1186 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1952 



(^) /(^) constant except at single discontinuity . Let 



^ ^ < ?o , 



fiO = 



'2F0 



(539) 



1 



^0 < ^ ^ 1 , 



2(1 - ^0) ' 



where ^o is some fixed number between and 1 but not, in the cases of 

 interest, extremely close to either or 1. Solutions of (532) satisfying 

 the boundary conditions (533) are obviously 



w(^) = A sin [A + iC/2^o]k, ^ ^ < ^o , 



(540) 

 iv(^) = B sin [A - iC/2(l - ^o)]Hl - ^), ^o < ^ ^ 1, 



where A and B are arbitrary constants. The requirements that w and 



dw/d^ be continuous* at ^ = ^o lead to the equations 



A sin [A + iC/2^of^o = B sin [A - iC/2(l - ^o)]^(l - ^o), 

 ^[A + iC /2^of cos [A + iC/2^of^o 



= -B[A - iC/2(l - ^o)f cos [A - iC/2{l - ^o)f{l - ^o), 

 which will be consistent if this characteristic equation is satisfied: 



(541) 



tan [A + iC/2^oho , tan [A - iC /2{1 - ^o)]'(l - go) ^ ^ 



(542) 



[A + iC/2^o]^ [A - iC/2{l - ^,)]i 



The roots in A of equation (542) are the eigenvalues of the problem; the 

 eigenfunction corresponding to any given eigenvalue is given by equa- 

 tions (540) after the ratio B/A is determined from either of equations 

 (541). 



It is easy to show that when C = 0, the roots of (542) are Ai = tt , 

 A2 = 47r , • • • . For large values of C, representing relatively great 

 differences between the two parts of the stack, physical considerations 

 lead us to expect that there will be pairs of modes, one member of each 

 pair being essentially confined to each part of the stack and having a 

 propagation constant determined approximately by the width of that 

 part. It may in fact be shown that the asymptotic expression for the 

 eigenvalue of the mode which is essentially confined to the region 

 ^ g < go is 



C 



TT- 



A ;^ 72 



1 



(1 - go) 

 goC J 



— I 



'^ ?r /(T 



.2go ~ ^0 V ~ 



- go) 



goC 



* The continuity of dw/d^ is a consequence of the continuity of Ez 

 that we neglect any discontinuity in ^ at ^ = ^0 . 



; (543) 



provided 



