(III-7c) 



III-4 



where P-^ is the m'^" order Legendre polynomial, and the radial functions jj^ 



COS (kr) 

 It should be noted that, for large kr, i^ behaves like — ; and n^ like 



*-^ 111 jrv 111 



sin(kr) 



k^ ■ 



We can now proceed to tackle the scattering of a plane wave (III- 7a) incident on a 

 fluid sphere consisting of a different material and imbedded in an infinite space of 

 water. * Let the density and sound velocity inside the fluid sphere be given by 

 Pi and Co . Any sound wave inside the fluid sphere will have to satisfy acoustic 

 equations just like (Ill-la) through (III-4a) but with Po and Co replaced by Po 

 and Co • Suppose the incident plane wave is approaching along the polar axis and 

 has pressure amplitude Pj^j^^,: 



/ \ ikr cos 9 - iuot ,„^ „. 



p. (x,t) = p. e (III- 8) 



mc — inc 



This incident wave causes both a scattered wave, pg^,, outside the sphere and an 

 internal wave, p', inside the sphere. The total acoustic pressure outside the fluid 

 sphere is therefore p^^^^, + Pg^,. 



Each of the three pressure distributions Pi^c' Psc ^^'^ P' "^^^ ^^ ^^~ 

 pressed in terms of the elementary wave functions (III-7b) since there is complete 

 <t> symmetry. In particular, the scattered wave must be strictly an outgoing wave, 



eikr 

 and since the r dependence must be like -j for large (kr), is therefore of the 



kr 



form: 



p (x,t)=y^A P (cose)rj (kr) + in (kr) 1 e"^"^^ (III-9a) 



sc - / J mm L m m J 



m=0 



*Our treatment will follow Anderson, Ref. Ill- 2. 



Arthur Sl.Ilittkllnt. 



S-7001-0307 



