Civil Engineering Reference
In-Depth Information
3
Im sin
θ
2
1
Re sin
θ
B
0
A
−
1
−
2
−
3
−
2
−
1
0
1
2
3
4
Figure 7.2
The complex sin
θ
plane. The solid line is the path of steepest descent for
θ
0
=
π/
4, both cuts are represented by short dash lines along the imaginary axis and for
|
|
1 along the real axis. The initial integration path is represented by dashed
lines parallel to the real axis.
Re
sin
θ
s
2
)
1
/
2
is equal to 0, and when one of the two lines is crossed, keeping
constant the real part of cos
θ
corresponds to a change of sign of the imaginary part.
The dependence of the reflected wave in
z
2
is exp
(jz
2
cos
θ)
. The amplitude of the
wave tends to 0 when
z
2
→−∞
if Im cos
θ<
0. The
s
plane is a superposition of two
planes, the physical Riemann sheet, where Im cos
θ<
0, and the second Riemann sheet,
where Im cos
θ>
0. As indicated previously, the communication between both sheets
can be performed without discontinuity by crossing the cuts. For a semi-infinite layer,
the surface impedance becomes
cos
θ
=
(
1
−
Z
s
=
Z/φ
cos
θ
1
(7.12)
and
Z
s
and
V
depend on the sign of cos
θ
1
. New cuts must be added in the
s
plane,
defined by
n
2
s
2
−
=
u
2
,
2
real
≥
0
(7.13)
7.4
The method of steepest descent (passage path method)
The method has been used by Brekhovskikh and Godin (1992) for a similar prob-
lem, the prediction of the monopole field reflected by a semi-infinite fluid. The method
provides predictions of
p
r
which are valid for
k
0
R
1
1. The same calculations are per-
formed in what follows, with the difference that, for layers of finite thickness, the surface
impedance in Equation (7.10) is given by Equation (7.8) and the cut related to cos
θ
1
is
removed because both choices of cos
θ
1
give the same impedance. The Hankel function
