IV-38 



2k^ 



Exponential: P -2<u2>a^k=L 



JiL 



2^2 



(l + a^yr l + 2a''k 



(IV-81a) 



Gaussian: P = ^/TTk^ L- <u=> dye * =ynk^ La 



sc 4 



2r 



ya 



<ru=>ll -e ^ J 



(IV-81b) 



We may take the high and low frequency limits of these expressions to obtain the 

 approximate expressions for the scattered power shown in Table IV- 3. These are 

 seen to agree with the qualitative discussion of Chapter II. Furthermore, the exact 

 expressions for the entire frequency range in (IV-8I) permit us to examine the 

 range of validity of the high and low frequency approximations. Thus, Table IV-3 

 also shows ranges of frequency for which the approximation can be tolerated. In 

 this context, we have chosen a relative error of approximately 1/3 as the maximum 

 allowable error for the approximation to be applicable. It should be emphasized 

 that all of these results are good only for the far field where the geometric approxi- 

 mations leading to (IV-45) are valid. Furthermore, the scattering volume must 

 not be too large if the single scattering approximation under which all of this was 

 devised is to remain tenable. In other words, the range L must be sufficiently 

 small that the scattered powers shown in Table IV-3 are small compared to unity. 



TABLE IV-3 



THEORETICAL SCATTERING FOR TWO TYPES OF 

 CORRELATION FUNCTION 





ka» 1 (highfreq.) 



ka << 1 (low freq.) 



Exponential 



P =-2ak^ <u^> L 

 sc 



P ^4a^k* <u^> L 

 sc 





applicable when ka > 1 



applicable when ka < ■£ 



Gaussian 



P *"aA^ ak^ < u^ > L 

 sc 



P -^ a=^k*<u=> L 

 sc 2 





applicable when ka > 2 





jarthur B.littleJnt. 



S-7001-0307 



