550 BELL SYSTEM TECHNICAL JOURNAL 



wave is 



hk — hk+\, (1) 



where hk denotes the reflection coefficient (assumed to be a real number) 

 between the impedance of the ^th elementary length and the average 

 impedance. 



However, if the current starts with unit value at the sending end, 

 then the wave has to be multiplied by the factor g-'^^/^ [^ reaching the 

 point of reflection, where P is the propagation constant per two ele- 

 mentary lengths. In returning to the sending end the reflected wave 

 is again multiplied by a like amount so that its value on arrival there 

 becomes 



(hk - hk+i)e-''P. (2) 



The totality of echoes returning to the sending end is 



£6 = - Ai + L (hk - hk+i)e-''P = Z hk(e~'^P - g-^-^+O- (3) 



Let 



g-p = g-e+i4> ^ Be^^ (4) 



When n is large, it is permissible to use the assumption that k has co 

 for its upper limit in the above summation. The real part of Eb is 

 accordingly 



00 



Ebr = L hk[B^ cos k(i> - 5*-i cos {k - 1)0]. . (5) 



k=l 



By the same method as described for the more complicated case in 

 Equation 15, Appendix II: 



00 



1^7 = ^ £ [52*= C0S2 k(t> - 252^-1 COS H COS { (/^ - 1)0} 



+ 52A:-2cos2 {(yfe - 1)0}]. (6) 



This series may next be evaluated, giving: 



pT^ _ fe^ / 1 - 2^ COS + ^2 1 _ ^2 \ 



"' 2\ 1-^2 "^ 1 + 25 COS + 52; ■ ^'^ 



In a similar manner it follows for Ehi, the imaginary part of Eh, that 

 P^ ^ ^2 / 1 - 2J3 COS (/> + 52 1 _ ^2 X 



'* 2V 1-52 1 +25COS0 + 52/ ■ ^^^ 



