Civil Engineering Reference
In-Depth Information
in Equation (7.7) is replaced by its asymptotic expression
1
/
2
exp
j
x
4
2
πx
π
H
0
(x)
=
−
(7.14)
and the new variable of integration
s
=
ξ/k
0
is used instead of
ξ
.Using
z
2
+
z
1
=
−
R
1
cos
θ
0
and
r
=
R
1
sin
θ
0
, Equation (7.7) can be rewritten
k
0
2
πr
1
/
2
exp
−
∞
jπ
4
p
r
=
F(s)
exp[
k
0
R
1
f(s)
]d
s
(7.15)
−∞
F
and
f
are given by
1
−
s
2
cos
θ
0
)
f(s)
=−
j(s
sin
θ
0
+
(7.16)
V(s)
s
1
−
F(s)
=
(7.17)
s
2
An asymptotic evaluation of the integral in Equation (7.15) is obtained in the following
way. The initial contour of integration in the complex sin
θ
plane for Equation (7.15) can
be modified within certain limits without modification of the result. The function
f
(s) is
rewritten
f(s)
=
f
1
(s)
+
jf
2
(s)
(7.18)
where
f
1
and
f
2
are real. A new contour
γ
is used, where
f
1
has a maximum at a point
s
M
, and decreases as rapidly as possible with
|
s
M
|
.If
f
is an analytic function, the line
of steepest descent of
f
1
is the line of constant value of
f
2
. The derivative d
f(s)/
d
s
s
−
=
0
for
s
s
M
,and
s
M
is the stationary point. The line of constant
f
2
including the stationary
point is the best choice for the path
γ
. This line is called the steepest descent path. For
k
0
R
1
1, the contribution on
γ
to the integral is restricted to a small domain around
s
M
. The stationary point in the s plane is located at
s
=
sin
θ
0
,where
θ
0
is the angle of
specular reflection represented in Figure 7.1. It has been shown by Brekhovskikh and
Godin (1992) that the path of steepest descent
γ
is specified by
=
s
2
)
1
/
2
cos
θ
0
=
1
−
ju
3
,
s
sin
θ
0
+
(
1
−
−∞
<u
3
<
∞
(7.19)
In the previous equations and in what follows,
√
1
−
s
2
=
cos
θ
, the choice of the deter-
mination depends on the location on the passage path. The passage path for
θ
0
=
π/
4
is represented in Figure 7.2. The choice for
√
1
−
is Im
√
1
−
s
2
s
2
≤
0, except between
A and B, where the path has once crossed the cos
θ
cut, and Im
√
1
−
s
2
≥
0. The station-
ary point is B where sin
θ
=
sin
θ
0
.For
θ
0
=
0 the passage path is the real s axis, i.e. the
initial path of integration and for
θ
0
=
π/
2, the passage path is the half axis defined by
Re
s
0 half plane. This axis is the dot - dash line in Figure 7.2.
If a pole of the reflection coefficient crosses the path of integration when it is deformed,
the pole residue must be added to the integral. The expression of residues is given in
Section 7.5.3. There is no pole contribution for
θ
0
=
1 located in the Ims
≤
0 because the path of integration is
not modified. As indicated in Section 7.5.3, the crossing is impossible for semi-infinite
=
