280 BELL SYSTEM TECHNICAL JOURNAL 



If we are to have 5 = b" , we must have 



So ^ s, s, + s. 



S3K S4K 



+ 





cosh r 



- O2 O2O4 — 'J3' 



_\ (52 + S,)S, (53 + ^4)^2 1 .... 



L 53(5i + S2)KK' (52 + 53)54i^'i^" J ' ^^'^ 



Since the term on the left is complex, while that on the right is a 

 numeric, each must separately vanish if we are to have this equality. 

 Similarly the terms within the bracket of the left hand side would have 

 to vanish. 



We see that the two terms on the left do not vanish simultaneously 

 unless we satisfy the progression equation 



25i532 - 252^54 + (53 - 52)(5i54 + 5253) = 0. (42) 



This equation is satisfied by a system whose area increases exponen- 

 tially with the distance. The terms involving 5i53 — 52^ and 5254 

 — 53^ are always very small no matter what the rate of progression. 

 Hence is is desirable to see if neglecting these terms we can still satisfy 

 the above conditions. The most useful value of the two terms on the 

 right hand side of equation (41) is 1. Hence setting each term equal 

 to 1 and solving for K' and K", we find that 



We see then that if we neglect second order quantities, we can repre- 

 sent with good approximation the pressure ratio of any tapered filter 

 by the expression 



P'2 



/_5^ 



\5„,: 



Pl \On+l 



where 5 is the propagation constant of a tapered structure. For a 

 complete solution, 8 is not constant except for a progression which 

 satisfies equation (42). 



A . Exponentially Tapered Filters and Horns 

 If we assume that the area of a given section is e-'^ times as large 

 as that of the section preceding, equation (40) reduces to 



2e-'[cosh r cosh /] = K'e-^ + (e"'' X -^ ) e^ (43) 



