Modulation of Short Wind Waves by Long Waves

Abstract

The behavior of short surface waves (wavelength less than a meter or so) riding on longer wind waves or swell has sparked interest for several decades now. In the problem as posed here, the energetic waves near the peak of a wind-wave or swell spectrum are treated as a large-scale, slowly varying “medium” in which short gravity-capillary waves evolve. This “WKB approximation” should be well founded, since the time and space scales of the long and short waves are widely separated, and no reflections of the short waves occur. Short wave “packets” or “components” are examined independently as they evolve under the influence of slowly varying winds and currents, and later reassembled into a larger picture. Resonant non-linear exchanges are not explicitly included, although (for example) a “packet” might be regarded as a set of tightly coupled wavenumbers rather than as a pure sine-wave component.