My PhD Thesis, PDF 3MB - Stanford University

namely, set the source function f in equation (2.1) to zero. Then, similar to the
formulation given in Chapter 2, we can obtain the radial and vertical components of the
displacements ( ?u k
( j )
( j )
and ?v k
) and the stresses ( ?
( j )
k
the liquid layer), the displacement and the stress are
?
(1 )
?u k
(1 )
k
( r )
( r )
(1 )
e 11
(1 )
e 21
(1)
( r ) e12 (1)
( r ) e 22
- 135 -
( r )
( r )
and ?
( j )
). For the first layer (i.e., j=1,
k
( 1)
c p
( 1)
c p
where the explicit expressions of the elements {e nm
( 1)
, (6.2a)
} are given in the Appendix A-4.
Note that the source term in (6.2a) is zero in this case. For j= 2, ..., N, the expressions for
displacements and stresses in a solid layer are
?u
?v
?
?
( j )
( r )
( j )
( r )
( j )
( j )
( r )
( r )
( j )
e11 ( j )
e 21
( j )
e 31
( j )
e 41
( j )
( j )
( j )
( r ) e ( r ) e ( r ) e ( r )
12
13
14
( j )
( j )
( j )
( r ) e ( r ) e ( r ) e ( r )
22
23
24
( j )
( j )
( j )
( r ) e ( r ) e ( r ) e ( r )
32
33
34
( j )
( r ) e 42
( j )
( r ) e 43
( j )
( r ) e 44
where the explicit expressions of the elements {e nm
( j )
( r )
( j )
c p
( j )
c s
( j )
c p
( j )
c s
. (6.2b)
} are given in the Appendix A-4. In
the outer most layer, i.e., j=N+1, there are only out going waves. Therefore, the
displacements and stresses in the outer most layer are
?u
?v
?
?
( N 1)
( r )
( N 1)
( r )
( N 1)
( N 1 )
( r )
( r )
( N 1)
e 13
( N 1)
e 23
( N 1)
e 33
( N 1)
e 43
( N 1)
( r ) e14 ( r ) e 24
( N 1)
( N 1)
( r ) e 34
( N 1)
( r ) e 44
( r )
( r )
( r )
( r )
( N 1 )
c p
( N 1 )
c s
, (6.2c)