Civil Engineering Reference
In-Depth Information
substitution, Equation (10.46) can be rewritten
A
6
q
z
+
A
5
q
z
+
A
4
q
z
+
A
3
q
z
+
A
2
q
z
+
A
1
q
z
+
A
0
=
0
(10.97)
The coefficients
A
i
are given in Appendix 10.B. For the waves polarized in the
direction
Y
, Equations (10.45), (10.51) are always valid with the same substitutions, 1/c
by
q
p
and
q
by
q
z
.
The slowness component
q
z
is now given by
q
z
(B
5
+
B
1
tan
2
ϕ)
−
2
q
z
B
1
q
x
sin
ϕ
cos
2
ϕ
B
1
q
y
+
ρ
(10.98)
q
x
cos
2
ϕ
ρ
0
C
1
+
−
−
=
0
10.7.2 Slowness vectors when the symmetry axis is parallel to the
surface
The previous model cannot be used. The axis
X
in the surface is chosen parallel to the
axis of symmetry
Z
. The slowness components
q
z
=
q
x
and
q
y
in the surface are known,
and the components of the new vector
q
p
perpendicular to the axis of symmetry
Z
are
q
y
and
q
z
. Equation (10.46) can be rewritten
S
0
q
P
+
S
1
q
P
+
S
2
q
P
+
S
3
=
0
(10.99)
T
34
S
1
=
T
23
q
x
+
T
33
S
2
=
T
12
q
x
+
T
22
q
x
+
T
32
S
0
=
(10.100)
T
0
q
x
+
T
11
q
x
+
T
21
q
x
+
S
3
=
T
31
The component
q
z
is given by
(q
P
−
q
y
)
1
/
2
q
z
=±
(10.101)
For the waves polarized perpendicular to the meridian plane, Equation (10.51) can be
rewritten
ρ
0
C
x
q
P
B
1
q
z
B
5
+
=
ρ
−
(10.102)
10.7.3 Description of the different waves
Let
a
i
and
b
i
,i
1
,
2
,
3, the quantities equivalent in the system
X
,Y
,Z
to the
a
i
and
b
i
in the system
X,Y,Z
. For instance
u
x
=
a
1
exp[
jω(t
−
q
x
x
−
q
z
z
)
], where
u
x
is
a displacement component in the direction
X
. The same normalization as in the system
X,Y,Z
can be used for the
a
i
and
b
i
, the wave number component in the direction
perpendicular to the plane
Z
X
being equal to 0. Equations (10.47) - (10.50) and (10.52)
can be used with the slowness components in the plane
X
Z
. Each wave is associated
with a specific system of axes
X
,Y
,Z
and the boundary conditions are expressed in
=
