1222 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1957 



1 



a certain point. In what follows the wave amplitudes a and h will always 

 carry mode subscripts, so they need never be confused with guide radius 

 and radius of curvature. The mode current and \'oltage are related to 

 the wave amplitudes by 



V = ICia + h), 



(33) 

 / = K-\a - h), ^ 



where K is the wave impedance. We have 



K{n) = h(n)/ooeo , m 



(34) 



K[„] = UfJLo/hln] , 



for TM and TE waves respectively, where /i(,o and /?[„] represent the 

 unperturbed phase constants, 



(35) 

 h[n] = (0' - xm'T, 



and 



/3 = CO fj.oeo . (36) 



For a cutoff mode, h is negative imaginary; but we shall deal onl}^ with 

 propagating modes in the present analysis. 



If we represent the currents and voltages in the generalized tele- 

 graphist's equations (23) in terms of the travehng-wave amplitudes, and 

 then perform some obvious additions and subtractions, we obtain the 

 following equations for coupled traveling waves: 



da 



— = — * zZ U(m) («)«(«) + K(m)(n)b(n) + K(m)[n]a[n] + K(m) [n]fe[n]J, 



= — i ^ [K[m]{n)Ci(n) + K[m](n)&(n) + K[m][n]fl[n] + ^'[m] [n]0[>i]J, 

 = +/ Zl lK[»i](n)Ci(n) + K[„,](n)h(n) + K(,«][n]0[«l + f^lm][n]h[„]]. 



dz 



dz n , . 



(3/) 



dh[m\ 



dz 



The k's are coupling coefficients defined in terms of the impedance and 

 admittance coefficients by 



*«(m)(n) = ■2[(-^(m)^(n))' Y (m)(n) ± {K(m)Kin)) ' Z(m)(»)], 



iK(m)[n] = 2[(-^(m)-^[n])' ^^("OI"! ± {K(m)K[n\) ' Z(„,)[h]], 



U[m](n) = |[(-K^[ml-K'(n))' 5" [„,](„) ± (K [m]-K^(n)) '2^[ml(n)], 



*'^[m][n] = ^[(i^[m]-K'[nl)' I^[m]ln] ± (/C[m]i^[„]) 'Z[m][„]]. 



(38) 



