Civil Engineering Reference
In-Depth Information
Measurements of the reflection coefficient of a thick layer of sand at oblique incidence
have been performed previously with the Tamura method by Allard
et al
. (2002). The
modulus of the reflection coefficient presents a minimum at cos
θ
close to the predicted
Re cos
θ
B
=−
Re cos
θ
p
.
7.5.3 Contribution of a pole to the reflected monopole pressure field
If a pole is crossed by the path of integration when the path is modified, the integral over
the initial path is equal to the integral over the modified path plus a pole contribution.
These poles are zeros of the first order of the denominator on the right-hand side of
Eqution (7.10). There is also an apparent problem with the term (
s
/
(
1
s
2
))
1
/
2
in
F(s)
in Equation (7.17), but the zero at the denominator disappears if cos
θ
is used instead of
sin
θ
as the variable of integration. The following expression is used for the reflection
coefficient
−
=
−
√
n
2
−
jn(Z/φZ
0
)
√
1
−
s
2
cotg
(k
0
l
√
n
2
−
s
2
−
s
2
)
V(s)
√
n
2
−
jn(Z/φZ
0
)
√
1
−
s
2
cotg
(k
0
l
√
n
2
(7.38)
−
s
2
−
s
2
)
At a pole location
s
=
s
p
, the following relation is fulfilled
n
2
φZ
0
1
−
s
p
cotg
(k
0
l
n
2
−
s
p
=
jn
Z
−
s
p
)
(7.39)
The derivative G
s
(s
p
)
of the denominator can be written
s
p
n
2
jnZ
φZ
0
G
s
(s
p
)
=−
−
s
p
−
−
(7.40)
1
−
cotg
(k
0
l
n
2
s
p
sin
2
(k
0
l
n
2
s
p
1
−
s
p
k
0
ls
p
n
2
×
−
s
p
)
+
−
s
p
)
−
s
p
Using relation (7.39), Equation (7.40) becomes
1
−
n
2
s
p
1
−
s
p
2
k
0
l
1
G
s
(s
p
)
s
p
)
=
−
s
p
(
1
−
n
2
−
s
p
sin 2
k
0
l
n
2
−
s
p
+
n
2
−
s
p
(7.41)
Using Eq. (7.15), the pole contribution can be written
−
2
n
2
s
p
1
−
s
p
=−
2
πj
k
0
2
πr
1
/
2
exp
−
−
s
p
jπ
4
SW(s
p
)
exp[
k
0
R
1
f(s
p
)
]
G
s
(s
p
)
(7.42)
