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BELL SYSTEM TECHNICAL JOURNAL 



that for the small-current case (curve 2) there are four intersections, giving 

 four real values of /3 and hence four unattenuated waves. However, for the 

 larger current (curve 1) there are only two intersections and hence two 

 unattenuated waves. The two additional values of /3 satisfying both the 

 circuit equation and the ballistical equation are complex conjugates, and 

 represent waves traveling at the same speed, but with equal positive and 

 negative attenuations. 



Fig. 2 — Curve 3 is a circuit curve similar to that of Fig. 1. Curves 1 and 2 are Ijased 

 on a ballistical equation telling how much charge density p is produced when the voltage 

 V acts to bunch a flow consisting of electrons of two velocities. The abscissa, /3//3o , 

 is proportional to phase constant. Intersections of the circuit curve with a ballistical 

 curve represent waves. Curve 2 is for a relatively small current. In this case inter- 

 sections occur for four real values of (3, so the four waves are unattenuated. For a larger 

 current (curve 1) there are two intersections (two unattenuated waves). For the other 

 two waves 13 is complex. There are an increasing and a decreasing wa\'e. 



Thus we deduce that, as the current densities in the electron streams are 

 raised, a wave with negative attenuation appears for current densities 

 above a certain critical value. 



We can learn a little more about these waves by assuming an appro.ximate 

 expression for the circuit curve of Fig. 1. Let us merely assume that over 

 the range of interest (near /3//3o = 1) we can use 



p = a-eoiS-F 



(16) 



Here a- is a factor greater than unity, which merely expresses the fact that 

 the charge density corresponding to a given voltage is somewhat greater 



