Civil Engineering Reference
In-Depth Information
heading (direction of propagation) in the plane (
x
1
x
3
). The example of a thick orthotropic
panel is discussed in Chapter 12.
11.4
Coupling transfer matrices
The transfer matrices of various types of layer were evaluated in the previous section. This
section is devoted to the continuity conditions between two adjacent layers of different
nature. Figure 11.3 illustrates a stratified medium, where two points
M
2
k
and
M
2
k
+
1
(
k
=
1
,n
−
1) are close to each other at each side of a boundary between layers (
k)
and
(
k
+
1). An interface matrix, which depends on the nature of the two layers, must be used
to relate the acoustic field vectors
V
(k)
(
M
2
k
)
and
V
(k
+
1
)
(
M
2
k
+
1
)
. For simplification, the
interface matrices are derived for the two first layers of Figure 11.3.
11.4.1 Two layers of the same nature
If the two adjacent layers have the same nature, the continuity conditions are exploited
to build a global transfer matrix which describes the acoustic propagation between
M
1
and
M
4
. If the two layers are not porous, the global transfer matrix is simply equal to
product of the transfer matrices of the two layers. But, if the two layers are porous, the
continuity conditions are affected by the porosities of the layers:
v
1
(M
2
)
v
1
(M
3
)
=
v
3
(M
2
)
v
3
(M
3
)
φ
1
(v
3
(M
2
)
=
φ
2
(v
3
(M
3
)
v
3
(M
3
))
σ
33
(M
2
)
+
σ
33
(M
2
)
=
σ
33
(M
3
)
+
σ
33
(M
3
)
σ
13
(M
2
)
=
σ
13
(M
3
)
σ
33
(M
2
)
φ
1
v
3
(M
2
))
−
=
−
(11.65)
σ
33
(M
3
)
φ
2
=
In this case, the global transfer matrix [
T
p
] is written as:
[
T
p
]
=
[
T
1
][
I
pp
][
T
2
]
(11.66)
Fluid 1
(1)
(2)
(n)
θ
M
1
M
2
M
3
M
4
M
2n-1
M
2n
A
Figure 11.3
Plane wave impinging on a multilayer domain.
