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 freewave 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.