Civil Engineering Reference
In-Depth Information
X
1
x
3
x
2
Rigid wall
L
1
L
2
Double porosity
material
Figure 13.15
Double porosity material in a waveguide backed by a rigid wall (Sgard
et al
. 2005). Reprinted from Sgard, F., Olny, X., Atalla, N. & Castel, F. On the use of
perforations to improve the sound absorption of porous materials.
Applied acoustics
66
,
625-651 (2005) with permission from Elsevier.
R
in the case of circular cross section and sides
a
and
b
in the case of rectangular
cross-sections.
The numerical model is based on a finite element modelling of the double porosity
material at the mesoscopic scale and a modal description for the wave-guide. The wave
propagation inside the microporous material is described by the first Biot - Allard theory
and inside the meso-pores by Helmholtz's equation. Note that, given the size of the holes,
the associated viscous characteristic frequency is very low and the dissipation can thus
be considered as negligible, justifying the use of Helmholtz's equation (established for
a perfect gas) in the meso-pores. The weak integral forms associated with each domain
are then discretized using finite elements. The coupling between the double porosity
material and the waveguide is accounted for explicitly using the modal behaviour of
the waveguide. Atalla
et al
. (2001b) provide the details in the simplified case where
the skeleton of the porous medium is assumed motionless. The following presentation
accounts for the elasticity of the material.
Combining Equations (13.8), (13.17) and (13.38), the weak integral form govern-
ing the acoustic behaviour of a porous layer coupled with the modal behaviour of the
waveguide of cross-section
L
1
×
L
2
, can be written
φ
ω
2
αρ
0
R
pδp
d
φ
2
∂p
∂x
i
∂(δp)
∂x
i
−
p
[
σ
ij
δe
s
ij
−
ω
2
ρ
u
s
i
δu
s
i
]d
+
p
α
δ
∂p
∂x
i
u
s
i
d
p
φ
1
δ(pu
s
i,i
)
d
Q
R
φ
−
−
+
p
(13.74)
1
jω
+
p
δ(u
n
p(
x
))
d
S
x
−
p
A(x,y)p(y)δp(
x
)
d
S
y
d
S
x
p
ε
jω
∀
(δu
s
i
,δp)
+
p
A(x,y)p
b
(y)δp(
x
)
d
S
y
d
S
x
=
0
p
For a double porosity medium, the waveguide is in contact with both the poroelastic
material and the perforations. Fluid cavities (holes) in contact with the waveguide are
modelled in the same way as poroelastic patches. The associated weak integral form
