CURVED WAVE GUIDES 25 



where d is the beiuHnj; angle of the wave guide. The power carried by 

 the TEoi wave is 



Fte = cos - . 0-4 



Minima of TEoi occur when this phase difference is an odd multiple of tt. 

 Hence, the bending angles producing minima of TEm amplitudes are: 



1.36Xo . (2n + 1)2.22. 



a{{ + 0.12.-> \k\ •*) /(«) 



The initial fluctuation ratio approaches 



Cl min 



which is a large value tending to intinity. 



The relative attenuation of the slightly weaker component during one 

 beat cycle is 



A2t^ _ -2T/p+g _ -v/St/M ^ 4 _ ■i^'i 



Ao \k\ 



which is a small reduction tending to zero. Hence, the fluctuations persist 

 through a large number of beats. The power is transformed back and 

 forth between the TEoi and the TMu modes. 



In Section 5, it was shown that in a long, uniformly curved wave guide 

 the attenuation is intermediate between that of the TEoi and TMn modes. 

 But from equations 1.2-7 and 8 it follows that the two modes contribute 

 to the attenuation in proportion to their relative power flow. Since at 

 the beginning of the bend the power of the TMn component is zero, 

 it is to be expected that the initial rate of attenuation equals that of the 

 TEoi wave alone. This is proved by differentiating with regard to s. One 

 finds for all values of k that 



— ae + oe : = — ai 



as \ l«=o 



Discussion of Results 



Equation 6-2 corresponds directly to an equation derived by S. O. Rice 

 and, after allowing for the different choice of variables, to M. Jouguet's 

 equation (75)'^. It differs from the results of these earlier calculations 

 by the factor /(k) = 1 + 0.125 | k j "* which is a reminder that the simplihed 

 form of the equations given by the earlier authors is an e.xtrapolation to 

 infinite conductivity or infinite curvature of the wave guide. 



* Reference 3, pg. 150 of Cables and Transmission, July 1947. 



