Ogilvie 



It is interesting to note the following about the symmetry: 

 It turns out that $| and ^2 ^-^^ o<^<^ with respect to Y, but #3 

 is neither even nor odd. The linear term in $3 , namely, 

 B3(x,z;e)Y, is even, since it turns out that B^ = - B3. Careful 

 study of the ^2 problem shows that it actually implies that there is 

 a generation of fluid in the body, but the rate of generation is higher 

 order than the #2 term. Physically, of course, there can be no 

 fluid generated, and so a compensating source-like term appears 

 in %, 



The far field is again the entire space except for the plane 

 y = 0. The relations (2-3) to (2-6) are again valid, as well as the 

 discussion of them. But now it will not suffice to provide only source 

 singularities on the centerplane; clearly we must also provide 

 singularities which lead to antisymmetric potential functions. In 

 fact, since the body has zero thickness, we shall expect the leading- 

 order approximation to be strictly antisymmetric. These require- 

 ments are all met by a distribution of dipoles which are oriented with 

 the y axis. The potential of such a sheet of dipoles can be expressed: 



f(x,y,z) = ^ CC '^^^^."^'^ .3/2 • <2-26) 



^''-'-ooJ^[(x-e) +y +(z-;)T 



The inner expansion of such an integral can be obtained by the same 

 Fourier-transform technique that was used before. One finds that: 



i^irJ- 2J/2 

 f (k;y;m) = -^ (sgn y)(x (k,m) e 



The exponential function can be expanded into a series, which Is then 

 inverted term-by-term. Define a new function (cf, (2-10)): 



^^ 

 (k +m ) 



The following relationships exist between the two functions fjL(x,z) 

 and •y(x,z): 



..1^ ,^ _ i r* r°° V(e,;) de dC . /2 28^ 



'^^-^^ - ^ J.00 J-co F73f7^7I7F^ ' ^'"''^ 



kOO />00 



^(x,z) = -i-r r [pxx^M ded^ ^ (2.29J 



^'^^-oo J-oo [(x-e)'+(x-;)']'/' 



692 



