Civil Engineering Reference
In-Depth Information
As for isotropic media, the poles are close with the Biot theory and for the frame
in vacuum in the whole audible frequency range. The same trends can be observed for
cases a and b. The general properties of the Rayleigh wave for axisymmetrical poroelastic
media are the same as those reported by Royer and Dieulesaint (1996) for axisymmetrical
elastic media. The specific properties, as for isotropic porous media, are related to the
large structural damping and the small rigidity of the porous frames of porous sound
absorbing media. The order of magnitude of the phase speed is 50 m/s. Measurements of
the speed are possible close to the line source, at distances of 5 cm or less at acoustical
frequencies. Layers having a thickness larger than 3 cm can generally be used for phase
speed measurements in the high frequency range, as shown in Section 8.4 for isotropic
media.
10.9
Transfer matrix representation of transversally
isotropic poroelastic media
Only the case where the axis of symmetry is parallel to the normal to the faces is consid-
ered and there are no contributions of the waves polarized perpendicular to the meridian
plane. A transfer matrix representation with a different normalization was performed
by Vashishth and Khurana (2004). Another matrix description obtained with the Dazel
representation of the Biot theory (see Appendix 6.A) is given by Khurana
et al
. (2009).
The transfer matrix
The displacement field and the stress field are described by the column vector
V
(z)
defined by
V
(z)
=
[
u
s
x
,u
s
z
,w
z
,σ
zz
,σ
zx
,p
]
T
(10.126)
The normalization factors
N
for the three waves which propagate downwards are
denoted as
N
i
,i
=
1
,
2
,
3, and the normalization factors for the three waves which
propagate upwards are denoted as
N
i
,i
=
1
,
2
,
3. The column vector
A
is defined by
[
N
1
N
1
,N
1
N
1
,N
2
N
2
,N
2
N
2
,N
3
N
3
,N
3
N
3
]
T
A
=
+
−
+
−
+
−
(10.127)
The vector
V
can be related to
A
by
V
(z)
=
[
(z)
]
A
(10.128)
Let
q
(1),
q
(2) and
q
(3) be the three solutions of Equation (10.46) with a positive real
part. Let
a
1
(i),a
3
(i),b
1
(i),b
3
(i)
the solutions of the set of Equations (10.47) - (10.50)
for
N
=
1and
q
=±
q(i),i
=
1
,
2
,
3. If
N
remains equal to 1 when
q(i)
→−
q(i),
a
1
a
1
,b
1
b
1
,a
3
a
3
and
b
3
b
3
. The matrix elements
kl
are given by
=−
=−
=
=
1
,
2
i
−
1
=−
ja
1
(i)
sin
ωq(i)z
(10.129)
a
1
(i)
cos
ωq(i)z
1
,
2
i
=
(10.130)
a
3
(i)
cos
ωq(i)z
2
,
2
i
−
1
=
(10.131)
