Distributions of Envelope and Phase in Wind Waves

A theoretical expression derived previously for describing the joint distribution of the envelope and phase of second-order nonlinear waves is verified with wind wave measurements gathered in the North Sea. The same distribution is explored further to obtain the marginal and conditional distributions of wave envelopes and phases. The nature and implications of these are examined, with emphasis on the occurrence of large waves and associated phases. It is shown that the wave phase distribution assumes two distinct forms depending on whether envelope heights exceed the significant envelope height. For envelope heights sufficiently larger than this threshold, the wave phase distribution approaches a simple limit form, indicating that large surface displacements can occur only above the mean sea level. Comparisons with the North Sea data confirm these theoretical results and indicate that large surface displacements and thus large waves result from the random superposition of elementary spectral components enhanced by second-order nonlinear interactions. Further, large waves with higher and sharper crests do not display any secondary maxima or minima. They appear more regular or "narrow banded" than relatively low waves, and their heights and crests do not violate the Miche-Stokes-type upper limits. The results also suggest that third-order nonlinearities do not affect the surface statistics in any discernable way.