Sixteen thousand breaking waves, decided against the laboratory.

Duncan (1981) towed a hydrofoil through a tank and measured the breaking region it made: a wedge inclined at 10–14.7° whose cross-sectional area was 0.11 of the square of its length. The authors' scripts draw both numbers over their histograms, and this page decides both event by event with no tolerance: an inclination is INSIDE, ON or OUTSIDE the band by an exact comparison of the literal with 10 and 14.7; an aspect ratio is above or below 0.11 by an exact sign. One caveat belongs to the comparison itself and is the authors' as much as ours: Duncan's 0.11 relates a vertical cross-section to a length, whereas the area measured from above a real sea is the whitecap's plan view. The script draws the line on that histogram; the page decides it as drawn, and says so.

If breaking waves were geometrically similar to one another, a length made from the speed (cb²/g) would be a fixed multiple of the measured length, and the area a fixed multiple of the length squared: each ratio would be a constant up to measurement noise. Here each quantile is an order statistic of the exact ratios — the k-th smallest value, so a number that some event actually has — and the spread factors are exact ratios of two of them. The middle half of the events spans a factor of two in every ratio; the central ninety per cent, a factor of four to six. The authors' script tests the same thing by marking the tail beyond the mean plus two standard deviations, a float procedure on a distribution with no reason to be normal; the spread factors above are the same finding in a form that does not depend on a distribution.

"No predictable relationship" is a sentence about correlation, and a rank correlation is a rational number: average ranks, doubled, are integers, and Pearson's formula on integers is a ratio of two integers. The speed of a breaker ranks with its length at ρ = 0.360, with its area at 0.375, with its vertical extent at 0.410, with its duration at 0.150 and with its inclination at 0.097. Length ranks with area at 0.902. The authors' scripts fit Lb = a·cb and Lb = a·cb²/g through the cloud by RANSAC and least squares; those are float procedures whose coefficients depend on the fitter's choices, and they are not reproduced here — the rank correlation is what the cloud itself says, and it says a breaker's speed explains at most ρ² = 0.168 of the rank variation in any property measured.

Each record is one stereo-video run from the Katsiveli platform, its wind, significant wave height and peak frequency constant within it (verified: no record carries two values). 3,320 of the events come from the busiest run and 40 from the quietest; the counts above are exact and the description's "over 16000" is 16,369.