Civil Engineering Reference
In-Depth Information
(
N
−
2) rows if it is fluid (or equivalent
fluid), thin plate or an impervious screen. The impedance conditions at the termination
side depend closely on the nature of the termination: hard wall or semi-infinite fluid
domain.
−
3) rows if this layer is elastic solid, and (
N
11.5.1 Hard wall termination condition
If the multilayer is backed by a hard wall (Figure 11.4) and if a component of the field
vector
V
(n)
(
M
2n
)
is a velocity, then this component is equal to zero (infinite impedance).
These conditions may be written in the form [
Y
(n)
]
V
(n)
(M
2
n
)
0, where [
Y
(n)
]and
=
V
(n)
are defined according to the nature of the layer (
n)
.
1000
0100
,
100000
010000
001000
[
Y
f
]
=
01
[
Y
p
]
=
,
[
Y
s
]
=
(11.81)
where the superscript refers to the nature of the layer in contact with the wall: p for
porous, s for elastic solid, and f for fluid, equivalent fluid, thin plate or impervious screen.
Otherwise, vector
V
(n)
is the field variable vector of the layer in contact with the wall.
Adding the new equations to the previous system, [
D
0
]
V
0
=
0, a new system is
obtained, whose matrix [
D
]has(
N
−
1) rows and
N
columns:
,
[
D
0
]
[
D
]
V
=
0:[
D
]
=
V
=
V
0
(11.82)
[
Y
(n)
]
[0]
···
[0]
11.5.2 Semi-infinite fluid termination condition
If the multilayer is terminated with a semi-infinite fluid layer (Figure 11.5), continuity
conditions may be written to relate the vectors
V
(n)
(M
2
n
)
and the semi-infinite fluid
vector
V
f
(B)
,where
B
is a point in the semi-infinite medium, close to the boundary.
These conditions are expressed as
[
I
(n)f
]
V
(n)
(M
2
n
)
+
[
J
(n)f
]
V
f
(B)
=
0
(11.83)
(1)
(2)
(n)
Fluid 1
A
M
1
M
2
M
3
M
4
M
2n
M
2n-1
Figure 11.4
A multilayer domain backed by a hard wall.
