Civil Engineering Reference
In-Depth Information
k
0
2
)
,and
D
2
=
2
µk
0
. If the origin of the
x
3
axis is fixed at point
M
, the vectors
V
s
(M)
and
V
s
(M
)
are expressed as
V
s
(M)
=
[
(
0
)
]
A
V
s
(M
)
=
[
(h)
]
A
λ(k
0
2
k
13
2
)
+
2
µk
13
2
µ(k
13
2
with
D
1
=
+
=
−
(11.18)
Then, the transfer matrix [
T
] which relates
V
s
(M)
and
V
s
(M
)
is equal to [
T
]
=
[
(
0
)
][
(h)
]
−
1
. In order to alleviate the instability that may arise from the inversion of
matrix [
(h)
], the origin of the
x
3
axis can be rather fixed at point
M
, and the transfer
matrix written:
[
T
s
]
=
[
(
−
h)
][
(
0
)
]
−
1
(11.19)
The inversion of the matrix [
(
0
)
] is calculated analytically
2
k
1
ωδ
3
1
µδ
3
0
−
0
k
33
−
k
1
ωk
13
δ
3
k
1
µk
13
δ
3
0
0
−
[
(
0
)
]
−
1
=
(11.20)
k
1
ωδ
3
1
µδ
3
0
0
k
33
−
k
1
ωk
33
δ
3
k
1
µk
33
δ
3
0
−
0
11.3.3 Poroelastic layer
The acoustic field in a layer of porous materials
In the context of Biot theory, three kinds of wave can propagate in a porous medium:
two compressional waves and a shear wave (Chapter 6). Let us denote by
k
1
,
k
2
,
k
3
,and
k
1
,
k
2
,
k
3
, the wave number vectors of the compressional waves (subscripts 1, 2) and the
shear wave (subscript 3). The nonprimed vectors correspond to waves propagating for-
ward while the primed vectors correspond to waves propagating backward. Let
δ
1
,δ
2
and
δ
3
, be the squared wave numbers of the two compressional waves and of the shear wave.
These quantities are given by Equations (6.67), (6.68) and (6.83). The
x
3
components of
the wave number vectors are
k
i
3
=
(δ
i
−
k
t
)
1
/
2
i
=
1
,
2
,
3
(11.21)
k
i
3
=−
k
i
3
i
=
1
,
2
,
3
The square root symbol
()
1
/
2
represents the root which yields a positive real part.
In the complex representation, the frame displacement potentials of the compressional
waves can be written as
ϕ
i
=
A
i
exp
(j(ωt
A
i
exp
(j(ωt
−
k
i
3
x
3
−
k
t
x
1
))
+
+
k
i
3
x
3
−
k
t
x
1
))
i
=
1
,
2
(11.22)
