Civil Engineering Reference
In-Depth Information
For materials having an impedance which does not depend significantly on
θ
the
modified simulation is similar to the initial formulation of Chien and Soroka. A detailed
description of the domain of validity of the modified formulation is beyond the scope
of the topic, due to the large number of parameters that characterize a porous layer.
Some trends can be shown with the following examples. A precise correction due to
the sphericity is necessary when the direct field exp
(
−
jk
0
R
2
)/R
2
and the reflected field
V(
sin
θ
0
)
exp
(
−
jk
0
R
1
)/R
1
almost cancel each other. This happens for
θ
0
≈
π/
2 because
at
θ
0
=
=−
1. In the first examples
θ
0
is close to
π
/2. Let
p
t
be
the exact total field above the layer given by
π/
2
,R
1
=
R
2
and
V
exp
(
−
jk
0
R
2
)
R
2
p
t
=
p
r
+
(7.66)
where the reflected field
p
r
is calculated with Equation (7.6). Let
p
t
be the total field
obtained with the modified formulation. In Figure 7.16,
|
(p
t
−
|
is represented as
a function of frequency for a layer of material 1 and a layer of material 2 of thickness
2 cm. The geometry is defined by
z
1
=
p
t
)/p
t
z
2
=−
2
.
5cm,
r
=
1m.
The error is small for both materials in the low frequency range. At low frequencies
there is one pole at
θ
p
close to
π
/2 and Equations (7.49) - (7.51) can be used. The modified
formulation corresponds to the same set of equations where cos
θ
p
=−
Z/Z
s
(
sin
θ
p
)
is
replaced by
Z/Z
s
(
sin
θ
0
)
. The modified formulation can also be used close to grazing
incidence for thin layers because
θ
0
and
θ
p
are close to each other and
Z
s
(θ
p
)
is close
to
Z
s
(θ
0
)
. The error is also small for the modified formulation at high frequencies for
both materials. This can be explained by using the passage path method to predict the
reflected pressure. The use of the passage path method can be justified at high frequencies
for media with a low flow resistivity when
θ
0
is close to
π
/2 because, in Equation
(7.25)
|
Z
0
/Z
s
(s
≈
1
)
|≈
1, there is no pole close to
π
/2. It is shown in Figure 7.17 that
in the high-frequency range there is a good agreement between the exact pressure
p
t
obtained with Equation (7.6) and the prediction
p
t
obtained with Equation (7.20) from
−
10
material 2
material 1
9
8
7
6
5
4
3
2
1
0
0
0.5
1
1.5
2
2.5
3
3.5
4
Frequency (kHz)
Figure 7.16
Modulus of the normalized error in the pressure evaluation with the mod-
ified formulation. Thickness
l
=
2cm,
z
1
=
z
2
=−
2
.
5cm,
r
=
1m.
