28 BELL SYSTEM TECHNICAL JOURNAL 



1 1 + >' 



r._.e = r^ + - lo.' i-^-y^i with 

 y = J >c-\i + gf) + .4 ^gr- + \^ (1 + ^r)^ 



This formal solution is hard to evaluate. It can be greatly simplified 

 for the subcritical and supercritical cases. 



1. Subcritical Curvature | k ] « 1 



o 



^ ~ 2 -TS 



J. - V f ^ ^ '^' ^ ~ •iA ^ r = r 



\ ^ 1 + grf/ 



For very low curvatures, the average attenuation approaches that of the 

 "fl" mode, and this in turn approaches that of the TE(t\ wave. 



2. Supercritical Curvature. 1^1^ 1 



The differential attenuation constant is small compared to the differential 

 phase constant. 



Substituting these values into 7-1 and 2, one finds 



-0.5j9m 



y = e 



Expressed as a function of 6: 



ye = cos i/' + j sin \}/ with 



^|y ^ M{d - 0.5 d„.) 7-3 



M has the value per eq. 6-18. 



The power ratio of the combined TMn and TEoi waves is 



We = tan^ ^/2 



In view of equation 1.2-7 the instantaneous rate of energy loss is 



ae = ai cos" - + a-jsm" - = ai + (as — aj sm - /-4 



2 2 ■^ 



"averuge = " / a. d, = — ag dO 



Sm Jo "vi •'0 



, . . /l sin MdA 



In view of .^-20 



Vv^^ + 1 v""^ - 1 sin M0„n 



a^verage '-''l 



Mdr 



