1286 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1957 



The expressions (21) are hard to evaluate, but for some special cases of 

 interest they can be simplified greatly. 



To compute the coupling effects between the TEoi wave and the 

 TMn wave we make use of | A7/ | « 1 and get the following approxima- 

 tions : 



T. ^[,-g .yV] exp - (.. - <§ Ay) I, 



[, + g ,yr 



rp _ 1 1 , -u - , 3,3 

 J 22 — 



2 ,2 



Co a 



exp - 72 + ~ At I, (22) 



105 



m m ■ ^0^ A 2,2 -yl 



T12 = T21 = J ^r^ Ay I e ^ . 

 15 



In (22) the condition | Tn - T^ | ' » 4 | TnT^i \ is satisfied; conse- 

 quently the wave propagation is described by (17), 



Em = exp - [y, - ^" At) nl + g^' Ay't sinh AT^fe^^"', 



Ein = j ,-V ^T^ sinhAT/i/e~^" . 

 15 



(23) 



In addition to small oscillations, which are negligible, the wave ampli- 

 tude El suff'ers an additional attenuation 



A«. = - ^' Aa. (24) 



105 



Physically this means that to a first approximation there is no net power 

 transfer from Ei to E2 . The power converted from Ei to E2 in one 

 section of the iterative serpentine bends is all reconverted in the same 

 section. But this power, which travels partly in the E2 wave, suffers 

 the E2 attenuation and consequently changes the Ei attenuation. 



To evaluate (24) w^e introduce the coupling coefficient and the differ- 

 ence in attenuation constants between the TEoi and TMu wave. Then, 

 the relative increase of TEoi attenuation is: 



— ^ = 6.39 X 10-^ <f i ("2.69 i-l), (25) 



aoi A" \ A" / 



where aoi = attenuation constant of TEoi , 



a = inner radius of pipe, 



X = free-space wavelength. 



