Most Probable Largest Wave from the Significant Wave Height

A wave buoy reports the significant wave height $H_s$, the mean of the highest third of the waves. A ship does not meet that average, though. It meets individual waves, and the biggest one in a record is taller than the significant height. This relation estimates how much taller. It gives the most probable height of the largest wave among $N$ successive waves, on the assumption that wave heights follow the Rayleigh distribution. It is a statistical consequence of that assumption, not an empirical fit.

$N$ is the number of waves in the record: the record length divided by the mean wave period, about 1,000 in three hours. The factor $\sqrt{\ln N / 2}$ grows very slowly, because the logarithm of $N$ does, so a longer record raises the expected largest wave only gradually. For about a thousand waves it comes to a little under twice the significant height, which is why a 5 m significant sea can hand a ship a 9 m wave. Note that this is the most probable largest wave, not a ceiling, and it carries all the uncertainty of the Rayleigh assumption.

Take $H_s = 5$ m over three hours of 10 s waves, which is 1,080 waves. The relation gives 9.35 m (the page shows 9.344 m). Solved for $N$, a 9.35 m largest wave in a sea with $H_s = 5$ m corresponds to about 1,080 waves.

The significant height alone does not fix the largest wave: the length of the record matters too, through $N$. The source is Longuet-Higgins, M. S. (1952), On the statistical distribution of the heights of sea waves, Journal of Marine Research, 11(3).