46 BELL SYSTEM TECHNICAL JOURNAL 



From (2), (3) and (4) it follows at once that 



dt ^ \ +r 3 (t-y)a is - n (y)+r i (t-y)a is+n (y)+ ... (5) 



provided the functions r (/), rx(t), r 2 (t) . . . satisfy, and are defined by, 

 the equations 



r 



<rp2 1 Ki Ki 



■^«*-T-*xirzrE+z? (6) 



f 



. ^ (4 « = -j- = ?ET ^ • ZT+z-, ' kTFTj etc - 



If the indicial admittance in any section of an infinitely long periodic 

 structure is determined, and equations (6) solved for r (t),ri(t), r 2 (t) . . . 

 (by aid of any of the methods discussed in the present paper), then 

 -<4nW is given by (5) by a single quadrature. The solution may 

 appear quite involved; as a matter of fact it is the simplest and most 

 easily interpreted and computed form of solution possible and its 

 complexity merely reflects the complicated character of reflection 

 effects due to terminal impedances. This considered statement is 

 made in the light of an extensive study of the whole problem and the 

 literature bearing on it and has been tested in many specific cases. 



When the terminal impedances Z\ and Z 2 are complicated and* en- 

 tirely unrelated to the corresponding characteristic impedances K\ and 

 K 2 , the solution of equations (6) and the numerical computations of 

 (5) are laborious but entirely possible, the only questions being as to 

 whether the importance of the problem justifies the necessary ex- 

 penditure of time and effort. In many cases, also, approximate 

 solutions are obtainable. Without any computations, however, the 

 solution (5) admits of considerable instructive interpretation by 

 inspection. The first term represents the current in the wth section 

 of an infinitely long structure when a unit e.m.f. is impressed through 

 a terminal impedance Z\. r (t) is the corresponding voltage which 

 exists across the terminals proper. The second term is a reflected 

 wave from the other terminals due to the terminal impedance irregu- 

 larity which exists there. The third term is a reflected wave from the 

 sending end terminals due to the corresponding terminal impedance 

 irregularity, etc. The solution, consequently, is expanded in a form 

 which corresponds exactly with the actual sequence of phenomena 

 which occur. 



