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Singularities in Surface Waves
G. Ruíz Chavarría and T. Rodriguez Luna
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 (dis-
locations). 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 dis-
locations. 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 simula-
tions, the Navier-Stokes and continuity equations were solved in polar coordinates
in the shallow water approximation.
1 Introduction
Waves are ubiquitous in nature. The light and the sound are two examples of them,
but possibly the most classical picture is that of a wave on the surface of a liquid.
They carry energy but not mass and exhibits a myriads of phenomena like reflection,
refraction, interference and diffraction. The linear waves are by far the most stud-
ied due to the fact that its properties can be deduced analytically. Usually a wave
is represented as having a sinusoidal shape, with constant amplitude and a defined
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