Wave Steepness

Height divided by length. It is the simplest formula on this shard and arguably the most informative, because steepness is what separates a swell from a sea, a stable wave from a breaking one, and a manageable loading from a destructive one. A long low swell might be $H/L = 0.005$; a developed wind sea runs 0.03 to 0.06; anything above about 0.07 is whitecapping hard.

The deep-water ceiling is one seventh. Stokes worked it out in 1880 from the geometry of the limiting wave: as a wave steepens, the crest sharpens, and at $H/L \approx 0.142$ the crest angle closes to exactly 120°. At that point the water at the crest is moving as fast as the wave form itself, so it cannot stay in the wave. It falls out of the front. A steeper progressive wave does not exist — not because it is unlikely, but because there is no solution.

That limit explains a great deal about why seas look the way they do. A long wave can be tall: a 200 m swell can physically stand 28 m. A short wave cannot: a 20 m wind wave tops out near 2.8 m no matter how hard it blows. Wind seas are therefore limited by their own shortness while they are young, and the only way for them to get big is to get long — which takes fetch and duration, and is the entire subject of the fetch-limited growth curves that Sverdrup and Munk began during the Second World War and that ran through the Shore Protection Manual into the Coastal Engineering Manual. It is also why an enclosed lake in a savage wind produces a short, steep, vicious and comparatively small sea, while an Atlantic gale with 2000 km of fetch produces something entirely different in kind.

In shallower water the limit tightens. Miche's 1944 criterion gives the maximum steepness as $0.142\tanh(kd)$, and the same hyperbolic tangent that appears everywhere else in this subject appears again: the bottom limits the orbital motion, so the crest reaches its critical speed sooner. As $kd$ becomes small this collapses towards the depth-limited criterion, and the two ways a wave can break — too steep, or too shallow — turn out to be a single continuous criterion seen from two ends.

Two cautions about using the ratio. Match the height and the length to the same place: a deep-water height divided by a local shallow-water length is not the steepness of anything. And a steepness computed from a significant height is a characteristic of the sea state, not a statement about any wave in it; the steepest individual waves in that record are steeper than the figure suggests, and those are the ones that break.

Where steepness matters most practically is loading. A steep, short-period sea puts far more punishment into a vertical wall, a moored vessel or a small boat than its bare height suggests, because the wave arrives more often, with more of its energy near the surface, and with a much steeper front face. This is why mariners fear a short, steep sea on a wind-against-tide day more than a larger swell, and why the design sea state for a structure is a height AND a period, never a height alone.