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using
×10 -3
a 13
θ 2 −Γ
1.5
s p 1 , p 2 , p 3 a 2 s
s p 1 , p 2 , p 3
ρ 1 θ
First longitudinal wave P1
1
2
a 23
θ 2 s p 1 , p 2 , p 3
s p 1 , p 2 , p 3 a 11 ρ s θ s
1
p 1 , p 2 , p 3 =
β
a 23
θ
a 12
θ
s p 1 , p 2 , p 3
s p 1 , p 2 , p 3
1 −Γ 1
2
a 22
θ 1 −Λ 1
a 13
θ 2 −Γ 2
3 167
0.5
s p 1 , p 2 , p 3
s p 1 , p 2 , p 3
0
a 12
θ 1 −Γ 1
s p 1 , p 2 , p 3 a 2 s ρ 1 θ 1
s p 1 , p 2 , p 3
a 22
θ 1 −Λ 1
100
s p 1 , p 2 , p 3 a 11 ρ s θ s
s p 1 , p 2 , p 3
Second longitudinal wave P2
II
p 1 , p 2 , p 3 =
β
a 22
θ
a 13
θ
80
s p 1 , p 2 , p 3
s p 1 , p 2 , p 3
1 −Λ
2 −Γ
1
2
a 12
θ
a 23
θ
s p 1 , p 2 , p 3
s p 1 , p 2 , p 3
1 −Γ 1
60
2
3 168
40
with
1 = A 11
R 11
i ωθ 1
Γ
θ 1
3 169
20
× 10 4
Γ 2 = A 22
θ
R 22
2
3 170
10
i
ωθ
2
Third longitudinal wave P3
ρ 1 θ 1 + R 11
i
8
Λ 1 =
3 171
ωθ 1
6
I p 1 , p 2 , p 3 are the factors of the three
compressional waves that link their amplitude in the
two fluid phases with the amplitude in the solid frame.
This shows that the amplitudes are dependent on the
velocity of each type of wave. The electric field corre-
sponding to the propagation of the three compressional
wave modes in a homogeneous mediumdrives a conduc-
tion current that exactly balances the streaming current.
Therefore,
p 1 , p 2 , p 3 and
where
β
β
4
2
0
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
0.1
0.2
Water saturation
Figure 3.2 Effect of water saturation on the attenuation of the
three P-wave modes calculated from the imaginary components
of the roots of the cubic Equation (3.163).
E+ Q 0
2
S 2 w 2 =0
J =
σ ω
3 172
strong attenuation of the second and third modes, mak-
ing the experimental detection and verification of the
two modes extremely difficult.
Next, we use the set of the relations introduced in the
matrix previously to reformulate the amplitude of the
relative displacement of the each of the fluid phases,
W 1,2 , as a function of the amplitude of the solid phase U :
and
Q V 2 ω
σ ω S 2 β
I p 1 , p 2 , p 3 ω
E= i
ω
U exp ikx
ω
t
3 173
Here, we reformulated the different parameters at low
frequency where the angular frequency of the wave
motions is much smaller than the critical frequency.
The critical frequency is equal to the inverse of the
intrinsic time scale in the two fluid system where
ω
I
p 1 , p 2 , p 3 U exp ikx
w 1 =
β
ω
t
3 165
<
ω c =
η 1 η 2 k s ρ 1 k r1 η 1 +
ρ 2 k r2 η 2 :
Q V 2 ω
Q 0
V 2
II
p 1 , p 2 , p 3 U exp ikx
w 2 =
β
ω
t
3 166
3 174
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