Civil Engineering Reference
In-Depth Information
been carried out by Ba nos (1966), and by Brekhovskikh (1960). The close similarities
between the acoustic and the electromagnetic case are shown in Allard and Henry (2006).
Some points are summarized. At sufficiently low frequencies, any layer having a finite
thickness can be replaced for
θ
0
close to
π
/2 by an impedance plane having the same pole
as the main pole of the layer. The contribution of the term [1
+
sgn
(
Re
(u))
]
√
πu
exp
(u
2
)
in Equation (7.48) corresponds to the contribution of a crossed pole in the passage
path method (see Appendix 7.C). This contribution does not exist if the porous layer
is semi-infinite, because in this case
Re
sin
θ
0
<
1. However, in this case, for small
|
u
|
,
there is a contribution equal to half the contribution of the pole if it were crossed.
7.7
Pole localization
7.7.1 Localization from the
r
dependence of the reflected field
Using Equations (7.49) - (7.50), Equation (7.51) can be rewritten
{
1
−
cos
θ
p
2
k
0
R
1
exp
−
3
jπ
4
√
π
exp
(u
2
)
erfc
(
exp
(
−
jk
0
R
1
)
R
1
p
r
=
−
u)
}
(7.52)
and cos
θ
p
is related to the reflected pressure by
R
1
p
r
exp
(jk
0
R
1
)
]
/
(
2
πk
0
R
1
)
1
/
2
exp
−
exp
(u
2
)
erfc
(
u)
3
πj
4
cos
θ
p
=
−
−
[1
(7.53)
Two simulated measurements of cos
θ
p
with Equation (7.53) are presented in Figures
7.13 and 7.14. The reflected pressure is calculated with Equation (7.6) and
θ
p
related to
the main pole is evaluated with an iterative method from Equation (7.53). A comparison
between the exact cos
θ
p
and cos
θ
p
evaluated with Equation (7.53) is performed, for a
layer of thickness
l
=
3 cm of material 1. In Figure 7.13,
r
=
1mand
z
1
=
z
2
=−
1cm.
In Figure 7.14,
z
1
=
|
also
increases and the systematic error increases. With thinner layers, the range of frequencies
where the systematic error can be neglected is larger. The angle of specular reflection is
larger in Figure 7.13 than in Figure 7.14, and the systematic error is larger. Measurements
z
2
=−
5 cm. In both figures, when frequency increases,
|
cos
θ
p
−
0.05
0
Exact
Simulated measurement
Exact
Simulated measurement
−
0.1
−
0.1
−
0.15
−
0.2
−
0.2
−
0.3
0.25
−
−
0.4
−
0.3
−
0.5
−
0.35
−
0.6
−
0.4
−
0.7
−
0.45
−
0.8
−
0.5
−
0.9
−
0.55
−
1
0 0.25 0.5 0.75 1 1.25 1.5 1.75 2 2.25 2.5 2.75 3
0 0.25 0.5 0.75 1 1.25 1.5 1.75 2 2.25 2.5 2.75 3
Frequency (kHz)
Frequency (kHz)
Figure 7.13
Comparison between the exact cos
θ
p
and a simulated measurement
obtained with Equation (7.53). Material 1,
l
=
3cm,
r
=
1m,
z
1
=
z
2
=−
2cm.
