Civil Engineering Reference
In-Depth Information
surface. The condition
τ
=
0 at the air - frame contact surface is due to the fact that the
density of the porous media used for sound absorption and damping is generally much
heavier than air, and the thermal exchanges with the saturating air do not modify the
temperature of the frame. This condition must be fulfilled for the previous description
of the bulk modulus to be valid. The bulk modulus of the saturating air depends on the
thermal permeability
q
via the averaged density. From Equation (4.40), the averaged
density
<ξ
>
is given by
q
(ω)jωp
φκ
ρ
0
P
0
p
ρ
0
T
0
<ξ>
=
−
(5.5)
From Equation (4.39), and using
P
0
/T
0
=
ρ
0
(c
p
−
c
v
)
, the bulk modulus of the sat-
urating air can be written
K(ω)
=
P
0
/
1
−
j
B
2
ωρ
0
q
(ω)
φη
γ
−
1
γ
(5.6)
where B
2
is the Prandtl number. When
ω
tends to zero,
q
tends to the static thermal
permeability
q
0
. It has been shown by Torquato (1990) that
q
0
≥
q
0
. A comparison of
Equations (5.6) and (4.102) shows that both parameters are equal for identical cylindrical
pores parallel to the direction of propagation. From Equations (4.45) - (4.46) and Equation
(5.6) it can be shown that
q
(ω)
q(B
2
ω)
for identical cylindrical pores. When the
density of the frame has the same order of magnitude as the air, the isothermal limit for
the bulk modulus cannot be reached (Lafarge
et al
. 1997), and the previous description
must be modified.
=
Static permeabilities and low frequency limits of
ρ
and K
The limit of the ratio
ρ/
[
φσ/(jω)
]is1when
ω
→
0 as shown in Chapter 4 for cylindrical
pores. Norris (1986) has shown that this relation was always valid. More precisely, the
limit when
ω
→
0 can be written
ηφ
jωq
0
+
ρ(ω)
=
cte
(5.7)
The nature of the constant is shown explicitly in Section (5.3.6).
The limit of the bulk modulus at the first order approximation in
ω
is obtained from
Equation (5.6) which can be rewritten
P
0
1
+
jB
2
ωρ
0
q
0
φη
γ
−
1
γ
K(ω)
=
(5.8)
This equation is the generalization of Equation (4.102) obtained for the case of parallel
identical cylindrical pores. The static thermal permeability
q
0
is the limit when
ω
tends
to zero of
C
Im
K(ω)
where
C
is given by
γφη
P
0
(γ
−
1
)B
2
ρ
0
ω
C(ω)
=
(5.9)
