The relation between wave length and wave period ditributions in random Gaussian waves

Wave data often describe the variation of sea elevation with time, at a fixed point. Wave period is the time between successive crossings of the mean level, while wave length is the distance between crossings at a fixed time along a fixed direction. For a deterministic, harmonic wave, the dispersion relation describes the relation between wave period and wave length. For random waves, wave period and wave length, as well as wave amplitude, are random quantities. We study these distributions for Gaussian waves and show how they can be calculated exactly from the general wave frequency spectrum and wave number spectrum, respectively. The results show that the dispersion relation leads to considerable underestimation of short waves for wide band spectra. We also consider the effect of truncation of high frequencies, to illustrate the sensitivity of the wave length distribution to spectrum truncation. Further, we give examples of wave period or wave length and wave amplitude distributions in sea states with unidirectional and directional spectrum. Received March 8, 1997: revised manuscript received by the editors September 22, 1998. The original version (prior to the final revised manuscript) was presented at the Seventh International Offshore and Polar Engineering Conference (ISOPE-97), Honolulu, USA, May 2530, 1997.