Deep-Water Wave Group Velocity (Swell Speed)

In deep water a train of waves does not move at the speed of its individual crests. The group, and the energy in it, advances at exactly half the crest speed. Watch one crest and it runs forward through the group: it appears at the back, travels to the front and vanishes there, so the group moves at half the speed of what it is made of. This is the speed that matters for swell, because it is how fast the energy of a distant storm reaches a coast or a buoy.

The formula is $c_g = \frac{g\,T}{4\pi}$, where $g$ is the acceleration due to gravity and $T$ is the wave period, the seconds between successive crests. It comes from linear wave theory for deep water (the treatment in chapter 3 of Dean and Dalrymple, 1991), so it is a theory result and not a fit, and it holds for waves small enough for the linear theory and water deep enough that the bottom plays no part. Read it as a proportion: the group velocity grows in step with the period, so a longer-period swell outruns a shorter one, and the period alone sets the speed.

A mariner's rule of thumb says the group speed in knots is about 1.5 times the period in seconds. For a 12 s swell that gives 18 kn, which is 9.26 m/s, and the page gives 9.36 m/s, about 1.1 % higher. The rule is only approximate, so a small difference is no surprise. Written with the single-crest constant $c_0 = 3.03\,P$ kn, where $P$ is the period in seconds, the group is half of that: 18.18 kn, or 9.35 m/s, which is essentially the page's answer.

Keep the speeds apart. The crest speed is twice this one, and the deep-water wavelength page gives the length of the wave whose period sets the speed. The relation holds in deep water only: as the water shoals the group velocity changes, which is why the shoaling coefficient is built from it. To see how long swell takes to cross a stated distance, the swell arrival time page puts this speed to work.