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

Yong Jun Cho

doi:10.3390/jmse8090654

Abstract

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.

Highlights

Offshore structures have to be designed robust enough to survive harsh environmental conditions and to secure sufficient safety against fatigue failures as well
The probability distribution of wave periods has been derived as a marginal distribution from the empirical joint distribution of wave crests and periods
These Longuet-Higgins models [2,3] are still frequently referred to even though their application should be limited to narrow banded waves, and the widespread perception such that the wave crest is a statistically independent random process with wave period for more massive waves [4] can be attributed to these Longuet-Higgins models [2,3]

Summary

Introduction

Offshore structures have to be designed robust enough to survive harsh environmental conditions and to secure sufficient safety against fatigue failures as well. Izadparast and Niedzwecki [21] in their study of a probability distribution of ocean wave power based on a three-parameter Rayleigh-Stokes model found some discrepancies between numerical results and the measured data over small waves and later attributed these differences to the fact that the narrow-banded assumption is not fully satisfied [21]. Based on the studies of Tayfun [18,24], Tung et al [23] proposed a simple but accurate expression for second-order nonlinear sea surface elevation for waves of finite bandwidth With this wave model being available, this study intends to theoretically derive the probability distribution of wave crests, the joint distribution of the wave crest and its associated period. Attention is centered on deep water waves only

Nonlinear Wave Model

Narrow Band Approximation

Finite Band Approximation

Wave Crest Distribution

Joint Distribution of Wave Crest and Its Associated Period