Civil Engineering Reference
In-Depth Information
The unit term provides a contribution exp(-j
k
0
R
1
)/R
1
to
p
r
andisdiscardedinwhat
follows. In Equation (7.15),
f(s),f
L
(s),F(s)
,and
F
L
(s)
, are now given by
1
−
s
2
cos
θ
0
)
f(s)
=
f
L
(s)
=−
j(s
sin
θ
0
+
(7.B.3)
2
s
1
−
s
2
n
2
s
2
)
−
1
s
2
n
2
φZ
0
1
s
2
cot
(k
0
l
n
2
j
Zn
F(s)
=−
−
−
s
2
−
−
−
(7.B.4)
=−
2
Z
0
s
1
−
s
2
[
Z
L
1
−
Z
0
]
−
1
s
2
F
L
(s)
+
(7.B.5)
Using Equations (A.3.9) -(A.3.14) of Brekhovskikh and Godin (1992) with the same
notations, the integral in Equation (7.15) can be written
exp
(k
0
R
1
f(s
0
))
a
F
1
(
1
,kR
1
,q
p
)
1
(
0
)
π
k
0
R
1
1
/
2
∞
F(s)
exp[
k
0
R
1
f(s)
]d
s
=
+
−∞
(7.B.6)
a
=
lim[
F(s)(s
−
s
P
)
]
(7.B.7)
s
→
s
0
1
/
2
q
p
={
j
[cos
(θ
p
−
θ
0
)
−
1]
}
(7.B.8)
Im
(q
p
)<
0
This equation can be replaced, for the case of thin layers and semi-infinite layers, by
q
p
=
exp
−
√
2sin
θ
p
−
jπ
4
θ
0
(7.B.9)
2
F
1
(
1
,k
0
R
1
,q
p
)
=−
jπ
exp
(
−
k
0
R
1
q
p
)
[erfc
(j
k
0
R
1
q
p
)
]
(7.B.10)
F(s
0
)
2
f
(s
0
)
+
−
1
(
0
)
=
a/q
p
(7.B.11)
where
s
0
=
sin
θ
0
. For the porous layer, a is given by
n
2
2
s
p
1
−
s
p
G
s
(s
p
)
−
a
=−
(7.B.12)
s
p
where
G
is given by Equation (7.40), and Equation (7.B.12) can be rewritten
2
1
−
−
1
s
p
s
p
2
k
0
l
1
s
p
)
a
=−
1
−
(
1
−
n
2
s
p
sin 2
k
0
l
n
2
s
p
+
n
2
s
p
−
−
−
(7.B.13)
