Civil Engineering Reference
In-Depth Information
The pole contribution
SW(s
p
)
can be written, if
s
p
is close to 1
1
=
4
π
k
0
2
πr
1
/
2
exp
jπ
4
−
s
p
SW(s
p
)
exp
(k
0
R
1
f(s
p
))
(7.43)
s
p
The same expression is obtained for a non-locally reacting medium. For the case of
thin layers, Figure 7.9 shows that Re sin
θ
p
>
1. However, Re sin
θ
p
remains close to 1
for ordinary reticulated foams and fibrous layers and the pole is crossed only for angles
of incidence close to
π
/2. For semi-infinite layers, expression (7.41) is simplified because
l/
sin
(
2
k
0
l
n
2
−
s
p
)
is replaced by 0. When the passage path method is used, there is
no pole contribution. In Figure 7.7 Re sin
θ
p
is always smaller than 1. When the initial
path of integration becomes the passage path, the part of the path located at Re sin
θ<
1
can cross the pole, but the pole is in the physical sheet and the path is not in the physical
sheet in the domain 0
<
Re sin
θ<
1, 0
>
Im sin
θ
, because it has crossed the cut. The
condition Re sin
θ<
1 has always been verified for semi-infinite layers when the causal
effective densities presented in Chapter 5 have been used. However, we have not found
any general proof of this condition.
7.6
The pole subtraction method
The passage path method is valid for
k
0
R
1
1 only if the poles and the stationary point
are sufficiently far from each other, allowing a slow variation of the reflection coefficient
close to the stationary point on the path of integration. If a pole is close to the stationary
point, the pole subtraction method can be used to predict the reflected pressure under the
same condition,
k
0
R
1
1. The passage path method remains valid for
|
u
|
1, where u
is the numerical distance defined by
2
k
0
R
1
exp
sin
θ
p
−
j
3
π
4
θ
0
u
=
−
(7.44)
2
This condition can be much restrictive than
k
0
R
1
1if
θ
p
and
θ
0
are close to each
other. The expressions for
p
r
obtained with the pole subtraction method are given for
one pole of first order close to the stationary point in Appendix 7.B. Two cases are
considered, a locally reacting surface with a constant impedance
Z
L
, and a porous layer
of finite thickness. The reference integral method of Brekhovskikh and Godin (1992) is
used. It is shown in Appendix 7.B that if one pole exists at
θ
p
close to
π
/2, for
θ
0
close
to
π
/2 the monopole reflected field is the same as the monopole reflected field above the
locally reacting surface with a surface impedance
Z
L
given by
Z
L
=−
Z
0
/
cos
θ
p
(7.45)
The pole of the reflection coefficient of the related locally reacting surface is located
at the same angle
θ
p
. The following approximation for
p
r
, obtained by setting
√
s
0
s
p
=
1
