Civil Engineering Reference
In-Depth Information
where
Z
c
(
1
)
is the characteristic impedance and
l
1
the thickness of layer 1 which is
fixed on a rigid impervious wall. Let
φ(
1
)
and
φ(
2
)
be the porosities of layer 1 and 2,
respectively. The pressures and the
x
3
components of the velocities at
M
1
and
M
1
for
the (
m,n)
mode are related by
φ(
1
)
φ(
2
)
υ
3
,m,n
(M
1
)
υ
3
,m,n
(M
1
)
=
(9.35)
p
m,n
(M
1
)
=
p
m,n
(M
1
)
(9.36)
and
Z
m,n
(M
1
)
is related to
Z
m,n
(M
1
)
by
φ(
2
)
φ(
1
)
Z
m,n
(M
1
)
Z
m,n
(M
1
)
=
(9.37)
Let
M
2
and
M
2
be two points located close to the boundary surface between layers 3
and 2,
M
2
in medium 2, and
M
2
in medium 3. Using the impedance translation formula
Equation (3.36), the impedance at
M
2
related to the mode (
m,n)
can be written
Z
c
(
2
)(k(
2
)/k
m,n
(
2
))
Z
m,n
(M
)
−
jZ
c
(
2
)(k(
2
)/k
m,n
(
2
))
cotg
k
m,n
(
2
)l
2
×
[
−
Z
m,n
(M
2
)
=
(9.38)
jZ
m,n
(M
1
)
cotg
k
m,n
(
2
)l
2
+
Z
c
(
2
)(k(
2
)/k
m,n
(
2
))
]
In this equation
l
2
is the thickness and
Z
c
(
2
)
the characteristic impedance of the second
layer, and
k
m,n
(
2
)
and
k(
2
)
are related by Equation (3.4). The impedances at the bound-
aries of the layers related to each mode can be obtained by this way up to
B
in the
L
th
layer in contact with the facing. The pressure field in the
L
th
layer can be written
p(x
1
,x
2
,x
3
)
=
[
A
m,n
(L)
exp
(
−
jk
m,n
(L)x
3
)
m,n
(9.39)
B
m,n
(L)
exp
(jk
m,n
(L)x
3
)
]cos
2
πmx
1
D
cos
2
πnx
2
D
+
The coefficients
A
m,n
(L)
and
B
m,n
(L)
are related to the impedance
Z
m,n
(B)
by
A
m,n
(L)
+
B
m,n
(L)
A
m,n
(L)
Z
c
(L)k(L)
k
m,n
(L)
Z
m,n
(B)
=
(9.40)
−
B
m,n
(L)
and
B
m,n
(L)
is related to
A
m,n
(L)
by
B
m,n
(L)
=
β
m,n
A
m,n
(L)
(9.41)
Z
m,n
(B)k
m,n
(L)
−
Z
c
(L)k(L)
Z
m,n
(B)k
m,n
(L)
β
m,n
=
(9.42)
+
Z
c
(L)k(L)
Equation (9.5) can be rewritten
φ(L)
=
m,n
Y
(R
−
r)
U
k
m,n
(L)
A
m,n
(L)(
1
−
β
m,n
)
Z
c
(L)k(L)
(9.43)
×
cos
2
πmx
1
D
cos
2
πnx
2
D
