Crest-height statistics in finite water depth. Part 1: The role of the nonlinear interactions in uni-directional seas

Abstract

This paper explores the competing nonlinear processes that define the largest crest heights in uni-directional random seas. In deep water, the third-order near-resonant interactions produce a focusing of the free-wave energy and hence larger crest elevations. However, as the effective water depth reduces, theoretical considerations, based upon the assumption that the frequency spectrum is narrow-banded, suggest that this process weakens and below kpd=1.363 (kp being the wavenumber of the spectral peak frequency and d the water depth) energy defocusing occurs. This paper first explores how the near-resonant interactions affect the crest heights arising in broad-banded, non-breaking, uni-directional seas in a wide range of effective water depths. It also quantifies the role of the bound-wave interactions. The numerical calculations conclude that kpd=1.363 indeed defines the boundary between energy focusing and defocusing for realistic jonswap sea-states, irrespective of the spectral bandwidth and steepness. However, for kpd<1.363, the bound-wave contributions increase the largest crest heights, while the near-resonant interactions reduce them. The tail of the crest-height distributions is therefore defined by two competing nonlinear processes. The present results have important implications for both the interpretation of laboratory data describing crest-height distributions and the appropriateness of second-order models for practical engineering calculations.