324 BELL SYSTEM TECHNICAL JOURNAL 



in which it is evident the current is not exph'citly involved. If we 

 write 



Q = RiP)m 



and 



w = Kr{pu 



R being the resistance per unit length and Kr the characteristic im- 

 pedance with respect to the mean square current {P)m we have in 

 addition 



a = RI2Kr. (38) 



Before continuing our discussion of attenuation, we shall, therefore, 

 have to calculate the losses in the sheath and the internal dielectric 

 medium. 



III. DissiPATivE Hollow Conducting Guides 



In the ideal case of the preceding section, where the conductivities 

 o"! and 0-2 of the dielectric and conductor are, respectively, zero and 

 infinity, the boundary conditions are simply that £z = £9 = at the 

 surface, p = a. When we take into account the dissipation which is 

 actually present in the conductor (and the dielectric as well) the 

 boundary conditions are the continuity of both the tangential electric 

 and tangential magnetic forces. This double set of boundary con- 

 ditions makes the problem inherently more difficult, of course. As we 

 are assuming a good conductor and dielectric, we shall treat the dissi- 

 pative case as a departure of the first order from the ideal case. Thus, 

 since the dissipation has a negligible first order effect upon the phase 

 velocity, the propagation constant 7 will now be 



7 = iwjv' -\- a, 



where a denotes the attenuation. 



We must now consider the field in the sheath as well as the field 

 in the inner dielectric medium. When necessary we distinguish be- 

 tween the electrical constants of the two media by the subscripts 2 

 and 1, respectively. We suppose that the sheath is electrically very 

 thick, a legitimate assumption at the very high frequencies in which 

 we are interested, and write for p > a, 



00 

 £j = X! Kn(p}^2)(An COS 7id + Bn siu vd) cxp. {icoi ± yz), 



II z = 2Z Kn{p\2){Cn COS lid + Z)„' sin nd) exp. (toj/'db 7s), 



n=0 



where 



X2" = 7" — /'2" 



