Environmental Engineering Reference
In-Depth Information
In region II occur some wave phenomena, noticeably, the interference. The pres-
ence of three rays give rise to the appearance of points where fully destructive in-
terference happens. This points are called dislocations because of its similarity with
dislocations in a crystal lattice. This kind of object is a true singularity, where the
phase becomes undefined. It is important to stress that dislocations appear not only
in the illuminated region, but a line of dislocations occur in the dark zone because
of the diffraction.
As we have stated before, the geometrical optics fails to predict the behavior in
the vicinity of a caustic. The divergence has been overcome for the first time with
the formulation of a theory by Airy in 1838. In order to describe the wave field near
a caustic Airy introduced a function called in his honor, which have some important
properties related to the existence and the absence of rays in both sides of a caustic.
This function is the solution of the differential equation:
d
2
w
dz
2
=
zw
.
(8)
The Airy function has a oscillating behavior for
z
0 the function
decays exponentially. It is important to mention that this theory is intended for simple
caustic, that is, if only two rays reach each point in the illuminated region. The wave
field produced by a parabolic wave maker differs from those studied by Airy because
in the illuminated region, the wave is the result of the interference of three rays.
The behavior of a linear wave in this configuration has been obtained by Pearcey
(
1946
). The work of Pearcey is based in the use of a diffraction integral, which is an
approximate solution of the wave equation. This integral is:
<
0 and for
z
>
+∞
dx
0
exp
(
ikd
(
x
0
,
x
,
y
))
h
(
x
,
y
)
=
√
d
,
(9)
cos
(ʸ(
x
0
))
(
x
0
,
,
)
x
y
−∞
where
is the angle between the tangent of parabola at point
x
0
and the x axis.
This quantity usually is small, implying that
cos
ʸ(
x
0
)
(ʸ(
x
0
))
≈
1. Because interest is
focused in the behavior around the Huygens cusp (its coordinates are
0
2
a
we
1
,
perform a Taylor expansion of
d
to first order around this position. The
final results is known as the Pearcey integral:
(
x
0
,
x
,
y
)
2
R
ka
2
1
/
4
+∞
exp
i
t
4
Vt
dt
k
i
2
exp
(
i
k
R
)
Ut
2
h
(
x
,
y
)
=
√
R
+
+
,
(10)
ˀ
−∞
2
R
3
/
4
x
.Ifthewavemaker
has a finite size and wavelength is not small when compared with R, the integral
must be carried over a finite domain.
2
2
R
1
/
2
√
a
1
k
2
k
=
=
(
−
)
=−
where
R
2
a
,
U
R
y
and
V
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