Deep-Water Wave Celerity

"Celerity" is the coastal engineer's word for the speed of a wave form, and it is used rather than "velocity" to keep a firm distinction from the velocity of the water itself. The water does not travel with the wave. In a linear deep-water wave each parcel of water runs round a closed circle and returns to where it started — that is the whole content of the small-amplitude assumption, and it is why a floating gull rises, falls and comes back rather than being carried out to sea. What travels is the form and the energy, not the fluid.

The formula $c_0 = gT/2\pi$ is the deep-water wavelength divided by the period, which is all a phase speed ever is. With $g/2\pi = 1.56$, a 10-second wave runs at 15.6 m/s — about 30 knots — and a 20-second swell at 31 m/s, or 60 knots. Longer waves genuinely do travel faster, and this dispersion is the defining property of deep-water gravity waves. It is why a storm's waves arrive sorted: the longest periods first, then progressively shorter ones over the following day or two.

The single most important sentence on this page is that the group travels at half the speed of the crests, and energy travels with the group. In deep water $c_g = c_0/2$ exactly. Watch a set of swell and you can see it happen: a crest appears at the rear of the group, runs forward through it growing and then shrinking, and disappears off the front. Each crest is doing twice the speed of the envelope it belongs to, so it spends its life migrating from the back to the front and dying there.

The consequence is practical and it is where forecasts go wrong. Swell propagation across an ocean basin has to be computed with $c_g$, not $c_0$. A 15-second swell crossing 4000 km travels at 11.7 m/s in group terms, taking about 95 hours; computed from the phase speed you would predict 47 hours and be two days early. Walter Munk and his colleagues turned this into a measuring instrument in the 1960s: by watching the frequency of arriving swell rise steadily over days at a station in Alaska, they were able to work backwards along the dispersion curve and locate individual storms in the Southern Ocean, half a world away.

Period and frequency are the other standing trap. They are reciprocals, so they run in opposite directions, and everything in wave work is quoted in one or the other depending on the discipline. A 10-second wave is 0.1 Hz. Oceanographers plot spectra against frequency, which squeezes all the long, energetic, dangerous swell into the cramped left-hand end of the axis and stretches the harmless short chop across the right. Mariners and coastal engineers talk in periods, where the important waves are on the right. The two plots of the same sea state look nothing alike, and reading one as though it were the other is a genuine and common source of confusion. When someone says "the period is going up", the frequency is going down and the waves are getting longer, faster, and — usually — more serious.

Two speeds are worth keeping apart from both of these. The orbital velocity is how fast the water itself moves in its circle, roughly $\pi H/T$ at the surface, which for a 3 m 10-second wave is about 0.94 m/s — sixteen times slower than the crest. And the wave breaks precisely when the orbital velocity at the crest catches up with the celerity: the water at the top is then travelling as fast as the form, so it can no longer stay in the wave, and it falls out of the front. Every kind of breaking, in deep water or shallow, is that same statement.