CHAPTER XI j 



BACKWARD WAVES j 



WE NOTED IN CHAPTER IV that, in filter-type circuits, there is an \ 

 infinite number of spatial harmonics which travel in both directions. 

 Usually, in a tube which is designed to make use of a given forward com- 

 ponent the velocity of other forward components is enough different from 

 that of the component chosen to avoid any appreciable interaction with the 

 electron stream. It may well be, however, that a backward-traveling com- 

 ponent has almost the same speed as a forward-traveling component. 



Suppose, for instance, that a tube is designed to make use of a given 

 forward-traveling component of a forward wave. Suppose that there is a 

 forward-traveling component of a backward wave, and this forward-travel- 

 ing component is also near synchronism with the electrons. Does this mean 

 that under these circumstances both the backward-traveling and the for- 

 ward-traveling waves will be amplitied? 



The question is essentially that of the interaction of an electron stream 

 with a circuit in which the phase velocity is in step with the electrons but 

 the group velocity and the energy flow are in a direction contrary to that of 

 electron motion. 



We can most easily evaluate such a situation by considering a distributed 

 circuit for which this is true. Such a circuit is shown in Fig. 11.1. Here the 

 series reactance A^ per unit length is negative as compared with the more 

 usual circuit of Fig. 11.2. In the circuit of Fig. 11.2, the phase shift is 0° 

 per section at zero frequency and assumes positive values as the frequency 

 is inci;eased. In the circuit of Fig. 11.1 the phase shift is —180° per section 

 at a lower cutoff frequency and approaches 0° per section as the frequency 

 approaches infinity. 



Suppose we consider the equations of Chapter II. In (2.9) we chose the 

 sign of X in such a manner as to make the series reactance positive, as in 

 Fig. 11.2, rather than negative, as in Fig. 11.1. All the other equations apply 

 equally well to either circuit. Thus, for the circuit of Fig. 11.1, we have, in- 

 stead of (2.10), 



V = (-^, (....) ' 



The sign is changed in the circuit equation relating the convection current 

 find the voltage. Similarly, we can modify the equations of Chapter VII, 



438 



