Civil Engineering Reference
In-Depth Information
where
φ(L)
is the porosity of the
L
th
medium, and Equation (9.11) becomes for the case
n
or
m
=
0
4
ν
m,n
URZ
c
(L)k(L)
J
1
[
(
2
πR/D)(m
2
+
n
2
)
1
/
2
]
A
m,n
(L)
=
φ(L)D(m
2
n
2
)
1
/
2
k
m,n
(L)(
1
−
+
β
m,n
)
(9.44)
ν
1
,
0
=
ν
0
,
1
=
1
/
2
,ν
m,n
=
1if
n
and
m
=
0
For the case
n
=
m
=
0
,A
0
,
0
(L)
is given by
πR
2
U
φ(L)(
1
Z
c
(L)sU
φ(L)(
1
A
0
,
0
(L)
=
β
0
,
0
)D
2
Z
c
(L)
=
(9.45)
−
−
β
0
,
0
)
Let
B
be a point located close to
B
, in a perforation above the
L
th layer. The impedance
Z(B
)
is
p/U
,where
p
is the average pressure over the aperture
πR
2
R
2
π
1
β
m,n
)
cos
2
πmx
1
D
cos
2
πnx
2
D
p
=
A
m,n
(L)(
1
+
r
d
r
d
θ
(9.46)
0
0
m,n
By the use of Equations (9.8) and (9.9),
Z(B
)
can be written
J
1
2
π
D
(m
2
n
2
)
sZ
c
(L)
1
π
m,n
1
φ(L)
+
β
0
,
0
4
ν
m,n
φ(L)
+
Z(B
)
=
+
1
−
β
0
,
0
(m
2
+
n
2
)
(9.47)
Z
c
(L)k(L)
k
m,n
(L)
(
1
+
β
m,n
)
×
(
1
−
β
m,n
)
This equation can be simplified by using Equation (9.40):
J
1
2
π
D
(m
2
+
n
2
)
Z
m,n
(B)
(m
2
π
m,n
sZ
0
,
0
(B)
φ(L)
+
4
v
m,n
φ(L)
Z(B
)
=
(9.48)
+
n
2
)
The inertial term j
(ε
e
d)ρ
0
ω
,where
ε
e
is given by Equation (9.17), must be taken into
account, but the effect of the viscous forces in the aperture and on the faces of the facing
around the aperture can be neglected when
Z(A)
is evaluated
+
Z(A)
=
Z(B
)
+
j(ε
e
+
d)ρ
0
ω
(9.49)
Finally, the impedance
Z
in the free air close to the facing is
Z
=
Z(A)/s
(9.50)
If the porous materials are transversally isotropic with a symmetry axis
Ox
3
, like the glass
wools described in Chapter 3, two wave numbers
k(i)
and
k
p
(i)
and two characteristic
impedances
Z
c
(i)
and
Z
p
(i)
in the
x
3
direction and the directions parallel to the plane
x
1
Ox
2
exist. The previous description is always valid but
k
m,n
(i)
is now given by
k(i)
1
1
/
2
2
πm
D
2
2
πn
D
2
1
k
p
(i)
−
1
k
p
(i)
k
m,n
(i)
=
−
(9.51)
