Singularities in Surface Waves - Selected Topics of Computational and Experimental Fluid Mechanics - page 206

ʺ ˁ c = /ʺ

The radius of curvature is the inverse of ). The method of stationary

phase allows us to obtain an expression for thewave amplitude in terms of the distance
traveled (d) by the ray and

( 1

ˁ (Paris and Kaminsky 2001 ):

A 0

ˁ c
ˁ c −

A =

d ,

(6)

where A 0 is the initial amplitude. The last equation predicts that amplitude diverge
for d

= ˁ c . The curve (or the surface in 3D waves) where optical geometry predicts
the divergence of wave amplitude is known as a caustic. In reality this does not
happen because ray theory is only an approximation in which the wave properties
are not considered. However, along a caustic we have a bright region (we use the
terminology of optics). In our system, we deal with a pair of caustics intersecting in a
point. This point is known as Huygens cusp and around it maximal wave amplitudes
take place. The equation of the caustics is:

a
3

y

3
2

4
3

1
2 a

x =± − . (7)

It is interesting to remark that other characteristics can be invoked for the definition
of a caustic. Note that along the caustic the wave fronts folds. This means that caustic
is the line (surface) separating illuminated from shaded regions. In this sense, a
caustic is the envelope of a ray family. An alternative definition of caustic follows
from Fig. 3 , in which some rays originating in wave maker have been drawn. The
caustic separates region I, where only an individual ray reaches each point, from
region II, where three rays reach each point.

Fig. 3 Rays originating in
the parabolic wavemaker. In
region I only a ray passes
through a point, while in
region II three rays reach
each point. The curve
separating both regions is
again the caustics

0.6

0.5

0.4

0.3

0.2

0.1

0

−0.4 −0.2 0 0.2 0.4

X