Civil Engineering Reference
In-Depth Information
from Equation (8.52) the cutoff frequencies are given by
f
c
=
(
2
m
+
1
)
V
T
4
l
or
f
c
=
(
2
m
+
1
)
V
L
4
l
,m
=
0
,
1
,
2
...
(8.53)
At the cutoff frequencies, the phase speed being infinite, the displacements are iden-
tical on the whole free surface. These displacement correspond to the
(
2
m
+
1
)λ
T
/
4and
(
2
m
1
)λ
L
/
4 resonances of the layer. For a medium with a small Poisson ratio the
first resonance is the
λ
T
/
4 resonance followed by the
λ
L
/
4 resonance. If the Poisson
coefficient is sufficiently close to 0.5, the second resonance is the 3
λ
T
/
4 resonance.
An experimental set-up for the case of bonded layers is described in Boeckx
et al
.
(2005) with phase speed measurements of the two first modes.
The main difference with the modes of a porous frame saturated by air is due to the
large loss angle. The phase speed does not tend to infinity when frequency tends to the
cutoff, but presents a maximum.
+
8.5.2 Excitation of the resonances by a point source in air
A heavy porous frame is not strongly coupled to the acoustic field in the surrounding
air. Moreover, the loss angle for usual sound absorbing porous layers is large; a current
order of magnitude is 1/10. However, the frame velocity created at the free surface of
a porous layer by a point source in air can be measured with a laser velocimeter, and
large peaks can appear in the velocity distributions. In contrast to the usual acoustic
impedance measurements, the chosen boundary conditions for the frame are important
and if the layer is bonded on a rigid backing, a careful gluing of the frame is mandatory.
The experimental set-up for the excitation and the measurement of the frame displacement
is shown in Figure 8.7. The velocity component
U
s
z
created by the monopole field over
the porous frame is given by
∞
J
0
(rξ)
µ
U
s
V(ξ/k
0
)
]
U
z
(ξ)
exp[
jµ(z
1
+
z
(r)
=−
j
[1
+
z
2
)
]
ξ
d
ξ
(8.54)
0
µ/k
0
and
U
z
is given by Equation (8.38).
where
V
is the reflection coefficient for cos
θ
=
Using
2
π
j
2
π
J
1
(z)
=
exp
(
−
jz
cos
θ)
cos
θ
d
θ
0
Laser Doppler vibrometer
Compression
driver
S
x
M
h
r
Sample
Rigid backing
z
Figure 8.7
Set-up for the measurement of the frame displacement (Geebelen
et al
.
2007). Reproduced by Permission of MedPharm Scientific Publishers - S. Hirzel Verlag.
