i(kr,n) = a J , (kr) 

 n n/v 



(16) 



where a are constants to be determined, 

 n 



13. Taking the finite cosine transform of Equation 11 and using Equa- 

 tion 16, we have 



VTT VTT 



/<*> cos — de = a J , (kr) - I i> cos — 

 r s v n n/v / o v 



de 



(17) 



j = a J , (kr) - 

 s n n/v 



(18) 



Then applying the operation lim /r(9/9r + ik) to both sides of Equa- 

 tion 18, and using the Sommerfeld radiation condition (Equation 6) we have 



lim /rl- 1- ik 



r-*x> \3r 



VTT 



) a J . (kr) - / $ 

 J n n/v / Y o 



cos — de 



(19) 



14. Equation 19 can be asymptotically evaluated to determine a 

 Firstly, the first term involving the Bessel function is evaluated. The func- 

 tion J , (kr) at r-*» behaves asymptotically (Abramowitz and Stegun 1964) 

 n/v 



as follows: 



J , (kr) ~ 

 n/v 



2 /, niT TT 

 TTk? C0S kr " 2v - 4 



(20) 



Hence , we have 



lim /r(|- + ik)J . (kr) 

 r-**. \3r / n/v 



2k i(kr-mr/2v+Tr/4) 



(21) 



