IV-15 



If the scatterer is very small compared to k, the quantity F? will be much less than 

 unity for any § located inside the scatterer.^ We note that ^^" ^ ~1 for r^« 1; 

 thus, in this case, the directivity pattern becomes approximately 1 in all directions. 

 Such a very small scatterer, which causes essentially isotropic scattering, is 

 called a classical or Rayleigh scatterer, and bears out our earlier conclusions that 

 very small scatterers will in general produce omnidirectional scattering. We find 

 very large scatterers producing a collimated beam. 



The scattering strengths and directivity patterns for a few typical spher- 

 ically symmetric scatters are listed in Table IV- 1. The directivity patterns are 

 also shown graphically in Figure IV- 3 on a decibel scale. We observe from Figure 

 IV-3 that the directivity pattern, D(9), drops by about a factor of 10 (20db) as Ta 

 is increased from zero to about 4. There is some variation, depending on the details 

 of the refractive index distribution, but all directivity patterns drop a factor of 10 

 between fa = and a value of Pa greater than 2^ and smaller than 6s. We may, 

 therefore, conclude that the principal scattered energy for a large scatterer (ka» 1) 

 is confined to an angle for which: 



fa = 2k a sin- <4, or equivalently 9< r— 



(IV-51) 



TABLE IV- 1 (after Skudrzyk) 



SCATTERING STRENGTHS AND DIRECTIVITY PATTERNS 

 OF SPHERICALLY SYMMETRIC SCATTERERS 



Deviation of Index of Refraction Scattering 



(Inhomogeneity) Strength Directivity Pattern 

 ^(?) S D(9) 



Remarks 



u(?) = 



u for ? <a 







for§>a 



3 



g (sinFa - Pa cos fa) 



spherical 

 scatterer with 

 constant sound 

 velocity 



u(§) = 



o 



4k^a^ u 

 o 



1 



exponentially 



decaying 



inhomogeneity 



l + F^a*^ 



u(?) - 



(1 - -f 



{ V ^ a' for ? <a 

 I for?>a 



k^a^ u. 



(4_6?i2£i+2cosFa: 

 "^0 ^ A 



parabolically 



decaying 



inhomogeneity 



15 



^^ ( Far 



u(?) = 



u 

 o 



- k^a"^ u 

 2 



-Fa 

 e 



inhomogeneity 

 decaying as 

 (?/a)-4 



(1 + {^Mff 



u(?) = 



u e-<^/^>^ 

 o 



l-^'^'-o 



-F2aV4 

 e 



Gaussian 

 inhomogeneity 



Note: I 



^ = 2 k sin |. 





artb 



ur m.Hittlc.Iljur. 



S-7001-0307 



