Singularities in Surface Waves - Selected Topics of Computational and Experimental Fluid Mechanics - page 217

is to plot the wave envelope along the symmetry axis. In Fig. 12 , we plot envelopes
in the following cases: (i) linear wave, (ii) non linear wave with initial amplitude
0.02H., (iii) Pearcey prediction for a finite size wave maker, (iv) Pearcey prediction
for an infinite wave maker, (v) envelope predicted by the stationary phase method.
In all cases amplitudes are normalized with initial amplitude ( h 0 ). It is important to
point out that interference in the region inside the caustics leads to oscillations of the
envelope. These oscillation are predicted by the Pearcey results. However, in a wave
field produced by a finite wave maker these oscillations are less important because
the zone where three rays reach a point is only a fraction of the area limited by the
caustics. Far from the cusp, and according to ray theory, only a ray reaches each
point. In this figure the asymmetries related with nonlinearities are evident. First, the
negative branch of linear wave envelope is exactly the reflection of the positive one
(black lines). The same does not apply for the non linear wave and only far from the
cusp both branches becomes symmetric.
Caustics are fictitious singularities appearing in the ray theory. On the other side,
dislocations are a kind of singularity which remains beyond the geometrical optics.
They are points of full destructive interference and can be recognized by two facts:
the wave amplitude is always zero and the phase is undetermined. According to the
last feature, in a dislocation the contours of constant phase cross. The Fig. 13 shows
both a diagram of the wave amplitude and curves of constant phase, calculated with
the Pearcey integral. The ray theory predicts that dislocations occurs only inside
the caustics, but due to diffraction two dislocations outside the caustics appear. The
numerical solution of wave equation (linear and nonlinear) predicts also the existence
of dislocations outside the caustics as we can see in Fig. 14 , where wave amplitude
as a function of (x, y) is shown. Dislocations are located in the blue regions of each
Fig. 13 Graph of wave amplitude and curves of constant phase obtained from the Pearcey integral,
assuming a finite wave maker. The phase is undetermined in the points of full destructive inter-
ference. Then the curves of constant phase cross in such points. These singularities are known as
dislocations