OgiZvie 



now also write down a|(x,zj€) by combining (2-7) and (2-11): 





% = °'^' • 



We have the two-term outer expansion -- with everything in it known 

 -- and the three-term inner expansion -- with the "constant" 

 A2(x,z;€) not yet determined. 



Solution of the higher- order problems : From the [ L] con- 

 dition, (2- 17), it can be seen that $2(x, Y,z;€) is not linear in Y. 

 However, the differential equation for $2 ^s easily solved, the body 

 boundary condition can be satisfied, and matching can be carried out 

 with the outer expansion. The result is: 



*3(x,Y,z;c) = A3(x,z;c) + B3(x,zj€) | Y | - ^^"^Y^^a^^^ "'"^izz^' 



where 



B3(x,z;€) = €^[(a, W + (a, H)J , 



A3(x,z;€) = af3(x,z;€). 

 We also obtain o-g, through the matching, 



<r2(x,z;c) = 2[(a,^h)^ + (a,^h)J , 



and this information also gives us 0*2 and Ag. 



Summary: Symmetrical Body, The results for both near- and 

 far -field expansion are stated in terms of the far-field coordinates 

 (the natural coordinates of the problem) in Table 2-1, In a sense, 

 the results are rather trivial. There could be difficulties near the 

 edges of H, but, barring such possibilities , the inner expansion 

 could be obtained from the outer expansion and then matched to the 

 body boundary condition. This is actually the classical thin-ship 

 approach. The outer expsinsion is uniformly valid near the thin 

 body, except possibly near the edges. 



In the classiccil approach to the thin-body problem, there is 

 usually a legitimate question concerning the analytic continuation of 

 the potential function into the region of space occupied by the body. 

 Sometimes one avoids the problem by restricting attention to bodies 

 which can legitimately be generated by a sheet of sources, but this 

 is not very satisfying. The nnethod of matched asymptotic expansions 

 avoids the question altogether by eliminating the need to ask it. What 



688 



