Shallow-Water Wave Celerity

In shallow water the whole character of a wave changes. The dispersion relation's $\tanh(kd)$ term, which was pinned at 1 in deep water, becomes simply $kd$ once $kd$ is small — $\tanh x \approx x$ to within 1% for $x < 0.17$ — and when you put that back, the wavenumber cancels out completely. What is left is $c = \sqrt{gd}$: a speed set by depth alone, with the period, the length and the height all absent.

That makes shallow-water waves non-dispersive. Every period travels at the same speed, so a shallow-water wave keeps its shape instead of spreading out into a sorted train the way a swell does. A wave group in shallow water travels at the same speed as its crests — $c_g = c$, with the deep-water factor of one half gone entirely — and the whole disturbance moves as a unit.

The numbers involved surprise people. In 4000 m of open ocean, $\sqrt{gd}$ is 198 m/s, or 713 km/h — the speed of an airliner. A tsunami has a wavelength of order 200 km, so even in the deepest water $d/L$ is about 0.02 and it is firmly a shallow-water wave; it therefore crosses the Pacific at jet speed with an amplitude of perhaps half a metre, invisible to a ship. As it reaches the shelf the depth falls, the speed falls with the square root of it, the front of the wave slows before the back, the enormous wavelength compresses, and all that energy has to go somewhere. It goes into height.

The same equation explains why waves arrive nearly square to the beach no matter which way the wind blew. If a crest approaches at an angle, the part of it in deeper water is travelling faster than the part in shallower water, so the crest pivots — exactly as a marching column wheels when the outer rank steps faster. This is refraction, and it is the mechanism behind a great deal of coastal behaviour: energy converging on headlands, where the contours bend around and focus it, and spreading in bays, where they diverge. It is why headlands erode and bays accumulate sand, and why a "sheltered" bay in the lee of a point can still receive a surprising amount of a long swell that has bent around into it.

Where linear theory starts to fail here is on the difference between the still-water depth and the depth under the crest. A real wave of finite height sits in water that is deeper under its crest than under its trough, so the crest travels faster than the trough and catches it up. The front face steepens. In an estuary that process produces a tidal bore; on a beach it produces the vertical wall of a breaking wave. Airy theory, which linearises exactly this term away, can tell you that it is coming and cannot describe it. Cnoidal theory, solitary-wave theory and the nonlinear shallow-water equations exist to take over at that point.

The practical caution is the same as everywhere else on this shard: check that you are actually in shallow water before using a shallow-water formula. The criterion is $d/L < 0.05$, which for a 10-second wave in the ocean means water shallower than about 5 m — a very small part of any coast. Most of the surf zone, most of a harbour approach and most of the ground a coastal engineer works on is intermediate depth, where neither limit is correct and the full dispersion relation is the only honest tool.