GUIDED WAVE PROPAGATION THROUGH GYROMAGNETIC MEDIA. HI 1163 



where in the definition of the xj/, Vr is replaced bj'' jr , the r^^ zero of J,{x). 

 Since the perturbations considered here do not couple modes ^ith dif- 

 ferent azimuthal number, s will be consideretl fixed hereafter, and only 

 the suffix /• will be retained. 



The field at 2 = —0 (just outside the ferrite) will consist, not only of 

 the incident field and its reflection p^ , but also of other modes excited 

 by the perturbation. Taking the incident E/ to have unit amplitude, Ave 

 have, from the continuity of tangential electric fields, 



(1 + Pr)Etr + X PnEtn = H i^nl + T„2) X PnlEti, 



^ (25) 



n.( 



where the r„i , r„2 are respectively the forward and backward traveling 

 amplitudes of the n^ perturbed mode. In the same way, we have for the 

 tangential magnetic field : 



(1 — Pr)Htr — 2I PnH tn = zl i^nl " Tn2)qntHtt . (26) 



n^r n,t 



The changes in sign of the coefficients of the backward waves have al- 

 ready been explained in Section 1.1. Let us now suppose that the incident 

 mode is the TEr mode. Multiplying both sides of (25) by V*^r and both 

 sides of (26) by ViAr , and integrating over the cross-section we have, 

 from the orthogonality relations of these functions, 



1 + Pr = X ("^nl + Tn-dVnr , 

 n 



/8r(l — Pt) = ^ i8r(T„l — T„2)9nr • 



n 



But in these series, all r except Tt will be small, as will be all p, q except 

 Prr , qrr ■ Heucc all terms, except those for which n = r, will be small of 

 second order, and can be neglected. Further, 



Vrr =1, Qrr = ^^ 



1^ 



/3r 



Thus we obtain finally, 



1 + pr = Trl -f Tr2 , 



1 — Pr = (ttI — Tt2) 



+ 



In A'iew of equation (23) for /3r , these equations can be written, to 



