И-Shaped surf beat understood in terms of transient forced long waves

Abstract

The observed И-shape of the long wave generated by a single short wave pulse is explained qualitatively by including the free waves which are generated along with the well known bowl-shaped forced wave. Free waves of complicated shapes are generally formed as the wave group encounters gradual depth changes. However, the main qualitative feature, which has so far lacked a simple explanation, namely the long wave being И-shaped rather than a single bowl, can be explained through the simple free wave configuration which results from an abrupt onset of forcing at constant depth. И-shaped transient stages then occur between the initial flat surface and the final stage where free and forced waves have become completely separated from the forcing due to their different speeds. The И-shape is however also the shape of the gradually growing, resonant long wave due to a single short wave pulse, which travels with phase speed c and group velocity c g close to the long wave speed g h . The latter result is derived and compared with detailed laboratory experiments. It may explain the lack of observed H 2-dependence observed for some incident long waves, as well as the qualitative shapes of the long wave and of the cross correlation function between short wave envelope and long wave elevation.