332 BELL SYSTEM TECHNICAL JOURNAL 



Hence at high frequencies the propagation takes place substantially 

 with the velocity of light appropriate to the substance of the guide. 

 The constant \2 being large, the field is concentrated largely in the 

 guide. 



Returning to the general w-th harmonic wave, we set 



An = SKn(\2a), Cn = SJn{\ia), 



Bn = TKn{\2a), Dn = TJ„(\,a). ^ ^^ 



Substitute in the last two equations of (56) and eliminate 5 and T. 

 Thus we obtain 



^'^ T J^ ( ^ M C • ( y^^JnKn lJilKnJn'\ ^ 



a \\i- X2- / V X2 Xi / 



O = J nJ^n 



X2 Xi / a \Xi^ X: 



Subsequently 



^■LlJ-i.Jn' _ ?(eiM2 + 111^2) J n'Kn' _ e2M2-K^>/" 

 pUn' PqJnKn q^Kr? 



and finally 



(66) 



(67; 



Kn-lKn+1 Jn-lJn+1 «(€l/X2 + Ml«2)/n Kn 



ei/^1 + e2M2 



(68) 



P'q' 



Allowing q to approach zero, we shall obtain in the limit an equation 

 whose roots in conjunction with (61) determine the critical frequencies. 



Thus if « > 1, we obtain 



(eiM2 + /X162) ^^TTTT^ = <'' - ^^)('"2 - ^^) + ^^ P'- (69) 

 Jn{p) n — I 



Since ordinarily m = fi-i, (69) becomes 



Jn-\{p) 62 



pJniP) {n - l)(€l + 62) 



(70) 



If the dielectric constant of the guide is very much higher than that of 

 the surrounding air, the first few roots of (70) are very close to those of 

 Jn-\{p) =0. As q increases indefinitely (68) degenerates into 



Jn-\{p)Jn+l{p) _ ^ ..... 



Jn'(P) ~ ^^ 



