26 BELL SYSTEM TECHNICAL JOURNAL 



From equations 6-3 and 6-4 it is seen that the TEoi wave is recovered by 

 bends which are an even multiple of dm\n ■ But such bends are efficient 

 transmitters of TEoi waves only over a narrow frequency range since 

 dmin varies with frequency. 



If the circular bend is followed by a long straight section, the TEoi and 

 TMu components existing at the end of the bend are carried over into the 

 straight section, but the TMu component dies down due to its greater 

 attenuation and constitutes a total loss. 



Numerical examples for first extinction angle. 



Using the same dimensions as in Table I of Section 5, one finds from 

 eq. 6-5 for: 



Example 1: dmin — 0.816 Radians — 46.8° 

 Example 2: dmin — 0.272 Radians — 15.6° 



7. Serpentine Bends 



Sections 5 and 6 dealt with bends continued with uniform curvature 

 over large angles. The present section considers the small random devia- 

 tions from a straight course which are unavoidable in field installations. 



Actual deviations are expected to be random both with regard to maximum 

 deflection angle and to curvature ; they are likely to approximate a sinusoidal 

 shape. For purposes of computation, the following analysis assumes as a 

 first case circular S-bends which consist of alternate regions of equal lengths 

 and equal but opposite curvatures. An exaggerated schematic of such 

 S-bends is shown on Fig. 5A. 



Each circular bend tends to produce a single mode with an attenuation 

 per equation 1.2-7. However, the discontinuous reversals of curvature 

 at the inflexion points produce mixed modes, and the initial part of each 

 region reduces the amplitude of the TM components produced in the 

 previous region. 



Each region may be treated as a discrete 4-terminal section of a trans- 

 mission network. Regardless of the wave composition at the input terminal, 

 differential attenuation will establish in a long serpentine wave guide a 

 steady state condition. In this steady state each region produces equal 

 attenuation. This attenuation per region and the resulting average 

 attenuation constant will now be derived. 



The TEoi and TMu waves each consist of "a" and "b" components with 

 separate amplitude ratios and propagation constants, as derived in Sections 

 1 and 4. In the first region (between points 1 and 2 of Fig. 5A) Ro is taken 

 as positive, and 



A - Vl + K-' - i^-' 1-2-6 



