Civil Engineering Reference
In-Depth Information
For a transversally isotropic medium, one of these directions is the axis of symmetry
Z
.
Any couple of orthogonal axes
X
and
Y
in the symmetry plane can complete the set. The
superscript
i,i
, ρ
i
12
, ρ
i
22
.
The viscous and inertial interactions between both phases when velocities are parallel to a
direction where the parameters are diagonal are the same as for an isotropic medium with
the parameters equal to the diagonal elements. The wave equations in the first representation
of the Biot theory can be written
∂σ
xi
x,y,z
will be used to define the diagonal elements
σ
i
,
i
,α
i
=
∞
∂σ
yi
∂y
+
∂σ
zi
ρ
i
12
u
i
)
ω
2
( ρ
i
11
u
s
i
+
∂x
+
∂z
=−
(10.28)
∂σ
yi
∂y
+
∂σ
xi
∂σ
zi
ω
2
( ρ
i
22
u
i
+
ρ
i
12
u
s
i
)
∂x
+
∂z
=−
(10.29)
i
=
x,y,z
where
ρ
i
22
=
ρ
i
12
φρ
0
−
(10.30)
ρ
i
11
=
ρ
i
12
ρ
1
−
(10.31)
In these equations
ρ
0
is the density of air,
φ
is the porosity, and
ρ
12
is defined similarly
as in Chapter 6. Using Equations (10.28) - (10.29) with the stresses and the strains of the
second representation gives the following wave equations (to alleviate the presentation we
use the equivalent notations:
σ
ij
,k
=
∂
σ
ij
/∂x
k
,i,j,k
=
1
→
x,
2
→
y,
3
→
z)
t
1
i,
1
t
2
i,
2
t
3
i,
3
=−
ω
2
(ρu
s
i
+
ρ
0
w
i
)
σ
+
σ
+
σ
(10.32)
where
ρ
=
ρ
1
+
φρ
0
,and
ω
2
ρ
0
u
s
i
+
φ
2
w
i
ρ
i
22
−
p
,i
=−
(10.33)
Using Equation (5.50), this equation can be rewritten
ω
2
(ρ
0
u
s
i
+
−
p
,i
=−
C
i
w
i
)
(10.34)
where
ρ
0
α
i
σ
i
G
i
(ω)
jω
is related to the effective density in the direction
i
by
C
i
=
ρ
i
ef
/φ
.
φ
+
C
i
=
10.4
Waves with a given slowness component in the
symmetry plane
10.4.1 General equations
The meridian plane is defined in the same way as in Section (10.2). It contains the
Z
axis
and the wave number vector. In this plane the direction perpendicular to
Z
is denoted by
