Civil Engineering Reference
In-Depth Information
where [
T
1
]et[
T
2
] are the transfer matrices of the two porous layers, and [
I
pp
]isa
6
×
6 interface matrix built from Equations (11.65)
1
0
0
0
0
0
0
1
0
0
0
0
1
φ
2
φ
1
φ
2
φ
1
0
−
00
0
1
−
[
I
pp
]
=
φ
1
φ
2
(11.67)
0
0
0
1
0
0
0
0
0
1
0
φ
1
φ
2
0
0
0
0
0
×
Note that this interface matrix is equal to the 6
6 unit matrix if the two layers have the
same porosity.
The transfer matrix of a porous layer using the second Biot representation (Appendix
6.A) is given in Section 10.9 of Chapter 10 in the case of a transversally isotropic
medium. The matrix for an isotropic medium is a special case. In this representation the
components of
V
(z)
[
u
s
x
,u
s
z
,w
z
,σ
zz
,σ
zx
,p
]
T
are equal at each side of the boundary
between two porous layers with frames bonded together. The interface matrix is a unit
matrix in this representation and the transfer matrix of a layered porous medium is simply
the product of the transfer matrices of each layer.
=
11.4.2 Interface between layers of different nature
When the adjacent layers have different natures, the continuity equations may be used
to relate the two interface matrices [
I
12
]and[
J
12
] to the field variable vectors at
M
1
and
M
2
[
I
12
]
V
(
1
)
(M
2
)
[
J
12
]
V
(
2
)
(M
3
)
+
=
0
(11.68)
Matrices [
I
12
]and[
J
12
] depend on the nature of the two interfacing layers. The
number of rows of the two matrices is equal to the number of continuity equations at
the interface. These interface matrices relate the field vectors at points
M
2
and
M
3
,by
Equation (11.68). Since
V
(
2
)
(M
3
)
=
[
T
(
2
)
]
V
(
2
)
(M
4
)
where [
T
(
2
)
] is the transfer matrix of
the second layer, the acoustic propagation between the points
M
2
and
M
4
is expressed by
[
I
12
]
V
(
1
)
(M
2
)
+
[
J
12
][
T
(
2
)
]
V
(
2
)
(M
4
)
=
0
(11.69)
The expressions for the interface matrices for various interfaces are given in the
following.
Solid - fluid interface
The continuity conditions are given by
v
3
(
M
2
)
v
3
(
M
3
)
=
σ
33
(
M
2
)
(11.70)
=−
p
(
M
3
)
σ
13
(
M
2
)
=
0
