Envelope and Phase Statistics of Large Waves

A theoretical expression derived previously for describing the joint distribution of the envelope and phase of nonlinear waves is verified with wind-wave measurements collected in the North Sea. The same distribution is explored further to derive 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 if envelope elevations exceed the significant envelope height or not. For envelope elevations 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 also suggest that large surface displacements and thus large wave heights arise from the constructive interference of spectral components with different amplitudes and phases. Further, large waves with high and sharp crests do not display any secondary maxima and 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.