Ill -6 



where 



and 



a (k'a)n (ka) - gh B (ka) j (k'a) 

 r = ^ "^ ^ m '•'m' ' 



m a (ka)j (ka) - gha (ka) j (k'a) uu-na; 



a^(kr) - (2m + 1) ^A- j^(kr) ; 6 ^(kr) = (2m -f 1) ^^ n^(kr) ; (Ill-Ub) 



Po Co 



^ Po Co 



The resulting expression for Pg^, is, of course, hopelessly complicated. How- 

 ever, we are interested only in the far field, for which the asymptotic form of 

 j + i Ujjj applies (showing the character of an outgoing spherical wave): 



(-i)™n (kr) + in^(kr)l =- (-1)"^ ^-j::^ for k r » 1 (III-12) 



i kr 

 i kr 



If we substitute (III- 11) and (III- 12) in (III-9a), we find for the far field scattered 

 sound the expression: 



. ikr-iuJt _ , ,vm. , 



P (x,t) = p. i^- y? (cose)^-gH±i) („i.i3) 



"^sc - inc kr / ■> m 1 + iC 



m 

 m=0 



The sum in (III- 13) has the effect of a directivity factor; it indicates the dependence 

 of the amplitude of the spreading wave on the polar angle 9 . For small values of 

 ka and k'a, i.e. , for a sphere with a circumference small compared to the wave- 

 length of the sound in either medium, the coefficients Cj^ may be approximated 

 quite conveniently by using the first term in the series expansions of j^^ and n^^ 

 (in the case of j o we need the first two terms to be able to compute a^ ): 



j (ka)--- — ^^ .. ^,. , m^l ; 



m 1 • 1 • 3 • • (2m + 1) 



... x^ -1 • 1 • 3 • • (2m+ 1) >. 



n (ka) ^ — , m s ; 



m ,, ,m+i 



(ka) 



Jo(ka)=-l 



(ka) 



Arthur Kliitth.Hnt. 



S-7001-0307 



