My PhD Thesis, PDF 3MB - Stanford University

normalized Hankel functions of n th order, respectively. It should be emphasized that the
normalized Hankel functions are used instead of the ordinary Hankel functions ( H n
( 2 )
and H ( x ) ). For a large argument x, the ordinary Hankel functions are approximated as
n
( 1)
H ( x )
n
( 2 )
H ( x )
n
2
e
x
2
e
x
n
i( x
2 4 )
,
n
i( x
2 4 )
.
When the argument x is complex, the imaginary part of x causes the exponential to
increase or decay, making the numerical procedure unstable (computer may overflow or
ix
underflow). If we normalize them by multiplying factors e
- 30 -
( 1 )
( x )
ix
and e , respectively, their
asymptotic behavior will be much improved. I will use the normalized Hankel functions
( 1 )
( 2 )
( H ( x ) n and H ( x ) n ) to express the solutions. Therefore, the algorithm is numerically
stable even for large argument (the high frequency problem). Section 2.5.1 gives an
example to show this.
where
In the fluid-filled borehole (j = 1), the solution is
? (1 )
(1 )
( 1) (1 )
( r , k , ) cJ ( r ) s H ( r )
o
o
and (see Appendix A-3)
( 1)
c e p
i ( 1 ) ( 1 )
( r r ) ( 2 ) ( 1)
(1 )
H ( r ) (c s )e o
p i ( 1 )
r ( 1) (1 )
H ( r ) (2.7a)
o
(1 )
c p
s
(1 )
c e p
i ( 1 ) ( 1 )
r
4 ( (1)
i
c
2
(1 )
2 )
, (2.7b)
F ( ) ;