A MATHEMATICAL THEORY OF LINEAR ARRAYS 87 



V$, = cos co/ + cosf ait -\- -- cos G ~ 7^) 



I ' (12) 



= COS wt + sin( co/ + - cos 



By just inspecting this equation, we find no evidence for existence of a 

 linear array with a space factor equal to the square of the space factor of 

 the couplet. Still less obvious is the method of obtaining proper amplitude 

 ratios. 



Arrays of Arrays 



The foregoing method of composition of space factors is in reality an 

 analytical expression of geometric construction of "arrays of arrays". 

 Consider, for instance, a pair of equiphase sources of equal strengths 



"*- 



-^ -I 1- — J — •+■ 



(A) (B) 



-^ 



Fig. 6 



(Fig. 6A). Take two such pairs as elements of an array of the same type 

 (Fig. 6B). The middle sources add up to a single source of strength two. 

 If the operation is repeated by taking (B) as elements of (A) or by taking 

 (A) as elements of (B), then (C) is obtained; the amplitudes of (C) are 

 proportional to 1, 3, 3, 1. 



Each shift of a source to the right through distance i is represented 

 analytically as multiplication by s. An algebraic identity 



(flo + Oiz + 02S-)2 = oos + aiz- + 022^ (13) 



is an expression of an obvious fact that each element of an array is shifted 

 through the same distance as the entire array. Similarly a given change 

 in the strength and the phase of the array is achieved by making the same 

 change in all its elements; this fact is expressed by the identity 



b(ao + oiz + 023') = bao + baiz + ba2?^. (14) 



