1192 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1952 



pressed, though less conveniently, m terms of Bessel functions of order 

 one-third.) It is easy to show that the boundary conditions (533) at 

 ^ = and ^ = 1 require 



AhM + Bhin) = 0, 

 Ahiir,) + Bh.2(T2) = 0, 



(556) 



where 



Tl 



(A + iC) 



(2cy ' 



Equations (55G) will be consistent if 



(A - iC) 



(2cy ■ 



/li(Ti)/l.2(r.>) - /ii(r2)/i2(ri) = 0; 



(557) 



(558) 



and this is the relation which must be satisfied by the eigenvalues 

 Ai , A2 , A3 , • • • , for any given value of C. 



Approximate values of Ai and A2 have been found using the analog 

 computer for the range ^ C ^ 100, with spot checks by numerical 

 solution of equation (558) ; and Ai/tt^ and A2/7r" are plotted in Fig. 27. 

 The eigenf unctions are qualitativelv similar to those shown in Fig. 23 

 for the stack with a symmetric discontinuity. As in the symmetric ex- 

 ample in case (i) above, we find that for small positive C, Ai is real and 

 greater than x-, while A2 is real and less than At-. The two eigenvalues 

 coincide at 



C ^A9. Ai = A2 :^ 29. (559) 



For larger values of C, Ai and A2 are complex conjugates. Their asymp- 



12 



10 



$3 



cr 



2 



6< 



E 



50 



c 



Fig. 27 — Real and imaginary parts of Aj/x- and Aj/'x- for a nonuniform stack 

 whose average properties vary linearly across the stack. 



