III-9 



2. The breathing mode also dominates if the fluid has approxi- 

 mately the same density as water (g=^ 1). Note that this 

 condition is necessary (at least on the average) if the scat- 

 terer is to remain suspended in the water. 



3. If the sphere is very hard (gh^ > > 1), the sloshing mode 

 will dominate. 



The arguments leading to a breathing and a sloshing mode may be ap- 

 plied quite generally to the first two terms of (III- 13), even in the case where ka 

 is not very small. The first term is an outgoing spherical wave corresponding 

 to a breathing motion, and the second term has the cos 9 dependence associated 

 with rigid body harmonic oscillation along the polar axis. The higher order 

 terms (m ^ 2) correspond to non- spherical deformations of the sphere and not- 

 quite- rigid -body motions due to the finite size of the sphere. 



In order to gain some insight in the general case (III- 13), we must re- 

 sort to numerical calculation. The properties of the scattering sphere which in- 

 terest us most are the directivity of the scattering and the fraction of the incident 

 power which is scattered. Both of these may be studied by comparison with uni- 

 form scattering from a perfectly reflecting sphere (commonly called geometrical 

 scattering). In this case, the total energy intercepted by a sphere of radius a 

 from an incident plane wave of pressure amplitude p^j^^, is considered to be scat- 

 tered uniformly through a solid angle of 4tt . The pressure amplitude of the geo- 

 metrically scattered wave at a distance r from the sphere may be written as 



unif 



= 1/2 



[_4nr^ J ^inc 2r ^ 



Using this as a standard, a reflectivity factor can be defined for large r by using 

 (III-13): 



R. = 



unif 



_2_ 

 ka 



P (cos 9) (-1) 

 m 



m=0 



m ( 2m + 1) 



(1 + iC ) 

 m 



(III- 16) 



This determines the directivity of the scattering. The total power scattered by 

 the sphere (in comparison to geometrical scattering) is also of interest. Since 

 the total power in a geometrically scattered wave is just that arriving at the sphere 

 through a cylinder of radius a, the ratio of total scattered power to geometrically 

 scattered power is 



n = 



1 



TT a' 



Pine 



i 



'dS 



where S is a large sphere. 



artbur B.Ilittb Jnt. 



S-7001-0307 



