Civil Engineering Reference
In-Depth Information
where
Z
0
is the characteristic impedance in the free air. The new set of equations for the
determination of the coefficients
N
i
is
#
$
&
'
(
=
0
q(k)a
1
(k)
−
c
a
3
(k)
r
ik
1
1
q(i)a
1
(i)
+
c
a
3
(i)
−
N
i
(10.89)
%
i
=
1
,
2
,
3
k
=
1
,
2
,
3
N
i
1
B
8
1
q(i)b
3
(i)
c
B
6
a
1
(i)
B
7
q(i)a
3
c
b
1
(i)
+
−
+
i
=
1
,
2
,
3
r
ik
1
c
B
6
a
1
(k)
B
7
q(k)a
3
(k)
+
−
k
=
1
,
2
,
3
B
8
1
q(k)b
3
(k)
(10.90)
Z
0
cos
θ
c
b
1
(k)
−
−
=
N
i
(a
3
(i)
b
3
(k))r
ik
,
b
3
(i))
(a
3
(k)
×
+
+
+
i
k
N
i
1
−
B
7
1
c
b
1
(i)
+
q(i)b
3
(i)
c
B
3
a
1
(i)
+
B
4
q(i)a
3
i
=
1
,
2
,
3
r
ik
1
c
B
3
a
1
(k)
B
4
q(k)a
3
(k)
+
−
k
=
1
,
2
,
3
N
i
(a
3
(i)
+
b
3
(i))
(10.91)
−
B
7
1
c
b
1
(k)
−
q(k)b
3
(k)
cos
θ
i
1
jω
−
Z
0
=−
b
3
(k))r
ik
.
(a
3
(k)
+
+
k
The displacement components
a
i
and
b
i
are given by Equations (10.47) - (10.50) and
the frame displacement components at the free boundary are given by Equations (10.70)
and (10.71). The velocity components induced by a circular field or by a line field can
be obtained with the same expressions as in Chapter 8.
10.7
Symmetry axis different from the normal to the
surface
10.7.1 Prediction of the slowness vector components of the different
waves
The waves are defined by the slowness vector components
q
x
and
q
y
at the free surface
of the porous layer. Let
Z
be the axis normal to the surface and
Z
be the symmetry
axis. The axis
Y
is chosen perpendicular to
Z
and
Z
. There is no loss of generality
because the
y
slowness component is not equal to 0 by definition as in the previous
Sections. The two sets of axes are
XYZ
and
X
YZ
(see Figure 10.6). The stress - strain
Equations (10.13) - (10.27) and the wave Equations (10.28) - (10.34) are always valid, but
