Ill -7 



Substituting these in (III- 11a), we obtain for the C : 



3gh' 



C 



(kar (1-gh)^ 



-1 [m + g(m+l)] [(2m+l)(l • 3 • • (2m - 1)] ' 



m (j^a)^™"^^ m(l - g) 



(III- 14) 



Returning now to the sum in (III- 13), we observe that for ka < < 1 and k'a < < 1, 

 the first two terms in the sum (m = and m = 1) are of order (ka)^ . All the 

 other terms are of higher order in ka; in fact, successive terms increase in 

 order by (ka)^. For small ka, therefore, the first two terms of (III- 13) should 

 suffice. Using (III- 14) to compute approximations for Co and Ci, and recogniz- 

 ing that Po =1, Pi (cos 9) = cos 6, we obtain: 



ikr-iuot 



p (x,t)-^-? p. (kaf 



sc - r inc 



-^ — -^ s_ cos 



3gh= 1 + 2g 



(III- 15) 



Fortunately, (III- 15) may be given a relatively simple physical interpretation if we 

 take the vantage point of the sphere. Since the sphere is very small compared to 

 a wavelength, the passing sound wave has the effect of slowly raising and lowering 

 the pressure in the entire vicinity of the sphere. In response, the sphere will ex- 

 pand and contract uniformly, as if it were breathing. However, since the mechan- 

 ical properties of the fluid sphere are different from those of the surrounding water, 

 the fluid sphere will expand and contract with a different amplitude from that which 

 would be experienced by an equivalent sphere of water. The result is the emission 

 of a pure spherical sound wave, corresponding to the first term in the square brack- 

 ets of (III- 15). 



artbur Sl.littlcJnir. 



S-7001-0307 



