IV-14 



are very large compared to the wave length of the incoming sound, and v^^hose index 

 of refraction changes very smoothly, will scatter the sound in the forward direction 

 in a narrow cone. This means that a large and smooth scatterer will cause a 

 collimated beam of scattered pressure. Conversely, if the spectrum of u (?) is 

 substantial for values of x of the same order as or larger than k , the scattering 

 will essentially be omnidirectional. 



3. It is instructive to examine the above in some detail for a spherically 

 symmetric pattern: 



u(i) = u(?) (IV-47) 



To evaluate (IV-45) in this case, it is convenient to introduce a spherical coordinate 

 system for the vector |^ using the vector d as the polar axis, 5 = (?, 9 ,0 ). 

 In this coordinate system, ° ° 



? • d = ? d cos 9 = 2 § sin -^ cos 9 (IV-48) 



- - o 2 



(IV-45), therefore, becomes: 



°= n 2t\ 



ikx /•/-/• o^9„ 



-2ik5 sin — cos 9 

 2 o 



Pi(x) = -^^r| I d?rd9 j d« ?=sine i,'(?)e' 



(IV- 49) 



oi2 ikx 

 2k e 



Tx 



o 



1 



d§ § sin(r§) u(?) 



where we have performed the and integrations and introduced T according to: 



r = 2k sin - (IV- 49a) 



In this case, the directivity pattern defined above becomes a function of 9 (or 

 equivalently of f) alone. We recall that the meaning of the directivity pattern is the 

 ratio of the scattering integral pi at a point x divided by the value of pi at a point 

 located at the same distance x in the forward direction. For the spherically 

 symmetric case, the directivity pattern is given by: 



D (9) = f d? § sin (r§) u (?)/r 1 d ? §^ u (?) (IV-50) 



'/'\ 



;artl)ur a.littlcjnt. 



S-7001-0307 



