330 BELL SYSTEM TECHNICAL JOURNAL 



Ee = An^-j- /n(Xip) + -Bn-T— V„'(Xip) sin n9, in the guide, 



L AiV Ai J 



He ^ - \ Anl^Jn'O^ip) + Bn^ Jn{\ip) COS iiO, in the guide, 



L XiC- Ai^p J 



C„ ^ X„(X2P) + i^n ^' X„'(X2P) 1 sin nd, in the air, 



Ao'P A2 J 



cos 7i6, in the air. 



(55) 



Ee- 

 He= - 



Cn ^' i^,/(X2p) +Dn^ KniXop) 

 KzC^ \2''P 



The boundary conditions require the continuity of the tangential 

 components of E and H. Hence if a is the radius of the guide, we have 



AnJn(\ia) = CnKn{\2a), 5„/„(Xia) = D„Kn{\2a), (56) 



Ai G- Ai A2 Q- A2 



^n^V„'(Xia) +5„-^7„(Xia) = C„'^i^„'(X2a) +Dn^Kn{\2a). 

 Mc^ Xi^o X2C- Wa 



This is a homogeneous set of linear equations in the coefficients A, B, C 

 and D from which only the ratios of these coefficients can be deter- 

 mined. But there are only three independent ratios and four equations ; 

 eliminating these ratios we shall obtain the characteristic equation of our 

 boundary value problem from which the propagation constant 7 can be 

 calculated in terms of the frequency, the radius of the guide and the 

 electromagnetic constants of the guide. 



If « = 0, the above set of equations breaks up into two independent 

 sets connecting the pairs A, C and B, D. Hence non-trivial solutions 

 are possible by letting A — C — QorB = D = 0. In one case Ez is 

 zero everywhere and in the other Hz vanishes. Thus in the circularly 

 symmetric case we have waves of either the £-type or iJ-type in the 

 sense previously defined. But if n 9^ 0, then E^ and Hz must be 

 present simultaneously. 



The case « = is so much simpler than the others that we shall 

 examine it separately. Thus the characteristic equation for an 

 £o-wave is 



ei/i(Xia) e2Ki(\2a) 



Xia/o(Xia) X2aiCo(X2a) ' 



and that for an Jfo-wave is 



Ati/i(Xig) ^ fxzK i(\2a) ^ 

 \iaJo(\ia) \2aKo(^2(i) 



(57) 



(58) 



