My PhD Thesis, PDF 3MB - Stanford University

x ( r , z ) are used. Since there exist only P and SV waves in this radially symmetric
problem, the displacement u can be written as (see Appendix A-1)
u ( r , z, t ) (e ) , (2.2)
where and are P wave and S wave potentials, respectively. Substituting equation
(2.2) into the elastodynamic equation (2.1) we obtain (see Appendix A-2)
??
( r ) ( z )
,
r
(2.3a)
0 . (2.3b)
( 2 ) 2
( 2
r 2 ) ??
Equations (2.3a) and (2.3b) apply to each separate layer. These equations plus the
application of boundaries at each interface constitute our mathematical problem. It is
easier to solve this problem in k domain, because partial differential equations (2.3a)
and (2.3b) in k domain become ordinary differential equations in r. Through the
Fourier transform pair
f
? i t ik z
(r , k , ) f (r , z , t )e dtdz
f ( r , z , t )
1
4 2
?
( i t ik z )
f ( r, k , ) e d dk
we can transform a function from z t domain to k domain and vice versa. From
equation (2.3) we obtain the elastodynamic equations in k domain as
d 2 ? ( j )
dr 2
1
r
d ? ( j )
dr
( j )
( k
2
k 2
) ? ( j)
- 28 -
,
F ( )
,
( r )
( j )
( j)
( 2 ) r
, (2.4a)