Singularities in Surface Waves

Abstract

In this paper we investigate the evolution of surface waves produced by a parabolic wave maker. This system exhibits, among other, spatial focusing, wave breaking, the presence of caustics and points of full destructive interference (dislocations). The first approximation to describe this system is the ray theory (also known as geometrical optics). According to it, the wave amplitude becomes infinite along a caustic. However this does not happen because geometrical optics is only an approximation which does not take into account the wave behavior of the system. Otherwise, in ray theory the wave breaking does not hold and interference occurs only in regions delimited by caustics. A second step is the use of a diffraction integral. For linear waves this task has been made by Pearcey (1946) (Pearcey, Philos Mag 37 (1946) 311–317) for electromagnetic waves. However the system under study is non linear and some features have not counterpart in the linear theory. In the paper our attention is focused on three types of singularities: caustics, wave breaking and dislocations. The study we made is both experimental and numerical. The experiments were conducted with two different methods, namely, Schlieren synthetic for small amplitudes and Fourier Transform Profilometry. With respect the numerical simulations, the Navier-Stokes and continuity equations were solved in polar coordinates in the shallow water approximation.