group velocities in engineered materials, and photonic devices such as waveguides and optical fibers, where dispersion management controls signal timing. While vg can exceed or appear to exceed the speed of light in some cases, the front velocity (leading edge) still limits information transmission, preserving causality.