Singularities in Surface Waves - Selected Topics of Computational and Experimental Fluid Mechanics - page 203

wavelength, however this is just an idealized image. Most of times the wave field
contains a continuum of wavelengths, so its shape takes a complicated form and often
it evolves in time. This paper deals with waves on the surface of a liquid, which are
governed by the Navier-Stokes and continuity equations. They share some features
with linear waves, but some of its properties are the result of nonlinear interactions.
An example is the waveform, which is no more symmetric as in a sinusoidal wave.
On the other side, the amplitude of a surface wave cannot grow indefinitely, after a
threshold the wave breaks. The energy carried is rapidly dissipated into turbulence,
the formation of air bubbles and drops, etc. (Babanin 2011 ). This phenomenon is com-
monly observed along the coasts, when the waves approach the shore. Under these
circumstances the wave amplitude grows essentially by a decrease in the sea depth,
until the slope of surface becomes infinite. This is the bathymetric breaking, which
has been studied extensively. In the open sea the wave breaking is also present, but
the mechanisms involved in its production are quite different. Let us consider a con-
tinuous wave field and the fact that surface waves are dispersive. Then, components
of different wavelengths moves with different phase velocity. The further evolution
could produce a rise in the wave amplitude and eventually to the development of
the breaking. Of course there are others mechanisms involved in the wave breaking,
among them, the interaction between the wind and the waves and the occurrence of
currents moving in opposite direction to the wave motion (Zemenzer 2009 ).
In the ocean there are three stages in the wave evolution (Babanin 2011 ). In the first
one thewind blows and deforms thewater-air interface, so an initially small amplitude
wave is produced. During a time scale covering thousand of periods energy is injected
to the wave, allowing to a slow growth of its amplitude. In this step the wave evolution
can be described with a weak nonlinear theory. In the second stage, which covers
only few periods, the wave becomes highly asymmetric either in the horizontal and
vertical directions. The peaks become steeper and the troughs retract. At a certain
time the free surface becomes multivaluated and breaking develops just in a fraction
of a period. The breaking is the mechanism to dissipate energy, which is converted
in heat, turbulence, bubble production, etc. The case we study has a different driving
mechanism, namely the spatial focusing. The aim to make experiments with this kind
of breaking is to study this phenomenon in laboratory, where waves cannot evolve
over thousand of wavelengths, but in which underlying nonlinear interactions are
still present. In addition the study of the wave field under spatial focusing reveals
the existence of other singularities apart breaking, like caustics and dislocations, and
phenomena as interference and diffraction.
As stated before a nonlinear wave is asymmetric. In order to quantify this asym-
metry two quantities are introduced, the skewness and the asymmetry, defined as
(Babanin 2011 ):

a 1
a 2 −

S = 1 , (1)

b 1
b 2 −

A s = 1 , (2)