44 BELL SYSTEM TECHNICAL JOURNAL 



This formula can be safely employed only when z considerably exceeds 

 the critical value »/Vl — v 2 . 



For v>l and z>n, the ultimate periodic terms are very small, and 

 may be omitted unless n is too small. Consequently in this region, 



and 



S H (,Z\p)= ~ 2 jJn{z), 



* -ql 



C n {z\v)= J' n {z), 



In the range of values for which the foregoing approximations are 

 valid we have also to the same approximation (see Appendix II) 



and 



Jn(z) = \ — cos (g«z-e»). 

 JM = -q»\l— sin ton*-©*). 



\ 7TZ 



Appendix IV 

 The Effects of Terminal Impedances 



In the text of this paper, the calculation of the wave-filter indicial 

 admittances is based on the assumption that the voltage is applied 

 directly to the filter at "mid-series" position and that the filter is 

 either infinitely long or else, what amounts to the same thing, is 

 terminated in its characteristic impedance. By virtue of these as- 

 sumptions, the disturbing effects of terminal reflections are eliminated, 

 and, as shown in the text, the solution is reducible to a relatively 

 simple form, which admits of considerable instructive interpretation 

 by inspection, and is rather easily computed. 



In the following the general solution will be given for the indicial 

 admittance A n {t) in the «th section of a wave-filter of 5 sections or 

 length, with the e.m.f. applied to the initial or zero-th section through 

 an impedance Zi{p)—Zi and the last or sth section closed by an 

 impedance Z 2 {p) =Z 2 . 



For any type of periodic structure, including as a limiting case, the 

 smooth line, it can readily be shown that 



1 1 e- nT -\- P2 e-^-^ T 



= °1F. T -lsY (1) 



Znip) Ki l- PlP2 e- 2 * r 



