Civil Engineering Reference
In-Depth Information
Thin plate - porous interface
For a thin plate in contact with a porous layer, the continuity conditions are
υ
3
(
M
2
)
=
υ
3
(
M
3
)
υ
3
(
M
2
)
=
υ
3
(
M
3
)
(11.76)
σ
33
(
M
2
)
σ
33
(
M
2
)
+
=−
p(
M
3
)
σ
13
(
M
2
)
=
0
with
p
σ
33
, the pressure at the plate's interface with the porous media. These con-
ditions can be rewritten as [
I
pi
]
V
p
(M
2
)
=−
+
[
J
pi
]
V
i
(M
3
)
=
0, with
010000
001000
000101
000010
0
−
1
0
−
1
10
00
[
I
pi
]
=−
,
[
J
pi
]
=
(11.77)
11.5
Assembling the global transfer matrix
A simple product of transfer and interface matrices cannot generally be used to calculate
the global transfer matrix of a stratified medium, since most of the interface matrices are
not square. On the other hand, Equation (11.69) relates the acoustic field vectors at the
right-hand-side boundaries of adjacent layers in the stratified medium. For the medium
illustrated by Figure 11.3, this equation leads to the following relations
[
I
f
1
]
V
f
(A)
[
J
f
1
][
T
(
1
)
]
V
(
1
)
(M
2
)
0
[
I
(k)(k
+
1
)
]
V
(k)
(M
2
k
)
+
[
J
(k)(k
+
1
)
][
T
(k
+
1
)
]
V
(k)
(M
2
(k
+
1
)
)
=
0
,k
=
1
,...,n
−
1
+
=
(11.78)
This set of equations can be rewritten in the form [
D
0
]
V
0
=
0, where
[
I
f
1
]
J
f1
][
T
(1)
]
[0]
···
[0]
[0]
[
J
12
][
T
(2)
]
[0]
[
I
12
]
···
[0]
[0]
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
[
D
0
]
=
[
J
(
n
−
2
)(
n
−
1
)
][
T
(
n-1
)
]
[0]
[0]
[0]
···
[0]
[
J
(
n-1
)(
n
)
][
T
(
n
)
]
···
[
I
(
n-1
)(
n
)
]
[0]
[0]
[0]
(11.79)
and
=
υ
f
(A) υ
(
1
)
(M
2
)υ
(
2
)
(M
4
)
υ
(
n
−
1
)
(M
2
n
−
2
)υ
(n)
(M
2
n
)
T
(11.80)
υ
···
0
Matrix [
D
0
] is rectangular. However, the global transfer matrix must be square since
the physical problem is well posed. At the excitation side, one impedance equation
linking the pressure to the normal velocity is missing. At the termination side, impedance
conditions relating the field variables are missing. These impedance conditions are three
for a porous layer, two for a solid layer and one for a fluid, thin plate or impervious layer.
Thus, if
N
is the dimension of
V
0
;[
D
0
]has(
N
−
4) rows if the last layer is porous,
