Civil Engineering Reference
In-Depth Information
A systematic method of calculating the surface impedance at oblique incidence is
based on transfer matrices. It is presented in Chapter 11.
6.6.3 Example: Fibrous material
The surface impedances at normal incidence calculated by Equation (6.107), of two
samples of different thicknesses made up of the material described in Section 6.5.4, are
represented for
l
=
10 cm and
l
=
·
4 cm in Figures 6.10 and 6.11, and compared with
measured values (Allard
et al
. 1991). Measurements were performed in a free field on
samples of large lateral dimensions.
The agreement between measurement and prediction by Equation (6.107) is good in
the entire range of frequencies where the measurement was performed. A peak appears
in the real and the imaginary parts of the impedance around 470 Hz for
l
5
=
10 cm and
860 Hz for
l
6 cm. The surface impedances calculated by Equation (6.107), and by
Equations (4.137), (6.86) and (6.87) for the same material with a rigid frame, are close
to each other, except around the peaks which are not predicted by the one-wave model.
The peaks appear around the
λ
/4 resonance of the frame-borne wave which is located at
the frequency
f
r
such that
=
5
·
π
2
l
Re
(δ
b
)
=
(6.109)
The quantity
δ
b
is very close to
δ
1
given by Equation (6.88), and
f
r
can be written as
Re
(K
c
)
ρ
1
1
4
l
f
r
=
(6.110)
2.5
2
1.5
Re
1
0.5
0
−
0.5
Im
−
1
−
1.5
−
2
−
2.5
0.2
0.4
0.6
0.8
1
1.2
1.4
Frequency (kHz)
Figure 6.10
Normalized surface impedance
Z/Z
0
of a layer of the fibrous material
described in Section 6.5.4. The thickness of the layer is
l
=
10 cm. Prediction with
Equation (6.107).
. Prediction for the same material with a rigid frame: ------.
Measurements:
•••
. (Measurement taken from Allard
et al
. 1991).
