Ogilvie 



allow z© -♦ 0. In physlccil variables, the result is: 



A(x,y,z) = --i- \ dke'»^''cr*(k)Ko(|k|r) 



- lim — =- \ dk a k V (k) \ . . , 



|i— 4Tr^J-a) *^-ooV(k2+i2)[g7(k2+i2)_(uk-l^./2r] 



(3-6) 



where fx denotes a fictitious Rayleigh viscosity, guaranteeing that 

 the proper radiation condition is satisfied, and c (k) is the Fourier 

 transform of (r(x) , the source density. 



The two-term outer expansion is: 



^(x,y,z) ~ Ux + <^|(x,y,z), 

 which has the two- term inner expansion: 



<j>(x,y,z) ~ Ux +- o-(x) log r - -^ H^) - g(x), (3-7) 



where 



poo 

 f(x) = \ 



J- 00 



de o-'(l) log 2|x-|| sgn (x-e) (3-8a) 



=-iI 



00 I I 



dk e'^V^k) log -^1^ ; (3 -8b) 



g(x) = lim ui r dk e^^-'k^^k) r , /^ . 



,1^0 4Tr2»^-oo '^-ooV(k^+i^)[gV(k2+i2)-(Uk-iHL/2)^] 



(3-9) 



The expansion should be compared with the corresponding expansion 

 for a line of sources in an infinite fluid, as given in (2-65). We now 

 have an extra term, g(x) , and the terms containing o-(x) ajid f(x) 

 differ by a factor of two from the earlier result. The latter variation 

 is not important; it results from the fact that the line of sources was 

 taken at z = zq < , and those sources merged with their images 

 when we let Zq— ^ 0. 



Define (r(x) = for values of x ahead of and behind the ship, 



734 



