Singularities in Surface Waves - Selected Topics of Computational and Experimental Fluid Mechanics - page 211

Fig. 4 Spatial domain used
in the numerical solution. It
is a annular ring
( r 1

40

30

Wavemaker

r 2 ). The wave
maker is outside this domain,
it intersects the outer
boundary in two points. The
boundary conditions are set
assuming that the wave front
evolves according to the
stationary phase method

< r <

20

10

DOMAIN

0

−10

−20

−30

−40

−50 0 50

Y

becomemultivaluated. However it is possible to study the focusing and the emergence
of nonlinearities appearing during the growth of waves. It is important to remark that
when non linear terms are dropped from Eqs. ( 13 )-( 15 ) we recover the equations
of a linear wave. The numerical solution is performed in a annular domain, for
r 1 < r 2 . The wave maker is outside this domain, it intersects the outer boundary

in two points (see Fig. 4 ). We consider that initially the fluid is at rest, that is, surface
is not deformed. For imposing the boundary conditions we approximate the values
of surface deformation assuming that the wave evolve from wave maker to the outer
boundary according to the stationary phase method. The numerical solution was
carried out using a mesh of 400 points in radial direction and 256 modes in the
angular variable

r <

01.
Numerical simulationwasmade under two conditions. In the first one the nonlinear
terms are dropped, then solution correspond to a linear wave. In the second case
nonlinear terms are retained. In both cases maximal amplitude is attained in the
vicinity of Huygens cusp, along the symmetry axis. Otherwise, due to finite size
of the wave maker we observe that interference in the region delimited by caustics
occurs only in a section near the cusp.

ʸ . Otherwise the time step is set to ʴ t = 0 .

5 Experimental and Numerical Results

As stated in Sect. 3 surface waves were produced with a parabolic wave maker. In
order to characterize the initial wave front we recall that the equation of a parabola
is y 0 = ax 0 . In our experiments and in numerical simulations the value of parameter

a is 2; thus the position of the Huygens cusp is R

1
2 a

= = .

25m away from the
parabola vertex. Most of results presented here correspond to waves with a frequency
f

0

= ʻ = .

82 cm. Experiments and numerical simulations
were conducted to study three types of singularities: wave breaking, caustics and

7Hz or equivalently 3