GUIDHD-WAVE PUOrAGATlOX TilKOlCill (I VliO.MAfiNETlC MKDIA 593 



The value of x" is then given by 



X1.2' = ve (1 — pe' — ) — /3(p£ + pn)M,i , (17a) 



\ VeVh/ 



or 



Xi/ = Vh(i - p/ - — ) + Kpe + Ph)Ai,2 , (17b) 



\ VeVh/ 



where \i and A2 are the roots of (IG) and xf, X2" are the corresponding x^- 

 The labehing of the roots is not important, but consistency must be 

 maintained. From (14) E, and Hz must satisfy 



E, + jAiH, = yp, , 

 and 



E, + jK^H, = ,^2 

 so that 



& = ^f ~ t"^' . (18a) 



A2 — Ai 



and 



H. = i -^^ . (18b) 



A2 — Al 



Solutions of (15) may now be sought in cyhndrical coordinates. To 

 satisfy the boundary conditions in circular guide it will be necessary to 

 assume the solutions to vary as e^" , where 6 is the polar angle and n is 

 any integer, positive, negative or zero. Equation (15) then becomes 



if r is the radius. Solutions which are regular within the guide ^^'ill have 

 the form of constant multiples of J„(xi.2^), where J„ is the n^^ order 

 Bessel function. The solutions of (15) are, then, 



V'1.2 = A^,2Jn(xl.2r)e'"\ (19) 



where the ^4's are constants. E^ and Hz can be found now from (18), but 

 further equations must be found to express Et and Ht . Using P to denote 

 the starring operation, (10a) and (10c) may be re-written as 



(JpfP - PEVE)Et + mt = - V7/. , 

 and 



