CHAPTER XIV 

 FIELD SOLUTIONS 



Synopsis of Cil^pter 



SO FAR, it has been assumed that the same a-c field acts on all elec- 

 trons. This has been very useful in getting results, but we wonder if 

 we are overlooking anything by this simplification. 



The more complicated situation in which the variation of field over the 

 electron stream is taken into account cannot be investigated with the same 

 generality we have achieved in the case of "thin" electron streams. The 

 chief importance we will attach to the work of this chapter is not that of 

 producing numerical results useful in designing tubes. Rather, the chapter 

 relates the appropriate field solutions to those we have been using and 

 exhibits and evaluates features of the "broad beam" case which are not 

 found in the "thin beam" case. 



To this end we shall examine with care the simplest system which can 

 reasonably be expected to exhibit new features. The writer believes that 

 this will show quaUtatively the general features of most or all "broad 

 beam" cases. 



The case is that of an electron stream of constant current density com- 

 pletely filling the opening of a double finned circuit structure, as shown in 

 Fig. 14.1. The susceptance looking into the slots between the fins is a func- 

 tion of frequency only and not of propagation constant. Thus, at a given 

 frequency, we can merely replace the slotted circuit members by suscept- 

 ance sheets relating the magnetic field to the electric field, as shown in 

 Fig. 14.2. The analysis is carried out with this susceptance as a parameter. 

 Only the mode of propagation with a symmetrical field pattern is con- 

 sidered. 



First, the case for zero current density is considered. The natural mode 

 of propagation will have a phase constant jS such that Hx/Ez for the central 

 region is the same as IIx/Ez for the finned circuit. The solid curve of Fig. 

 14.3 shows a quantity proportional to IIx/Ez for the central space vs ^ = 

 /3J {d defined by Fig. 14.1), a quantity proportional to /?. The dashed fine 

 P represents Ux|E^ for a given finned structure. The intersections specify 

 values of B for the natural active modes of propagation to the left and to the 

 right, and, hence, values of the natural phase constants. 



The structure also has j)assive modes of propagation. If we assume 

 fields which vary in the z direction as exp (^/f/)^, Ih/Ez for the central 



()3(l 



