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The theoretical treatment of statistical properties relevant to nonlinear random waves of finite bandwidth, such as the joint distribution of wave crest and its associated wave period, is an overdue task hampered by the complicated form of the analytical model for sea surface elevation. In this study, we first derive the wave crest distribution based on the simplified version of the Longuet-Higgins’ wave model and proceed to derive the joint distribution of the wave crest and its associated period, and the conditional wave period distribution with a given wave crest, which are of great engineering value. It is shown that the bandwidth of the wave spectrum has a significant influence on the crest distribution, and the significant wave crest is getting larger in an increasing manner as nonlinearity is increased as expected. It also turns out that the positive correlation of wave crest height with its associated period is extended to more massive waves as nonlinearity is enhanced contrary to the general perception in the coastal engineering community that the wave crest is a statistically independent random process with wave period over large waves. The peak period decreases due to the destructive interference of second-order free harmonics.

Joint Distribution of the Wave Crest and Its Associated Period for Nonlinear Random Waves of Finite Bandwidth