REGULAR COMBINATION OF ACOUSTIC ELEMENTS 285 



stituting these values in equation (40), we obtain 



(«2+ («+ 1)2) ((„ + 1)2+ („ + 2)2) 



2(w)2(w + 2)2 



ff{n + 2Y - (w + 1)^ 

 ^ 2{nf{n + 2)2 



cosh r 



^ ^/ / W2 + (« + 1)2 



+ 



2(n)2 

 (« + 1)2 + (w + 2)2 \ e« 



2(« + 2)2 ) K 



rr 1 • T^/ /•52 W + 1 , „ /.Si W. , 



If we substitute A = \ -?r = — r— ?; and K = \ br = — ;— r and 



03« + 2 \62« + l 



neglect 1 as compared with n^, we have 

 cosh 8 — 



"'' + '\o.kr ' 



2^2 / 2n2 



(50) 



If again our changes in areas are very small and hence n very large, 

 we can neglect 1 compared with 2^2 and obtain 



cosh 8 — cosh r , or 5 = F. 



Either of these solutions will hold for any other pair of sections if we 

 neglect 1 as compared with n^ for the first of 1 compared with n~ for 

 the second. Hence for either solution, the propagation constant is 

 little affected for this type of taper. The specific characteristic im- 

 pedances Zji^ and Zji, become 



Zo sinh r 



'Ri 



Zh. = 



Zo smh r 



^ 1 -> I 1 \/f T^; cosh r — ;r-^ ) — 1 



n 2n? + 1 \ \ 2^2 2w' 



If we neglect 1 as compared with n-, these expressions reduce to 



„ _ Ztifi sinh r _ Zow sinh r . 



^' ~ 1 + n sinh r ' "^^^ ~ - 1 + « sinh V ' ^^"^^ 



These impedances represent the impedances per square cm. looking 

 in both directions at the input junction of the filter, whose area is n-E. 

 As we move in either direction these impedances change since n 

 itself changes. If n becomes sufficiently large and F is not zero, the 

 two characteristic impedances approach the value Zq. 



