GENKRALIZKI) TKLIXiHArillST S EQIATIONS 787 



wiiere x is the separation constant and Ci , Co are the scale factors in 

 the expression for the elementary distance 



ds' = e-i (Jii + e-j dv . (5) 



In the case of TM wa\'es the V'-fnnction must vanisii on the boundary 

 of zero impedance. This boundary condition I'estricts x to a doubly in- 

 finite set of ^•alues Xmn witii the con-espondinj^- functions 7'm„ . In the 

 case of TE wa\'es the normal dei-i\'ati\'e of the 7'-fvuiction must \'anisli 

 on the boundary of zero impedance. Since we have to consider both 

 types of ^va^'es simultaneously, we shall distinguish between them by 

 enclosing the subscripts in parentheses for TM waves and in brackets 

 for TE wa\'es. The double subscript designation of various modes has 

 been standardized only for rectangular and circular waveguides. For 

 wa\'(^guides of other shapes the standard is to use a single subscript by 

 ai'ranging the modes in the order of their cutoff frequencies. For con- 

 venience, we shall use this convention in the general case and denote 

 T]\I modes b.y T'(„)(;/, r), and TE modes by T[„]{u, v). The correspond- 

 ing cutoff constants will be xm and X[«] • Ii^ what follows it is under- 

 stood that whenever the T-functions should be designated by double 

 subscripts, our single letter subscripts should be considered as symbols 

 for ordered double subscripts. 



The transverse field components may be derived from the potential 

 and stream functions, V and 11 for TM waves and U and ^ for TE 

 waves. Thus 



Et = - grad T^ - flux ^, Hi = flux n - grad U, (G) 



where the components of the gradient and flux of a scalar fiuiction W 

 are 



dW dW 



grad„ W = —— , grad, W = — — , 



ei du eo dv 



(7) 



dW dW 



flux„ W = ^^ , flux„ W = -^^ . 



€■> dv ei du 



The T-fiuictions corresponding to the various modes of the same va- 

 riety are orthogonal ; that is, the followhig integrals over the cross-section 

 vanish, 



\JT(n)T^m) dS = 0, JJTmTi,,,] dS = 0, if m ^ n. (8) 



It should be stressed that 7'(„) and !/'[,„] are not, in general, orthogonal. 

 ^ See Reference 6. 



