1360 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1956 



Table 1(d) — M = 12.930, e' = 4, e" variable 



Table 1(e) — fi.a = 12.930, e' = e", both variable 



Table 1(f) — fi^a = 6.465, e' = 4, e" variable j 



The three values of ^oa correspond to waveguides of diameter 2 inches, 

 I inch, and ye inch at Xo = 5.4 mm. The jacket materials (mostly carbon- 

 loaded resins) which have been tested to date show a range of relative 

 permittivities roughly from 4 to 100. There is some indication that the 

 permittivity of a carbon-loaded resin increases as its conductivity in- 

 creases; this suggested consideration of the case e = e . 



The tables cover the range from e" = 1000 down to e" = 1 at small 

 enough intervals so that the general course of each mode can be followed. 

 It is worth noting that at 5.4 mm a resistivity (l/o-) of 1 ohm cm cor- 

 responds to e' =32. Copper at this frequency has an e of approxi- 

 mately 2 X 10^ 



In general the tables include the modes derived from Fo(^ia) whose 

 limits are TMoi , TM02 , and TM03 , and the modes derived from /^i(fia) 

 whose limits are TEu , TMn , TE12 , TM12 , and TE13 (except that in 

 the i^-inch guide TM03, TM12 , and TE13 are cut off). Some results are 

 given for the TMis-limit mode, namely those which satisfy the arbitrary 

 criterion Re fia ^ 10; but these results are incomplete because for large 

 e" the corresponding root of Fi(^ia) approaches 10.173. Furthermore for 

 small values of e" the attenuation constants of a few of the TM-limit 

 modes become quite large and the corresponding values of ^la move far 

 away from the origin. Since our object was to make a general survey 

 rather than to investigate any particular mode exhaustively, we did not 

 attempt to pursue these modes outside the region originally proposed 

 for study. 



The results of the IBM calculations are recorded in Table I to three 

 decimal places. Since the roots f la were obtained by cubic interpolation 

 in a square of side 0.5, the last place is not entirely reliable; but spot 

 checks on a few of the roots by successive approximations indicate that 

 it is probably not off by more than one or two units. The propagation 

 constants of some of the relatively low-loss modes (especially TE12 and 

 TE13 , whose wall currents are largely circumferential) were calculated 

 from the approximate formulas,* as noted in the tables. The attenuation 



(Text continued on page 1375) 



* The formulas used were (A9) and (AlO) of the appendix, which are slightly 

 more accurate than (7) of the text. 



