Depth-Limited Breaking Wave Height

A wave running up a beach eventually reaches water so shallow that it cannot stand any more, and it breaks. The criterion is beautifully simple: the height at breaking is a fixed fraction of the water depth, $H_b = \gamma_b d_b$. McCowan derived $\gamma_b = 0.78$ in 1894 for a solitary wave on a horizontal bottom, and that number has been quoted ever since.

It should be treated as a central estimate, not a constant of nature. Measured breaker indices run from about 0.6 on a gentle dissipative beach taking steep incoming waves, to 1.2 and beyond on a steep reef face taking long swell — and higher still in some reported cases. The two controlling variables are the beach slope, where steeper gives higher $\gamma$, and the deep-water steepness, where steeper waves give lower $\gamma$. Over the range 0.6 to 1.2, the breaking height in a given depth spans a factor of two, and no formula can narrow it for you. Local measurement or a validated model can.

The consequence for design is that the wave reaching a structure is often limited by the water in front of it rather than by the storm offshore. Put a seawall in 3 m of water and, at $\gamma = 0.78$, the largest wave that can arrive is about 2.3 m no matter what the offshore sea state does — the bigger ones have already broken further out and spent themselves. This is depth limitation, and it is genuinely useful: it puts a ceiling on the design wave that is independent of the return period of the storm.

Two things ruin that reasoning if you forget them, and both raise the water level. $d_b$ is the still-water depth at the break point, which on a storm day includes the astronomical tide, the storm surge, and the wave setup — the mean water level inside the surf zone is itself raised by the breaking waves, typically by 10 to 20% of the breaking height. A 3 m nominal depth can easily be 5 m during the event that matters, which raises the depth-limited wave from 2.3 m to 3.9 m, and a structure sized for the first is in serious trouble against the second. Neglecting surge is the classic route to an under-designed seawall.

The second thing is the difference between one wave and a sea state. This criterion describes an individual wave; a real sea is a distribution of heights. The larger waves break further out, the smaller ones carry on past, and the result inside a saturated surf zone is a significant height that settles near 0.4 to 0.5 of the local depth rather than 0.78 of it. Models such as Battjes and Janssen's exist precisely to handle that difference, and quoting 0.78 for a significant height inside the breakers overstates it substantially.

Finally, the still-water depth at breaking is not where the wave lands. A plunging breaker throws its crest forward and downward, and the impact happens some distance shoreward of the break point and considerably below the crest level. Impact pressures from a plunging wave hitting a vertical face are the largest short-duration loads in coastal engineering — hundreds of kilopascals for milliseconds, with a strong dependence on trapped air — and they are the reason breaking-wave impact on vertical structures is treated as its own subject rather than as a case of hydrostatics.