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BELL SYSTEM TECHNICAL JOURNAL 



to rewrite it in terms of other parameters, and we will introduce them 

 now. Let 



Kh/Wo = O 



(2.28) 



C is usually quite small and is typically often around .02. Instead of ^ we 

 will use a quantity or a parameter b 



In terms of b and C, (2.27) becomes 



i-jY" = ( 



J(2^~l|•2)1r^l/3 



(2.29) 



(2.30) 



This has three roots which will be called 5i , ^2 and 63 , and these represent 

 three forward waves. They are 



8, = e-'"" = V3/2 - j/2 



h = 



-:bir/6 



7V/2 



= -V3/2-J/2 



(2.31) 



83 = e' = J 



Figure 2.4 shows the three values of 8. Equation (2.23) was of the fourth 

 degree, and we see that a wave is missing. The missing root was eliminated 



-0.866 -J 0.5 



0.866 -J 0.5 



Fig. 2.4 — There are three forward waves, with fields which vary with distance as 

 exp(— jjSe -\- 0eC5)z. The three values of 8 for the case discussed, in which the circuit is 

 lossless and the electrons move with the phase velocity of the unperturbed circuit wave, 

 are shown in the figure. 



by the approximations made above, which are valid for forward waves only. 

 The other wave is a backward wave and its propagation constant is found 

 to be 



r = j^e 



(■-?) 



(2.32) 



As C is a small quantity, C^ is even smaller, and indeed the backward wave 

 given by (2.32) is practically the same as the backward wave in the absence 

 of electrons. This is to be e.xpected. In the forward direction, there is a cumu- 

 lative interaction between wave and the electrons because both are moving 



