The statistical distribution of wave-heights is derived theoretically on the assumptions (a) that the wave spectrum contains a single narrow band of frequencies, and (b) that the wave energy is being received from a large number of different sources whose phases are random. Theoretical relations are found between the root-meansquare wave-height, the mean height of the highest one-third (or highest one-tenth) waves and the most probable height of the largest wave in a given interval of time. There is close agreement with observation.
Longuet-Higgins, M. S.. 1953. "On the statistical distribution of the heights of sea waves." Journal of Marine Research 11, (3). https://elischolar.library.yale.edu/journal_of_marine_research/774