Civil Engineering Reference
In-Depth Information
impedance at
A
can be obtained by Equation (9.21) modified in the following way. The
impedance
Z
c
in Equation (9.21) must be replaced by the impedance at
B
at normal
incidence of the layered material, including the layer of air, with the facing removed.
The second case is represented in Figure 9.6(b). A porous layer is in contact with the
facing. The distortion of the flux under the facing is now located in a porous layer,
and this distortion creates not only an inertial effect, but also a resistive effect. The
same procedure as in Section 9.2 can be used to evaluate these different effects. The
difference is the presence of porous layers, the finite length of the elementary cell, and
the presence of two waves in every medium with opposite values of the
x
3
component
of the wave number vector.
The different layers of porous material are represented in Figure 9.7.
Let
k(
1
)
be the wave number in layer 1. The pressure field in this layer can be written
=
m,n
p(x
1
,x
2
,x
3
)
A
m,n
(
1
)
[exp
(
−
jx
3
k
m,n
(
1
))
+
exp
(
−
j(
2
X
3
−
x
3
)k
m,n
(
1
))
]
(9.32)
×
cos
2
mπx
1
D
cos
2
nπx
2
D
In this equation,
k
m,n
(
1
)
is given by Equation (9.3) which can be rewritten
k
2
(
1
)
1
/
2
4
m
2
π
2
D
2
4
n
2
π
2
D
2
k
m,n
(
1
)
=
−
−
(9.33)
and
X
3
is the total thickness of the layered material. The modes related to different sets
(
m,n)
propagate independently in the cell.
The field
υ
3
,m,n
(x
1
,x
2
,x
3
)
of the
x
3
velocity component related to the (
m
,
n
) mode
is equal to zero in layer 1 at the contact surface with the backing. Let
M
1
and
M
1
be two
points located close to the contact surface between the layers 1 and 2,
M
1
in medium 1
and
M
1
in medium 2. Let
p
m,n
(M)
and
υ
3
,m,n
(M)
be the pressure and the
x
3
velocity
component at
M
related to the (
m,n)
mode. The impedance at
M
1
related to the (
m,n)
mode is given by
p
m,n
(M
1
)
υ
3
,m,n
(M
1
)
=−
jZ
c
(
1
)
k(
1
)
Z
m,n
(M
1
)
=
k
m,n
(
1
)
cot
k
m,n
(
1
)l
1
(9.34)
A
X
1
B
(L)
I
L
M
′
L
−
1
X
3
=
I
L
M
L
−
1
M
′
2
M
2
M
′
1
M
1
X
3
=
I
L
+ ... +
I
2
(1)
I
1
X
3
=
I
L
+ ... +
I
1
X
3
Figure 9.7
The different layers of the stratified material in the elementary cell.
