1200 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1952 



will certainly be qualitatively the same for uniform and slightly nonuniform 

 stacks, so long as the nonuniformity does not seriously distort the field pat- 

 tern of the operating mode. 



Some idea of how the effects of rapid random fluctuations in the average 

 properties of the stack may be expected to average out is given by equa- 

 tion (570), which assumes for the function f(^) a i^-cycle cosine ^^ariation 

 across the stack. As a numerical example, suppose that with this variation 

 of e We have 



'"^^ ~ '"'° = 0.01 (579) 



in a line designed to give an attenuation reduction ratio of 



r = Ko- (580) 



Assuming polyethylene insulating layers, we have for this line 



8i/a = 1/192.7 = 0.00519, (581) 



and from (578) the corresponding value of C is 



C = 247.6. (582) 



The value of v for which the relative increase in Re Ai due to the fluctua- 

 tions is, say, one-quarter is given by 



C' 1 C 



4' V2 



= 17.7. (583) 



Thus a 1 per cent fluctuation in e, repeated at intervals of about one- 

 eighteenth of the stack width, will cause only a 25 per cent increase in 

 attenuation, even for a Clogston line which is designed to have only one- 

 tenth of the attenuation constant of a conventional line of the same size. 

 Finally there is the question of the effects of variations in the average 

 properties of the stack in both directions parallel to the layers. Mathemati- 

 cal analysis of even a simple case of longitudinal variation would be much 

 more difficult than what has been done here; yet on physical grounds it 

 seems very likely that such variations A\ill add an appreciable amount to 

 the total attenuation of the line. If we consider two cross sections of a 

 laminated cable separated by a certain distance and having different trans- 

 verse nonuniformities, the field pattern of the lowest mode vnW be differ- 

 ent at the two cross sections, and so in traversing the intervening distance 

 the power will be partly reflected and partly converted to higher modes 

 with higher attenuation constants. The reflected or mode converted power 

 Avill be at least partly lost, with a consequent increase in the overall at- 



