Group velocity (waves)

Group velocity is the speed at which the energy and the envelope of a wave group propagate, equal to the derivative of angular frequency with respect to wave number, c_g = d-omega over d-k. In deep water it is exactly half the phase speed (celerity), c_g = c over 2, because the dispersion relation gives omega squared = g times k; in shallow water it tends to the celerity, so c_g approaches c. It governs how swell energy, the wave-energy flux, and the position of a ship’s Kelvin wake group line travel, and it sets the rate at which a wave train’s leading edge advances relative to individual crests.