528 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1951 



The quantity on the right of equation (B-8) is of the order of d/h. We 

 write, therefore, 



{2aJ) coth (2a«,^) = f ^ j 



and 



{2awd) 



Thus we have approximately 



Furthermore, from equation (III-34) 



cosh a„ (If + /) = 1 - ^-^ti^ {jj + i (1 + 2P) (^J (B-12) 



We can now equate equations (B-11) and (B-12) and after suitable manip- 

 ulation obtain 



,.[i+o0); 



(B-9) 



(B-10) 



(B-11) 



-\/wVo< / 1 ~ 



■-<7t^)[H-^)©^KTy])'B 



-13) 



where we have now dropped a term of order ( ~ I compared to unity. 



Equation (B-13) is- of considerable interest. It is observed that for 

 ( — j <3C I - I the attenuation becomes that given in equation (V-8) for 



infinitely thin laminae. For f — j » ( -- j, on the other hand, the attenuation 



approaches that given in equation (A-2) for an unbounded array of finite 

 laminae. This can be considered in two ways. Let us ask for the condition 

 that the two terms contributing to the attenuation in (B-13) are equal. We 

 find for this to be true that 



'^ = fV3(i + A)(i)a 



(B-14) 



In other words, at a given frequency, the attenuation of the line can be 

 little reduced by making d larger than the value given in equation (B-14) 



