Falttnsen 
APPENDIX B 
Inner expansion of far-field source solution for the zero-speed pro- 
blem 
We will show how a two-term inner expansion of (17) can be 
written as (18). 
We let y be of order ¢€ , and we reorder the terms in (17). 
By expanding the integrands in (17) we obtain 
oes ikx ¢*(k) 
@ (x, y, Z) = = Bip = / dk e SSS 
ey V2 v [ic]. 
enlLa.8,) 
vZ ‘3 
ve ikx 
+ a y dk e o*(k) 
__ sist) 
0 : 
v iux g*(ut+2v ) 
r ve i2vx du e a 
2 V2» [ul 
-(1-6,) 
12 l 
é-(1-t0) 
Mealy Te / iux o*(ut2y 
= e due 
an 2 PU 
0 
+ higher order terms 
1816 
