Civil Engineering Reference
In-Depth Information
9.3.3 Examples
The validity of similar models was verified by comparing prediction and measurement
of the impedance at normal incidence for different configurations (Bolt 1947, Zwikker
and Kosten 1949, Ingard and Bolt 1951, Brillouin 1949, Callaway and Ramer 1952,
Ingard 1954, Velizhanina 1968, Davern 1977, Byrne 1980). The trends predicted by
the model appear clearly in the measured impedances. Simple configurations of layered
media covered by perforated facings having circular apertures are modelled in what
follows, showing the effect of the open area ratio, the diameter of the holes, and the
flow resistivity of the material in contact with the facing. For the first example, the
normalized impedance and the absorption coefficient calculated from Equation (9.50)
are represented as a function of frequency in Figures 9.8 and 9.9 for different open
area ratios. An isotropic porous material, set on a rigid impervious wall, is in contact
with the facing at its upper face. The porous material is characterized by the following
parameters:
50 000 N m
−
4
s,
thickness
e
=
2
cm,
flow
resistivity
σ
=
characteristic
0
.
034 mm and
=
dimensions
1
.
5.
The Johnson
et al
. model is used for the effective density and the Champoux and
Allard model is used for the bulk modulus (see Chapter 5). The facing has a thickness
d
=
0
.
13 mm, porosity
φ
=
0
.
98, tortuosity
α
∞
=
=
1 mm, and circular apertures of radius
R
=
0
.
5 mm. The open area ratio takes the
following values:
s
0
.
4, 0.1, 0.025 and 0.005.
The real part of the impedance strongly increases when the open area ratio decreases,
due to the resistive effects in the porous material close to the facing. The imaginary part
also increases when the open area ratio decreases, due to the added masses at both sides
of the facing that are multiplied by 1/
s
. The absorption coefficient
A
0
increases at low
frequencies and decreases at high frequencies; the maximum value of
A
0
simultaneously
decreases because the real part of the impedance becomes much larger than
Z
c
.
In the second example, an air gap and a screen are successively inserted between
the facing and the porous material. The three configurations are represented in Figure
9.10.
=
(d)
(d)
14
40
13
35
12
30
11
25
10
20
9
15
8
10
7
(c)
5
6
(b
)
0
5
(a)
(c)
1
−
5
4
−
3
(b)
−
15
2
(a)
1
−
20
0
−
25
0.1
1
5
0.1
1
5
Frequency (kHz)
Frequency (kHz)
Figure 9.8
Influence of the open area ratio on the normalized impedance
Z/Z
c
of an
isotropic porous material covered by a facing. Parameters for the porous material: thick-
ness
e
50 000 N m
−
4
s, tortuosity
α
∞
=
=
2 cm, flow resistivity
σ
=
1
.
5, characteristic
0
.
034 mm and
=
dimensions
=
0
.
13 mm, porosity
φ
=
0
.
98. Parameters for the
facing: thickness
d
=
1 mm, circular apertures of radius
R
=
0
.
5 mm, open area ratio
s
=
(a) 0.4, (b) 0.1, (c) 0.025, (d) 0.005.
