982 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1954 



consider the case in which the ferrite cylinder fills the waveguide com- 

 pletely. E^ is then proportional to Tlf xi(2«2^), and so ^ is determined from 

 the condition 



Mxi(2a2ro) = 0. 



When "normal" propagation prevails (l3 less than the natural propaga- 

 tion constant of the medium co\/ei/x(l — pa^)) both 02 and x are imagi- 

 nary, equal to jai' and jx', say. Under these circumstances little is known 

 about the zeros of M. However, it is possible to say something about 

 the solution for large radial mode numbers. It follows from Erdelyi 

 et al.^ 1, p. 278, formula (2), that for large argument 



MjY.ii^M'n) = const •sin[a2'ro + x'log a2% + x'log 2 + $ (xO - 7r/4]. 



where $(x') = arg r(% -f jx')- The zeros of this expression are at 



^2 To + X log Q!2 ro = — - — X log 2 — $(x ). n = a large integer 



This equation may be solved graphically by setting 0:2^0 = u, assigning 

 values to ^, ps , u (and hence to x', 0C2). From a solution u one then 

 finds 



n = u/a2. 



M also has zeros for real a^ , x, if X is large enough. Thus the wave- 

 guide Mdll support waves with a 13^ greater than coVei(l — p/)- It is 

 shown in Reference 2 (1, p. 289) that when x is between ^ and %, M 

 has one zero, when x is between % and J^, M has two zeros and so on. 

 Suppose that pn is negative, = — \ ph\ . Then 



^ /3|ph| 

 For real positive j8, this equation has a solution for ii \ ph \ < x' 



Vx' - Ph' ' 



^^ % < X < /4i M will have a zero ?/(x) depending on the value of x- 

 Thus the equations 



Vx^ - PH^ V/5^ - iS2^ I Ph I 182 



solve the propagation problem parametrically. Similarly when x is 

 between % and J^, there are two zeros of M given by two functions 



