Civil Engineering Reference
In-Depth Information
β
+d
β
β
AIR
FRAME
Figure 5.B.1
The frame -air boundary in a porous material.
At high frequencies, the spatial dependence of temperature close to the surface of the
pore is the same as if the surface were an infinite plane. Equation (4.33) can be rewritten
∂
2
τ
∂β
2
−
j
ω
ρ
o
τ
η
j
ω
v
ρ
o
p
κη
=−
(5.B.1)
The solution which vanishes at
β
=
0is
v
p
κ
(
1
τ
=
−
exp
(
−
jβgB))
(5.B.2)
where
g
is given by
1
−
j
g
=
(5.B.3)
δ
In a volume of homogenization
V,
τ
is given by
τ
d
V
d
V
=
1
−
exp
(
−
jβgB)
d
V
v
p
κ
1
V
τ
=
(5.B.4)
V
Very close to the frame - air boundary, the volume d
V
of air related to d
β
is equal
to
A
d
β
,
A
being the area of the frame that is in contact with the air in the volume
V
of porous material. The quantity exp
(
jβgB)
decreases very quickly when the distance
β
from M to the surface increases, and the integral in the right-hand side of Equation
(5.B.4) is rewritten in Champoux and Allard (1991)
−
A
jBg
=
Aδ(
1
−
j)
exp
(
−
jβgB)A
d
β
=
(5.B.5)
2
B
v
and
<τ>
is given by
1
−
v
p
κ
j)
δ
B
τ
=
(
1
−
(5.B.6)
where 2/
=
A/V.
