Civil Engineering Reference
In-Depth Information
s
0
)
cos
(
2
k
0
l
n
2
s
0
)
2
k
0
ls
0
(
1
−
−
s
0
+
2
(n
2
s
0
))
s
0
(
1
−
−
×
n
sin
2
[2
k
0
l
n
2
−
(7.A.3)
n
sin[2
k
0
l(n
2
s
0
)
]
−
−
s
0
]
where
M(s
0
)
is given by
M(s
0
)
=
m
(s
0
)(
1
−
n
2
)
{
2
m
(s
0
)(n
2
−
1
)
+
(
1
−
s
0
)
1
/
2
n
2
×
[3
m
(s
0
)(
1
−
s
0
)
1
/
2
−
m
(s
0
)(
1
−
s
0
)
3
/
2
−
s
0
(
2
n
2
+
s
0
)
]
}
+
(7.A.4)
×
(m
(s
0
)
1
−
s
0
+
n
2
−
s
0
)
−
3
(n
2
−
s
0
)
−
3
/
2
Locally reacting medium of constant impedance Z
L
The pole is located at
θ
p
satisfying
1
−
s
p
=−
Z/Z
L
cos
θ
p
=
(7.A.5)
The reflection coefficient
V
L
can be written
cos
θ
+
cos
θ
p
V
L
(s)
=
(7.A.6)
cos
θ
−
cos
θ
p
Equation (7.21) gives
2
1
−
1
−
s
0
1
−
s
p
(
1
−
s
p
)
N
=
(
1
−
(7.A.7)
1
−
s
0
−
s
p
)
3
Appendix 7.B Evaluation of
p
r
by the pole subtraction
method
We follow the reference integral method described in Appendix A in Brekhovskikh and
Godin (1992). The expressions for
p
r
are simultaneously obtained for a porous layer and
for a locally reacting surface with an impedance
Z
L
. A subscript
L
is used for the locally
reacting surface. The following expressions are used for the reflection coefficients
2
√
n
2
s
2
−
V(q)
=
1
−
√
n
2
j
√
1
−
s
2
cot
(
√
n
2
(7.B.1)
s
2
s
2
k
0
l)nZ/(φZ
0
)
−
−
−
2
Z
0
Z
L
√
1
−
V
L
(s)
=
1
−
(7.B.2)
s
2
+
Z
0
