Phase Velocity

Phase velocity (wave celerity) c is the speed at which a single wave crest advances, c = L/T = omega/k. From the dispersion relation c = sqrt((g/k) tanh kh), so it depends on depth. In deep water (h greater than L/2) tanh kh approaches one and c = sqrt(g/k) = gT/(2 pi), about 1.56 T meters per second, longer-period waves running faster: this is why swell outruns the storm that made it. In shallow water (h less than L/20) c reduces to sqrt(gh), independent of period, so all long waves travel at the same depth-set speed. Energy travels at the group velocity, half of c in deep water, not at the phase velocity.