GENERALIZED TELEGRAPHIST'S EQUATIONS 



^lml(n) = jW«. J J 

 ^'[mllnl = 0, if 71 5^ 77/, 



797 



dT(n) dTlm] , d 



_€] du d du e-i 



2 dv 62 dv J 



6162 du dv; 



= /a)6 4- -P^— '- , if /i = 7W.; 



Y[m]{n) = 0, all m, n. 



We note that Z(;„)[„', = — Z[n](m) ; -^[m][n] = — -^[n][m] , {n 9^m). 



In rectangular waveguides we choose cartesian coordinates; 

 then 61= 62 = 1, H = X, V = y and 



_i . .-1/2 . pirx . Qir// 



^(P9) = lp«X(P«)(a^) s"i ^— - sill V^ , 



a 



rr 1 -1 / 7\-i/2 sirx tiry 



Tisi] = IstXistMb) cos — cos -T^ , 



a 



22 2 "^ 

 2 _2 _P^,?'r"^2 

 X(pg) ~ X[P9] ~ ^ r —TT — Xp3 > 



where Ipq = 2 if neither p nor g is equal to zero and log = Ipo = \/2. 

 Hence 



1 1 ^ 



(53) 



K^ ff [ ru/ \ ■ ^^^ p^^ ^'^y ■ Q'^y 



X // {h/a)sp sin — cos - — cos —^ sin 2— x 

 Jj [_ a a b 



f / /i.\, «^^ • P'TX- . tTry qwy 



■+■ {a/b)tq cos — sm ^— sin -i- cos ^-r- 

 a a b b 



= J<^tixu 



i,,hm[is/ay + (t/by][i - i-y^'m - i-y^] 



Xp.Xst(s' - p'){q' - f-) 

 a s 9^ p, q 9^ t, 

 = 0, if s = p or g = ^; 



ip,i..(pV - gV)[i - {-y-"'][i - (-)"+'] 



dx dy 



(54) 



'[p<i]Ut] = JWMx!/ 



XpqXsiis' - p-)iq~ - i^)ab 

 ii s 9^ p, q ^ t, 

 = 0, a s = p or q = t, but not if s = p and q = t, 

 = jccfx, li s = p and q = t. 



