1262 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1957 



20 log 



E, 



10 



E 



max _ 17 qc ^ 



(19) 



The attenuation constant of Ei is modified by the presence of the coupled 

 wave. Compared to the uncoupled attenuation constant it has been 

 changed by 



iV (20) 



Aq!c 



OCX 



') 



The amplitude E2 varies sinusoidally. From our point of view it is an 

 unwanted mode. The power level compared to the Ei power is 



20 log,. %^ = 20 log,. I 



(21) 



So far we have considered only a constant value of the coupling co- 

 efficient, c, corresponding to a uniform bend. The attenuation m such a 

 uniform bend is increased according to (20), and the worst condition we 

 can encounter at the end of the bend is a mode conversion loss (19) and 

 a spurious mode level (21). 



A practical case of changing curvature and consequently changing 

 coupling coefficient is the serpentine bend. A waveguide with equally 

 spaced supports deforms into serpentine bends under its own weight. 

 The curve between two particular supports is well known from the theory 

 of elasticity. An analysis of circular electric wave transmission through 

 serpentine bends^ shows that mode conversion becomes seriously high at 

 certain critical frequencies when the supporting distance is a multiple 

 of the beat wavelength between the TEoi and a particular coupled mode. 

 The beat wavelength is here defined as 



^b = — 



'2t 

 AiS 



(22) 



In serpentine bends formed by elastic curves, mode conversion at the 

 critical frequencies causes an increase in TEoi attenuation 



aoi 



W 



Co 



EI A/3-aoi_ 



Aa 



and a spurious mode level in the particular coupled mode 



E2 

 E, 



w 



Co 



EI Af^-'m 



OOl 



Aa 



(23) 



(24) 



where w = weight per unit length of the pipe, 

 E = modulus of elasticity, 

 I = moment of inertia. 



