Civil Engineering Reference
In-Depth Information
ja
3
(i)
sin
ωq(i)z
2
,
2
i
=−
(10.132)
b
3
(i)
cos
ωq(i)z
3
,
2
i
−
1
=
(10.133)
3
,
2
i
=−
jb
3
(i)
sin
ωq(i)z
(10.134)
B
7
j
ω
c
jb
1
(i)
sin
ωq(i)z
B
4
jωq(i)a
3
(i)j
sin
ωq(i)z
4
,
2
i
−
1
=−
+
(10.135)
B
3
j
ω
B
7
jωq(i)b
3
(i)j
sin
ωq(i)z
c
ja
1
(i)
sin
ωq(i)z
−
+
4
,
2
i
=
B
7
j
ω
c
b
1
(i)
cos
ωq(i)z
−
B
4
jωq(i)a
3
(i)
cos
ωq(i)z
+
B
7
jωq(i)b
3
(i)
cos
ωq(i)z
−
B
3
j
ω
(10.136)
c
a
1
(i)
cos
ωq(i)z
B
5
jω
jωqa
1
cos
ωq(i)z
c
a
3
(i)
5
,
2
i
−
1
=−
+
(10.137)
B
5
jω
jωqa
1
j
sin
ωq(i)z
c
a
3
(i)
5
,
2
i
=
+
(10.138)
6
,
2
i
−
1
=−
B
8
j
ω
c
jb
1
(i)
sin
ωq(i)z
+
B
7
jωq(i)a
3
(i)j
sin
ωq(i)z
−
B
8
jωq(i)b
3
(i)j
sin
ωq(i)z
+
B
6
j
ω
(10.139)
c
ja
1
(i)
sin
ωq(i)z
B
8
j
ω
c
b
1
(i)
cos
ωq(i)z
B
7
jωq(i)a
3
(i)
cos
ωq(i)z
6
,
2
i
=
−
(10.140)
B
6
j
ω
B
8
jωq(i)b
3
(i)
cos
ωq(i)z
c
a
1
(i)
cos
ωq(i)z
+
−
The vector
V
(l)
at the upper surface is related to the vector
V
(
0
)
at the lower surface
of a layer of thickness
l
by
=
[
(l)
][
(
0
)
]
−
1
V
(
0
)
V
(l)
(10.141)
The
transfer
matrix
for
the
porous
layer
in
the
second
Biot
representation
is
[
(l)
][
(
0
)
]
−
1
. All the components of
V
are equal at each side of the boundary
between two porous layer with frames bonded together and the transfer matrix of a
layered medium is the product of the transfer matrices of each layer.
[
T
]
=
Surface impedance Z
s
of a layered porous medium
The porous medium is glued to a rigid impervious layer. Three components of
V
(l)
are
equal to 0,
u
x
,u
z
,and
w
z
. In the free air close to the upper face, the
z
component of the
air velocity
v
z
is related to the pressure by
p
=
Z
s
v
z
with
v
z
=
(w
z
+
u
z
)jω
, the total
stress component
σ
zz
=−
p
and
σ
zx
=
0. The three components equal to 0 are related to
the components of
V
at the upper surface by, respectively
t
13
t
16
p
1
jωZ
s
−
t
11
u
s
x
+
t
13
)u
s
z
+
0
=
(t
12
−
t
14
+
(10.142)
