1258 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1957 



case of the TMn wave in a 2-inch pipe at X = 5.4 mm this hmit is 

 reached with A/3//3ii - 0.45 X 10"' or 5 = 0.75 X 10"' with e = 2.^ 

 To evaluate (1) beyond this Hmit we may use the approximations: 



1 Wn{x,px) 



px U„{x,px) 



l^ Vn{x,px) 

 px Zn{x,px) 



cot (1 — p)x = cot 8x, 

 = — tan (1 — p) = — tan 8x. 



(11) 



The approximations (11) require x » n, and 8x < x/2, as shown in 

 Appendix I. These requirements are usually satisfied for the lower order 

 modes in a multimode waveguide with a thin dielectric layer. 



Equation (1) has been evaluated for the TMn mode using the approxi- 

 mations (11) and a method which is described in Appendix I. The results 

 are shown in Fig. 2 for e = 2.5 and several values of a/A. 



In introducing a dielectric coat we have to be aware of the change in 

 attenuation constant the TEoi mode will suffer. Not only the loss factor 

 of the dielectric material will cause additional losses, but the concentra- 

 tion of more field energy into the dielectric will increase the wall currents 

 and so the wall current losses. 



To calculate the wall current attenuation of the TEoi mode in the 

 round waveguide with the dielectric layer, we proceed in the usual 



0123456789 

 COAT THICKNESS IN PER CENT OF THE RADIUS 



10 



Fig. 2(a) — Change in phase constant of the TMu wave in the dielectric-coated 

 waveguide, e = 2.5; a/X = 1.03. 



