Nonlinear Effects on Wave Envelope and Phase

Abstract

The effects of second‐order nonlinearities on the envelope and phase statistics of long‐crested sea waves are examined theoretically via the characteristic function expansion technique. Various cumulants typically required in this technique are derived explicitly to the first order of accuracy in the spectral bandwidth of sea surface elevations. The resulting joint and marginal probability densities are then examined in detail and checked with simulations. It is found that, except for certain minor modifications, the statistical distribution of envelope heights normalized by their rms‐value is described on the whole fairly well by the conventional Rayleigh law. In comparison, the wave phase and phase statistics are affected more significantly by nonlinearities, showing a systematic excess of values near the mean phase. The eventual comparison of the theoretical predictions with the actual wave data gathered from an offshore platform in the Gulf of Mexico in 1969 supports these conclusions surprisingly well, particularly when nonlinearities are predominantly manifest.