Civil Engineering Reference
In-Depth Information
10.4.2 Waves polarized in a meridian plane
A nontrivial solution exists if the determinant of the square matrix of Equation (10.44) is
equal to 0. Expansion of the determinant gives a cubic polynomial equation in
q
2
which
can be written with the notations of Vashishth and Khurana (2004)
T
0
q
6
T
1
q
4
T
2
q
2
+
+
+
T
3
=
0
(10.46)
The coefficients
T
i
are given in Appendix 10.A. For a given
x
slowness component 1/
c
,
Equation (10.46) gives three different
q
2
and each
q
2
is related to a wave propagating
toward the increasing z with Re
q>
0 and a wave propagating toward the decreasing
z
. The respective amplitudes
a
1
,a
3
,b
1
,b
3
can be obtained with the three first equations
of the set (10.44). For instance, the waves can be normalized with a coefficient
N
=
a
3
+
b
3
which is the amplitude of the displacement component
(
1
−
φ)u
s
z
+
φu
z
.The
displacement components are solutions of the following equations
ρ
a
1
−
(B
3
+
c
a
3
2
(B
1
+
B
2
)
B
6
)
q
B
5
q
2
−
−
B
5
+
c
2
B
6
c
2
ρ
0
b
1
(10.47)
B
6
q
+
+
=−
c
N
ρ
−
ρ
0
−
(B
4
+
B
7
)q
2
c
2
a
3
c
a
1
+
−
(B
3
+
B
5
)
q
B
5
−
(10.48)
B
7
q
c
b
1
=−
B
7
q
2
)
+
N(ρ
0
+
B
6
c
2
ρ
0
a
1
(B
7
c
a
3
B
8
)
q
+
+
+
C
1
b
1
=
(10.49)
B
8
c
2
B
8
q
+
−
+
c
N
b
3
=
N
−
a
3
(10.50)
10.4.3 Waves with polarization perpendicular to the meridian plane
A nontrivial solution of Equation (10.45) exists if the determinant of the square matrix
is equal to 0 and
q
2
is given by
ρ
B
5
ρ
0
B
1
c
2
q
2
=
−
C
1
−
(10.51)
For instance, these waves can be normalized with
a
2
=
N
. The displacement compo-
nent b
2
is given by
b
2
=−
Nρ
0
/C
1
(10.52)
10.4.4 Nature of the different waves
For porous media with the frame much heavier than air, the partial decoupling has the
same consequences for anisotropic and isotropic media. For isotropic media two kinds
