Civil Engineering Reference
In-Depth Information
V(
sin
θ
0
))(
1
+
√
πu
exp
(u
2
)
erfc
(
jk
0
R
1
)/R
1
of the modified formulation to
p
L
noticeably decreases the difference at low frequencies.
The large peak in the medium frequency range at
θ
p
close to
π
/2 has disappeared. The
reflected field is close to
p
L
and the contribution of the correction is less important than
for
θ
0
close to
π
/2.
In summary, the modified formulation can provide reliable prediction, except for
layers having a small thickness of media of low flow resistivity close to grazing incidence.
In this case it is better to use the exact expression (Equation 7.6) if the conditions of a
precise evaluation of the integral are fulfilled.
Adding the correction
(
1
−
−
u))
exp
(
−
Appendix 7.A Evaluation of
N
Layer of finite thickness
1
+
2
√
n
2
2
m
(s
0
)s(
1
n
2
)
s
2
)k
0
l
∂V
∂s
=
−
−
s
2
(
1
−
√
n
2
s
2
√
1
s
2
(m
(s
0
)
√
1
+
√
n
2
sin
(
2
l
√
n
2
−
−
−
s
2
−
s
2
)
2
−
s
2
(
1
−
n
2
)
φ
cot
(k
0
l
n
2
m
(s)
=−
j
m
−
s
2
)
(7.A.1)
∂
2
V
∂s
2
n
2
)m
(s
0
)s
2
(
1
−
2
k
0
nl
(
1
=
√
n
2
s
2
√
1
−
s
2
(m
(s
0
)
√
1
−
+
√
n
2
−
n
2
)
s
2
s
2
)
2
−
−
2
k
0
nl(
1
−
s
2
)s
cos
(
2
k
0
l
√
n
2
s((
1
−
s
2
)/
√
n
2
+
2
√
n
2
−
s
2
)
−
s
2
−
s
2
))
×
n
2
sin
2
(
2
k
0
l
√
n
2
−
n
sin 2
k
0
l
√
n
2
−
s
2
)
−
s
2
1
+
s
2
)k
0
l
√
n
2
s
2
n
2
)s
2
(
1
−
−
2
((
1
−
+
n
2
)
sin
2
(
2
k
0
l
n
2
√
n
2
s
2
√
1
−
s
2
(m
(s
0
)
√
1
−
+
√
n
2
s
2
s
2
)
2
(
1
−
−
s
2
)
−
−
s
2
)
3
n
2
)m
2
(s
0
)s
2
m
k
0
ls
4
(
1
−
−
√
n
2
s
2
√
1
s
2
(m
(s
0
)
√
1
+
√
n
2
√
n
2
−
s
2
sin
(
2
k
0
l
n
2
−
−
−
s
2
−
−
s
2
)
(7.A.2)
The coefficient
N
can be written
s
0
)
n
2
s
0
2
k
0
l(
1
−
−
N
=
1
+
n
2
)
sin
(
2
k
0
l
n
2
s
0
)
(
1
−
−
2
(
1
−
s
0
)(
1
−
n
2
)s
0
(
−
m
(s
0
)
1
−
s
0
+
n
2
−
s
0
)m
(s
0
)k
0
l
×
M(s
0
)
+
s
0
)
1
s
0
(m
(s
0
)
1
n
2
s
0
)
3
sin
(
2
k
0
l
n
2
s
0
+
s
0
)
(n
2
−
−
−
−
−
s
0
2
m
(s
0
)s
0
(
1
−
n
2
)
1
−
2
k
0
l
+
n
2
s
0
1
s
0
(m
(s
0
)
1
n
2
n
2
)
n
(
1
−
2
s
0
+
s
0
)
2
−
−
−
−
