Civil Engineering Reference
In-Depth Information
6.2.5 Comparison with previous models of sound propagation in
porous sound-absorbing materials
It has been shown by Depollier
et al
. (1988) that, in the simple monodirectional case,
the stress - strain equations (6.2) and (6.3), used with the simplified evaluation at
P
,
Q
and
R
by Equations (6.26) - (6.28), are very similar to the stress - strain equations from
Beranek (1947) and Lambert (1983), but are not compatible with the equation proposed
by Zwikker and Kosten (1949), (Equation 3.05 in Zwikker and Kosten 1949 is incorrect,
due to a term
P
o
which must be removed.)
6.3
Inertial forces in the Biot theory
Biot introduced an inertial interaction between the frame and the fluid, which is not related
to the viscosity of the fluid, but to the inertial forces. The porous frame is saturated by
an nonviscous fluid. Denoting the velocity in the medium by
u
and the density by
ρ
0
,
the components of the inertial force per unit volume can be written as
∂
∂t
∂E
c
∂
u
i
F
i
=
i
=
1
,
2
,
3
(6.30)
For the case of a porous material, similar expressions can be used to calculate the
inertial forces, but the kinetic energy is not obtained with the summation of the two terms
2
ρ
1
u
s
2
φρ
o
u
f
2
2
1
1
+
(6.31)
where
ρ
0
is the density of air. This is because the velocity
u
f
is not the true velocity of
the air in the material, but is a macroscopic velocity. In the context of a linear model,
the kinetic energy has been given by Biot:
2
ρ
11
u
s
2
ρ
22
u
f
2
2
1
ρ
12
u
s
u
f
1
E
c
=
+
·
+
(6.32)
ρ
11
,
ρ
12
and
ρ
22
being parameters depending on the nature and the geometry of the
porous medium and the density of the fluid. This expression presents the invariance of
the kinetic energy, and, by using a Lagrangian formulation, leads to inertial forces that
do not contain derivatives higher than second order in
t
. The components of the inertial
forces acting on the frame and on the air are, respectively
∂
∂t
∂E
c
∂ u
s
i
=
ρ
12
u
i
q
i
=
ρ
11
u
s
i
+
i
=
1
,
2
,
3
(6.33)
∂
∂t
∂E
c
∂ u
i
=
q
i
=
ρ
22
u
i
ρ
12
u
s
i
+
i
=
1
,
2
,
3
(6.34)
An inertial interaction exists between the frame and the air that creates an inertial
force on one element due to the acceleration of the other element. This interaction can
appear in the absence of viscosity. It has been described by Landau and Lifschitz (1959)
for the case of a sphere moving in a fluid. The interaction creates an apparent increase
in the mass of the sphere. The coefficients
ρ
11
,
ρ
12
and
ρ
22
are related to the geometry
of the frame, and do not depend on the frequency. The following 'gedanken experiment'
