Faltinsen 
eikx - yWVk(k- 2) 
@(x,y,z) = His o* (k) 
p25 
dk oikx - iyVk (2v -k) * 
i. | A (25, 2k) 
js (17) 
e i du el* + i2vx - iyV(ut2 v)(=u) o*(u+2 v) 
Y (u+2 v )(-u) 
(ut+2 pv ) 
| Bes jiux + i2vx - yV(ut2v)u * 
(u+2 v)u 
(17) is valid for y = 0(1)> 0. 6, in (17) is some very small posi- 
tive number. It will be evident in Appendix B why it is convenient to 
have ¢~(1-91) as an integration limit in (17). 
We will now find a two-term inner expansion of the far-field 
source solution. We then let y be of order € , and we reorder the 
terms in (17). The procedure is shown in Appendix B. We get 
o (x) 
@ (x,y,z) ~ - ve e 
V2rv |x- Ver Tedd (18) 
vz -ir/2 o(x) 
i 2 
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