IV-12 



We note that the vector ^^ is a unit vector in the x direction, i.e. , pointed from 

 the origin at the observer. For the far field, the magnitude of the vector ? must 

 be small compared to x. In other words: 



— << 1 (IV-41) 



We may, therefore, approximate r, as given in (IV-40), to first order in_by: 



r ^x I 1 



In order to obtain an approximation to (IV- 39) which is correct to the zeroth order 

 of 2. , we must approximate the exponent (r + ?i) of (IV- 39) in a fashion which is 



X ff 



accurate to first order in^ , but we may use the zeroth order approximation 

 r ~x to approximate the denominator in (IV-39). The first order approximation 

 to the exponent to (IV-39) may be rearranged by introducing a unit vector e ^ ) 

 pointed in the x\ direction. In that case we may, using (IV- 42), approximate the 

 distance r + 5i by: 



?^^'^-i • (^ - £^'^) -^-? • d 



(IV- 43) 



In the above, we found it convenient to introduce a vector d which is the difference 

 between a unit vector pointing at the observer and a unit vector pointing in the xi 

 direction: 



d =^ - e (^) (IV-44a) 



— X — 



Recall that the Xi direction is the direction of propagation of the incident wave so 

 that d represents a measure of the difference between the observer position and 

 the forward direction of scattering. If we introduce the angle 6 as the angle between 

 the observer direction x and the Xi direction (see Figure III- 2), we may write the 

 length of the vector d as: 



1 



. , / X (i)f r l4 e 



d=ld| =(2-2-=- e ) = [2 (1 - cos 6) J = sin - (IV-44b) 



artbur 2l.1LittlcJnr. 



S-7001-Q307 



