Civil Engineering Reference
In-Depth Information
2
1.5
Re
1
0.5
0
Im
−
0.5
−
1
1.5
−
−
2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
1.3
1.4
1.5
Frequency (kHz)
Figure 6.11
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
=
5
·
6 cm. Prediction with
Equation (6.107). . Prediction for the same material with a rigid frame: ------.
Measurements:
•••
. (Measurement taken from Allard
et al
.1991).
K
c
is given by Eq. (1.76). In terms of the shear modulus and the Poisson ratio, it is
given by
2
(
1
−
ν)N
K
c
=
(6.111)
(
1
−
2
ν)
The value of the shear modulus
N
in Table 6.1 for the equivalent isotropic porous
material has been chosen to adjust predictions by Equation (6.107), with the Poisson coef-
ficient equal to zero, and measurements. The same value can be used for both thicknesses.
This fact is a good evidence of the validity of this evaluation. The interpretation of these
results is very simple. Because of the stiffness and the density of the frame, the acoustic
field in the air can generate a noticeable frame-borne wave only at the
λ
/4 resonance of
the frame. The velocity of the frame at this resonance is equal to zero at the frame - wall
contact, where the material is glued, and reaches a maximum at the boundary frame-free
air, where the impedance is modified by the frame-borne wave. It should be noticed that
another model has been proposed by Kawasima (1960). In the context of this model, the
peaks in the surface impedance are interpreted by Dahl
et al
. (1990) as local resonances
of the fibres. The dependence on frequency of the location of the peaks as a function
of the thickness
l
, and the absence of peaks when the material is not glued to the wall,
could be more favourable to the hypothesis of a resonance of the whole frame in our
measurements. The observation of a quarter compressional wavelength resonance was
possible in a free field on a large sample. It seems difficult to observe the compressional
resonance in a Kundt tube, due to the contact with the lateral surface of the tube, and
the geometry of the porous samples in the tube. A numerical study on the effect of the
