Short waves on long waves

Abstract

Spatial changes in the wavelength and direction of short‐wavelength surface gravity waves, caused by their interaction with much longer wavelength gravity waves, are calculated by applying geometrical optics to the short waves in the reference frame that reduces the long waves to a steady current, which is horizontally variable but unidirectional. The wave rays are very nearly parallel to the direction of the current. Along a ray the wavelength and the angle between the propagation direction of the short waves and the ray direction are both maximum at the long wave troughs (where the steady current speed is maximum) and minimum at the crests (where the current speed is minimum), whether the short waves travel with or against the long waves. The short waves can never be reflected by the long waves. The calculated magnitudes of the wavelength and direction changes of the short waves are valid for arbitrary surface slope of both short and long waves, assuming no wave breaking occurs. Therefore these results are more general than any known ones, but they agree with earlier work when all wave slopes are taken small compared to unity and the wave number directions of the short and long waves are parallel.