Iribarren Number (Surf-Similarity Parameter)

Ramón Iribarren and Casto Nogales published this parameter in 1949, and it has turned out to be one of the most productive dimensionless groups in the whole of coastal engineering. It is the beach slope divided by the square root of the wave steepness: $\xi = \tan\beta / \sqrt{H/L_0}$. Battjes gave it the name "surf similarity parameter" in the 1970s and showed how much of the surf zone it organises.

What it really compares is two lengths — how fast the bottom rises against how fast the wave would need to steepen — and that comparison decides what kind of break you get. Below about $\xi = 0.5$ the wave SPILLS: the crest tumbles gently down the front face, the wave dissipates over a wide surf zone, and almost nothing reflects. Between 0.5 and 3.3 it PLUNGES: the crest curls over and lands ahead of the wave in one violent event. Between about 3.3 and 5 it COLLAPSES, and above that it SURGES — running up the slope and back down without really breaking at all, reflecting much of its energy seaward.

Plunging is the regime coastal engineering cares about most. The entire energy of the wave is released over a metre or two of travel, the impact pressures on anything caught in the plunge are the highest recorded in the field, and the vertical jet of water and entrained air scours a beach harder than any other kind of break. It is also, not coincidentally, the surfer's barrel: the same geometry that makes a wave rideable makes it dangerous to a structure.

The parameter reaches well past breaker type. Wave runup on a smooth impermeable slope scales close to linearly with $\xi$ through the plunging range — Hunt's 1959 formula is essentially $R/H = \xi$ — so it sets crest heights. The reflection coefficient rises steeply with it. And van der Meer's rock-armour stability equations split into a plunging branch and a surging branch at a critical $\xi$, which is why they come in two forms rather than one: the failure mechanism itself changes.

Two input traps are worth naming. The $L_0$ is the DEEP-water wavelength from the period, $gT^{2}/2\pi$ — not the local wavelength at the structure. Substituting a shortened shallow-water length gives a larger steepness, a smaller $\xi$, and a breaker classification that does not match what you can see from the beach. And the slope is $\tan\beta$, the tangent, so a 1:20 beach is 0.05 and a 1:2 armour slope is 0.5. Entering the "20" of "1 in 20" gives a nonsense answer that is 400 times too large, and it is an easy slip because the beach is spoken of as "one in twenty".

The most useful thing the parameter says is that breaker type is not a property of a beach. On a fixed slope, $\xi$ falls as the waves get bigger, because height is in the denominator under the root. So the same beach spills in a winter storm and surges under a small long-period summer swell — and a bank that produces a clean plunging break at one size produces a mushy spilling one at another. Beach and sea state together determine the surf zone, and neither alone says anything.