Civil Engineering Reference
In-Depth Information
The waves have a given slowness component 1/
c
in the
X
direction. The symbol
q
is
used to denote the
z
slowness vector component. This should not lead to any confusion
with the dynamic viscous permeability defined in Chapter 5 which does not appear in
this chapter except in Table 10.1. Two kinds of waves exist with a polarization in the
meridian plane, a pseudo-compressional wave (pseudo-P wave) and a pseudo-shear wave
(pseudo-SV wave). The components of the slowness vector must satisfy the relation (see
Equation 4.49 in Royer and Dieulesaint 1996)
G
q
2
c
2
1
2
G
+
A
Cq
2
2
ρ
1
=
+
+
+
c
2
2
G
1
/
2
C
q
2
2
(10.11)
G
+
G
−
4
F
G
2
q
c
+
A
−
±
+
+
c
2
where
ρ
1
is the density of the porous frame. The slowest wave is the pseudo-SV wave.
A transverse wave polarized perpendicular to the meridian plane (SH wave) can
also propagate with components of the slowness vector satisfying the relation (see
Equation 4.47 in Royer and Dieulesaint 1996)
G
c
2
+
G
q
2
ρ
1
=
(10.12)
The pseudo-transverse wave is purely transverse when the wave number vector is parallel
to the axis of symmetry or perpendicular to this axis. The pseudo-compressional wave
is purely compressional in the same conditions. It can be guessed that, if the frame is
much heavier than air, three frame-borne waves similar to the P wave, the SV wave and
the SH wave must propagate in the air saturated porous medium, plus an air borne wave
similar to the wave which propagates in the air saturating the motionless frame.
10.3
Transversally isotropic poroelastic layer
10.3.1 Stress - strain equations
The bulk modulus and the acoustical parameters related to the bulk modulus, thermal
characteristic length and viscous permeability, are scalar. As in Chapter 6, the material
the frame is made of is supposed to be incompressible, and the stress - strain relations
for the saturating air are given by Equation (6.3) with
Q
given by Equation (6.25) and
R
given by Equation (6.26)
σ
ij
=−
φpδ
ij
=
K
f
[
φθ
f
+
(
1
−
φ)θ
s
]
(10.13)
where
θ
f
is the air dilatation and
θ
s
is the frame dilatation. The second representation of
the Biot (1962) theory (Appendix 6.A) is used. In this representation, the pressure and the
total stress tensor acting on the porous medium are used. The total stress tensor is the sum
of three tensors: the stress tensor of the frame in vacuum considered as an elastic solid,
whose components
σ
ij
are given by Equations (10.1) -(10.6); the stress tensor components
σ
ij
φ)σ
ij
/φ
added to the
=−
φpδ
ij
; and the stress components
−
p(
1
−
φ)δ
ij
=
(
1
−
