1198 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1952 



while from equation (529) or (530), C is proportional to frequency for a 

 given stack, we see that our plots of Re Ai/tt'' versus C need only the in- 

 troduction of appropriate scale factors to read directly the variation of 

 attenuation with frequency due to nonuniformity in the stack. Although 

 a nonuniform Clogston line may still be better under some conditions 

 than a conventional line of the same size, the crossover frequency will be 

 higher and the improvement at any given frequency will be less than 

 if the stack were uniform. 



Among the various interpretations which may be given to our nu- 

 merical results, we shall consider here only the following: Suppose we 

 have a plane Clogston 2 line which, if it were perfectly uniform, would 

 have an attenuation constant smaller, at a certain frequency, than the 

 attenuation constant of the corresponding conventional line by a given 

 factor, say one-half, one-fifth, or one-tenth. For these particular attenua- 

 tion reduction factors the ratio of a to bi may be calculated from (574), 

 or from (577) if the msulation is polyethylene. The question is: What 

 variation in e across the stack is permissible if we are willing to have the 

 actual attenuation constant of the Clogston line be double its ideal value; 

 in other words, if we will settle for attenuation reduction factors of unity 

 (no improvement), two-fifths, or one-fifth instead of the ideal values one- 

 half, one-fifth, or one-tenth? 



To answer this question for any particular type of nonuniformity, we 

 have only to find, from the plot of Re Ai/tt^ versus C, the value of C for 

 which Re Ai/tt = 2. Then the fractional difference between the maximmn 

 and minimum values of e corresponding to this value of C is given by equa- 

 tions (530) and (531) to be 



€inax Cmin oLiO\ ,. f \ fK^Q\ 



Umax Jmin^ , K'^^o) 



where we have taken ^o ^ 2/3, and/^ax and /,„!„ are the extreme values of 

 the function /(^) which describes the type of nonuniformity bemg con- 

 sidered. 



The special types of nonuniformity which have been studied above fall 

 roughly into three different classes. In four of the cases, namely the sym- 

 metric and unsymmetric single discontinuities, the linear variation, and 

 the half-cycle cosine variation, the function /(^) varies monotonically from 

 one side of the stack to the other. In the symmetric rectangular step and the 

 one-cycle cosine variation, /(^) oscillates from one extreme value to the 

 other and back again, while in the three-cycle cosine variation, f(^) ex- 

 hibits three complete oscillations across the stack. The following table 



