FIELD SOLUTIONS 



639 



For instance, increasing P to values larger than Pi changes d for the cir- 

 cuit waves a great deal but scarcely alters the two "electronic wave" values 

 of 6, near 6 = deil ± 0.1). On the other hand, for large values of P the values 

 of d for the electronic waves are approximately 



6 = de±\^A (14.34) 



Thus, changing A alters these values, but changing A has little effect on the 

 values of 6 for the circuit waves. 



Now, the larger the P the slower the circuit wave travels; and, hence, for 

 large values of P the electrons travel faster than the circuit wave. Our 

 narrow-beam analysis also indicated two circuit waves and two unatten- 

 uated electronic waves for cases in which the electron speed is much larger 

 than the speed of the increasing wave. It also showed, however, that, as 

 the difference between the electron speed and the speed of the unperturbed 



Fig. 14.5 — When electrons are present in the open space of the circuit of Fig. 14.1, the 

 curves of Fig. 14.3 are modified as shown here. The nature of the waves depends on the 

 relative magnitude of the susceptance of the finned structure, which is represented by 

 the dashed horizontal lines. For Pi , there are four unattenuated waves, for P3 , two 

 unattenuated waves and an increasing wave and a decreasing wave. Line Pi represents a 

 transition between the two cases. 



wave was made less, a pair of waves appeared, one increasing and one 

 decreasing. This is also the case in the broad beam case. 



In Fig. 14.5, when P is given the value indicated by P2 , an "electronic" 

 wave and a "circuit" wave coalesce; this corresponds to yi and y-i running 

 together at 6 = (3/2) (2)''^ in Fig. 8.1. For a somewhat smaller value of P, 

 such as P3 , there will be a pair of complex values of 6 corresponding to an 

 increasing wave and a decreasing wave. We may expect the rate of increase 

 at first to rise and then to fall as P is gradually decreased from the value P2 , 

 corresponding to the rise and fall of .Ti as b is decreased from (3/2) (2) in 

 Fig. 8.1. 



It is interesting to know whether or not these increasing waves persist 

 down to P = —CO (no inductance in the walls). When P = — <», the 

 only way (14.25) can be satisfied is by 



coth ((€i/€)i/2(^' - Gly) = 



(14.35) 



