Group Velocity

Group velocity Cg is the speed at which a wave group, and the energy it carries, travels, given by Cg = d(omega)/dk, the slope of the dispersion relation. For surface gravity waves Cg = n c, where c is the phase velocity and n = 1/2 [1 + 2kh/sinh(2kh)]. In deep water n = 1/2, so energy moves at half the speed of the individual crests, which seem to appear at the back of a group and vanish at the front. In shallow water n approaches 1 and group and phase speed converge at sqrt(gh). Group velocity, not phase velocity, governs how fast wave energy and swell forecasts propagate across an ocean basin.