Singularities in Surface Waves - Selected Topics of Computational and Experimental Fluid Mechanics - page 204

where a 1 is the amplitude of the peak, a 2 is the amplitude of the trough, b 1 is the
distance from the peak to the next point of zero amplitude, and b 2 is the distance from
the peak to the previous point of zero amplitude. For a linear wave both quantities
vanish. In a nonlinear wave the first one is positive while the second one is negative.
Most of analytical results about surface waves have been obtained when the am-
plitude is small and consequently non linear terms are neglected in the governing
equations. In addition, another hypothesis are made, namely, the velocity field is
assumed irrotational and viscosity is neglected. Under all these assumptions, it is
possible to derive a dispersion relation, which in the general case is given by the
following equation (Elmore and Heald 1969 ):

gk tanh

k 3
ˁ

+ ˃

2

ˉ = ( kH ), (3)

where k is the wavenumber, ˃ the surface tension coefficient, ˁ the fluid density and

H is the liquid depth. In the limit of deep waters ( ʻ H ) the term tanh ( kH ) ≈ 1.

Then, waves are dispersive, that is, the phase velocity c = ˉ/ k is dependent on the

wavenumber k. The opp osit e limit is the shallow water case ( ʻ H ) for which the

= √ gH , irrespective the wavelength.
In Fig. 1 the phase velocity for waves in the deep water approximation is plotted
as a function of the wavelength

phase velocity is c

. The wavelength lies in the range 1-200 cm. In the
figure it is clear that wa ves are dispersive and that phase velocity attains a minimal
value for

ʻ

ˁ g

70 cm. It is important to note that the dependence of
phase velocity on wavelength in deep waters is a key feature for the time focusing.
This paper is organized as follow. Section 2 is devoted to describe the wave field
produced by a parabolic wave maker, the ray theory, the theories of Airy and Pearcey
and the singularities in this wave field (caustics and dislocations). In Sect. 3 we
describe the optical methods to study surface waves and the experimental setup. In

ʻ = 2 ˀ = 1 .

Fig. 1 Phase velocity (c) of
surface wave versus
wavelength (

180

160

) in the deep
water approximation
(

ʻ

140

H ). The phase velocity
depends upon the
wavelength. In experiments
and numerical simulations,
the wavelength lies in the
range 1 <ʻ< 10 cm

ʻ

120

100

80

60

40

20

10 −1 10 0 10 1 10 2

λ (cm)