My PhD Thesis, PDF 3MB - Stanford University

? (1 )
(1 )
( 1) (1 )
( r , k , ) cJ ( r ) s H ( r ) (A-3.2)
o
o
(1)
The first term cJ ( r ) represents the incoming wave. Constant c is determined by the
o
boundary conditions. Everywhere within the borehole 2
( 1) 2
J J 0 , even at r=0,
o o
since J o ( 0 ) =1, The second term s H o
(1)
(1 )
( r ) is outgoing wave generated by the point
source. It remains to determine the constant s . Substituting (A-3.2) and integrating
both sides over a small area 2
, we have
Since
s ( 2
2
0
0
( 1)
H ( x ) i o
2
ln
x
2
integrating equation (A-3.3) gives
where Green's Theorem
4 s i F ( ) 1
( 1) (1 ) 2 (1 )
H H 0
0
with u 1 is used. From (A-3.4) we obtain
- 154 -
) rd dr F ( ) 1
for x 0 ,
. (A-3.3)
for 0 , (A-3.4)
( u 2
G G 2
G u
u ) dS ( u
S
G ) dL (A-3.5)
L n n