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BELL SYSTEM TECHNICAL JOURNAL 



for the reflection coefficient on a transmission line]. In passing across the 

 disturbance, a wave undergoes a small phase shift 50(aj). Since the total 

 phase shift around the entire system must remain Ivn, one may write : <^(co) + 

 5(^(w) = lirn, in which </)(co) is the phase shift around the system not in- 

 cluding the disturbance. If there is no disturbance, w = w„ (a)„ is used here 

 for the coo of the nth mode), 5</)(w) = 0, and </)(co„) = l-wn. For waves inci- 

 dent upon eFo from either direction, the respective amplitudes a and h at 

 cFo must satisfy the equations: 





i>+ 



•-!-;.+ 



1 - 



1 - 





Writing Itth — 



d<f> 



(co — Un) for </>(co) = lirn — 50 (co) in these equations 



and keeping only first order terms, one obtains the following pair of ho- 

 mogeneous Unear equations for a and b: 



d<f>\ 

 dec L„ 



«») - y 



a--6 = 



-r + 



Mr 



{(C - iCn) - -\b = 0. 



These equations have a solution if and only if the determinant of the co- 

 efficients vanishes, that is, if: 





(co- 



^n) [i 





(co — co„) — e = 0. 



The two solutions are thus: co = co„ , for which a = —b, and w = Un — 



- , for which a = b. 

 d(p 



From these facts concerning the ampUtudes, namely, that at the disturb- 

 ance eFo the amplitudes of the two oppositely traveUng waves must either 

 be equal or equal and opposite in sign for all time, one may conclude that 

 the waves are of equal ampHtude and that the standing waves resulting for 

 each frequency must have a node at the disturbance for the co = a)„ solution 



and an antinode there for the co = w„ — 



d4> 

 dco 



solution. 



What this means in terms of the general solution for the degenerate mode 



