18 BELL SYSTEM TECHNICAL JOURNAL 



4. Interaction Between TEoi and TMn Modes 



Since the separate modes of propagation behave Uke traveUng waves in 

 transmission lines-, their interaction can be derived from the coupHng 

 equations derived in Section 1 of this analysis. 



The uncoupled propagation constants Fi and Vi are known (equations 

 3-15 and 3-19). In order to find the coupling discriminant one must derive 

 the coupling coefficient from the field equations 2-1 to 2-6. The coupling 

 coefficient is defined as 



In computing the coupling coefficient one may neglect the small attenua- 

 tion constant. The energy stored by the TEoi wave per unit length is 



Pi = / t±.2Trrdr= / Hlrj-lTr dr 4-1 



Jo v Jo 



This expression is not affected by the cutoflf factor v because one may 

 consider the field inside the guide as composed of slanting plane waves with 

 the electric field strength £i . The energy stored by the TMn wave per 

 unit length is 



Jniv /%o -p2 »2jr na 



' I — r dtp dr = \ I HI r)r dip dr The inter- 

 Jq 1] Jo Jq 



changed energy: 



I ^^^LE^dipdr + HnH.vdipdr 



Jo 7] Jo Jo 



Combining equation (4-1) with 3-3, 3-8 and 3-lU 



2 2 »3.832 



' 3.832^77 Jo ^ ^ ^ 

 lage 146 of the book 



' ySUiiy) dy = "^ /o(3.832) 

 ^ 



3.8322,7 

 From reference 1, page 146 of the book 



Hence 



Pi = °:^^ 4-2 



In a similar manner one finds 



2(1 - v-)r] 



^ Loc. cit. 



