664 



BELL SYSTEM TECHNICAL JOURNAL 



We see from (16.12) and (16.13) that, if we plot p/V vs. ^/^e for real values 

 of |S, p/V will be constant for small values of ^ and will rise as /S^ for large 

 values of ^, approximately as shown in Fig. 16.1. 



Now, we have assumed that the charge is produced by the action of the 

 voltage, according to the baUistical equation (16.11). This relation is plotted 

 in Fig. 2, for a relatively large value of Jo/utsV^ (curve 1) and for a smaller 



value of /o/«o^o (curve 2). There are poles at jS//3, = 1 



- , and a minimum 



between the poles. The height of the minimum increases as J^lu{/Vt is in- 

 creased. 



A circuit curve similar to that of Fig. 16.1 is also plotted on Fig. 16.2. 

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

 giving /oMr real values of /3 and hence /owr unattenuated waves. However, for 



1 





^^0— 



Fig. 16.1 — Circuit curves, in which the ordinate is proportional to the ratio of the charge 

 per unit length to the voltage which it produces. Curve 1 is for an infinitely broad beam; 

 curve 2 is for a narrow beam in a narrow tube. Curve 3 is the sum of 1 and 2, and approxi- 

 mates an actual curve. 



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

 unattenuated waves. The two additional values of ^ satisfying both the 

 circuit equation and the baUistical equation are complex conjugates, and 

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

 tive attenuations. 



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 approximate 

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

 the range of interest (near /3/i8, = 1) we can use 



P = a'i^W 



(16.14) 



