FIELD SOLUTIONS 631 



opening varies with $ as shown in part in Fig. 14.4. A horizontal line repre- 

 senting a given susceptance of the finned structure will intersect the curve 

 at an infinite number of points. Each intersection represents a passive 

 mode which decays at a particular rate in the z direction and varies sinu- 

 soidally with a particular period in the y direction. 



If the effect of the electrons in the central space is included, Hx/Ez for 

 the central space no longer varies as shown in Fig. 14.3, but as shown in 

 Fig. 14.5 instead. The curve goes off to + co near a value of d correspond- 

 ing to a phase velocity near to the electron velocity. The nature of the modes 

 depends on the susceptance of the finned structure. If this is represented 

 by Pi , there are four unattenuated waves; for P^ there are two unattenu- 

 ated waves and an increasing and a decreasing wave. P^ represents a tran- 

 sitional case. 



Not the whole of the curve for the central space is shown on Fig. 14.5. 

 In Fig. 14.6 we see on an expanded scale part of the region about d = \, 

 between the points where the curve goes through 0. The curve goes to + oo 

 and repeatedly from — oo to + oo , crossing the axis an infinite number of 

 times as 6 approaches unity. For any susceptance of the finned structure, 

 this leads to an infinite number of unattenuated modes, which are space- 

 charge waves; for these the amplitude varies sinusoidally with different 

 periods across the beam. Not all of them have any physical meaning, for 

 near ^ = 1 the period of cyclic variation across the beam will become small 

 even compared to the space between electrons. 



Returning to Fig. 14.1, we may consider a case in which the central space 

 between the finned structures is very narrow {d very small). This will have 

 the effect of pushing the solid curve of Fig. 14.5 up toward the horizontal 

 axis, so that for a reasonable value of P (say, Pi , Pi or P^ of Fig. 14.5) there 

 is no intersection. That is, the circuit does not propagate any unattenuated 

 waves. In this case there are still an increasing and a decreasing wave. The 

 behavior is like that of a multi-resonator klystron carried to the extreme of 

 an infinite number of resonators. If we add resonator loss, the behavior of 

 gain per wavelength with frequency near the resonant frequency of the 

 slots is as shown in Fig. 14.7. 



One purpose of this treatment of a broad electron stream is to compare 

 its results with those of the previous chapters. There, the treatment con- 

 sidered two aspects separately: the circuit and the effect of the electrons. 



Suppose that at j = <<? in Fig. 14.1 we evaluate not H^ for the finned 

 structure and for the central space separately, but, rather, the difference 

 or discontinuity in Hx . This can be thought of as giving the driving current 

 necessary to establish the field E^ with a specified phase constant. In Fig. 

 14.8, yi is proportional to this Hx or driving current divided by Ez. The 

 dashed curve T2 is the variation of driving current with 6 or ^ which we have 



