FIELD SOLUTIONS 



637 



wave we have considered. If we increase P {\ P \ decreasing; the inductive 

 reactance of the walls increasing) this finally results in the propagation of a 

 wave. There are two intersections, at ^ = ±0i , representing propagation 

 to the right and propagation to the left. The variation of di with P is such 

 that as P is increased (made less negative J di is increased; that is, the greater 

 is P (the smaller | P \), the more slowly the wave travels. 



There is another set of waves for which d is imaginary; these represent 

 passive modes which do not transmit energy but merely decay with distance. 

 In investigating these modes we will let 



= jf^ 



so that the waves vary with z as 



(*w« 



(14.30) 



(14.31) 



Fig. 14.3 — The structure of Fig. 14.1 is first analyzefi in the absence of an electron 

 stream. Here a quantity proportional to Ux/Ei at the susceptance sheet is plotted vs 

 B = /3rf, a quantity proportional to the phase constant /3. The solid curve is for the inner 

 open space; the dashed line is for the susceptance sheet. The two intersections at ±di 

 correspond to transmission of a forward and a backward wave. 



Now (14.28) becomes 



p = -tan (^ + elyiyi^ + elyi'^ 



(14.32) 



In Fig. 14.4 the right-hand side of (14.28) has been plotted vs $, again for 

 e, = 1/10. 



Here there will be a number of intersections with any horizontal line 

 representing a particular value of P (a particular value of wall susceptance), 

 and these will occur at paired values of $ which we shall call db^n . The 

 corresponding waves vary with distance as exp (± ^r^/d). 



Suppose we increase P. As P passes the point — (tan ^o)/^o , $" for 

 a pair of these passive waves goes to zero; then for P just greater than 

 — (tan d()/dn we have two active unattenuated waves, as may be seen 

 by comparing Figs. 14.4 and 14.3. 



