Group velocity
The speed v_g = dω/dk at which a wave packet's envelope — and therefore its energy and information — propagates.
Definition
The group velocity is the derivative of the dispersion relation ω(k) evaluated at the carrier wavenumber: v_g = dω/dk. It is the speed at which the envelope of a narrow-band wave packet moves, and therefore the speed at which any physical signal carried by the wave actually travels.
For causal signals, v_g is bounded by the speed of light. In a non-dispersive medium, v_g = v_p. In a normally dispersive medium (shorter wavelengths slower) v_g < v_p. In deep-water gravity waves v_g = v_p/2 — the crests within an ocean-swell packet appear at the trailing edge and vanish at the leading edge. For a free Schrödinger particle v_g = 2 v_p.
History
First written down by William Rowan Hamilton in 1839, generalised by Rayleigh in 1877. Stokes had observed the v_g = v_p/2 behaviour of water waves in 1847.