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In-Depth Information
−
ρ
f
ρ
T
f
=
C
∇
u
s
I
+
M
∇
w I
+2
C
G
t
d
s
2
A
2
=
Cl
2
A
1
Ml
2
A
2
F
ρ
f
−
ω
−
−
2 144
+2
M
G
t
d
w
,
+
ρ
f
ρ
l
2
A
1
+
Cl
2
A
2
+
i
H
ω
ω
A
2
b
ω
G
U
t
=FT
−
1
G
U
ω
,
2 145
2 155
C
G
t
=FT
−
1
C
G
ω
,
2 146
Eliminating
A
1
and
A
2
between these two equations
yields
the
following
equation
for
the
speed:
M
G
t
=FT
−
1
M
G
ω
,
2 147
az
p
+
bz
p
+
c
= 0, where
z
p
=
ω
l
is the complex speed and
where the deviators
d
s
and
d
w
have been defined previ-
ously (see Eqs. 2.53 and 2.55), where
d
s
is the mean
strain deviator of the solid phase, and the coefficients
G
U
(
t
),
G
U
(
ρ
−
ρ
f
F
+
i
b
ω
ω
ρ
ρ
f
a
=
,
2 156
ω
),
C
G
(
t
),
C
G
(
ω
),
M
G
(
t
), and
M
G
(
ω
) are defined
M
ρ
ρ
f
2
C
F
+
i
b
ω
ωρ
f
b
=
−
1
+
H
ω
−
H
ω
,
2 157
in Section 2.1.3.1.
We use the classical decomposition of the displace-
ments into dilatational components
c
=
1
ρ
f
H
ω
M
−
C
2
2 158
∇
u
s
=
e
,
2 148
The two roots of this equation are
∇
w
=
ς
2 149
=
−
b
+
b
2
−
4
ac
2
z
p
,
2 159
The dilatational wave propagation is obtained by
applying the divergence operator to Equations (2.141)
and (2.142) and then inserting Equations (2.148) and
(2.149) into the resulting equation. This yields
2
a
=
−
b
−
b
2
−
4
ac
2
z
I
p
,
2 160
2
a
ρ
−
ρ
f
F
1
F
∇
with
z
p
>
z
I
p
. These two waves correspond to the complex
wave speeds associated with the fast P-wave when the
solid and the fluid move in phase and the slow dilata-
tional wave (slow wave) when the solid and the fluid
move out of phase. The phase speeds are given by
2
H e
+
C
2
Ce
+
M
e
=
∇
ς
−
ς
2 150
+
bt
F
ς
,
−
ρ
f
ρ
ς
−
ρ
f
2
Ce
+
M
2
H e
+
C
F
ρ
f
ς
=
∇
ρ
∇
ς
2 151
−
1
−
1
1
c
p
=Re
z
p
and 1
c
I
p
=Re
z
I
p
, and the
−
bt
ς
,
2
2
inverse quality factors are 1
Q
I
p
=Im
z
p
Re
z
p
where
H
is the stiffness coefficient of the Biot
Frenkel
theory (
H
=
K
U
+43
G
U
). We assume plane wave prop-
agation in the
x
-direction:
-
2
2
and 1
Q
I
p
=Im
z
I
p
Re
z
I
p
.
Using the decomposition of the displacements into
rotational components
e
=
A
1
exp
ilx
−
ω
t
,
2 152
∇
×
u
s
=
Ω
,
2 161
ς
=
A
2
exp
ilx
−
ω
t
,
2 153
∇
×
w
=
Ψ
,
2 162
where
is the angular frequency and
l
is the complex
wave number. Using Equations (2.152) and (2.153) into
(2.150) and (2.151), we obtain
ρ
−
ρ
f
F
ω
applying thecurl operator toEquations (2.141)and(2.142),
and using Equations (2.161) and (2.162), we obtain
2
A
1
=
H
ω
−
A
1
l
2
−
CA
2
l
2
−
ω
ρ
−
ρ
f
1
F
C
G
2
2
2
F
Ω
=
G
U
∇
Ω
+
C
G
∇
Ψ
−
∇
Ω
+
1
F
b
F
CA
1
l
2
+
MA
2
l
2
−
i
ω
A
2
,
2 154
2 163
1
F
M
G
+
bt
2
−
∇
Ψ
F
Ψ
,
