LAMINATED TUANSMISSIOX LINES. II 



1195 



Fig. 29 — Real and imaginary parts of A,/7r- for a nt)nuiiiform stack whose 

 avorago propcM-ties vary as one cycle of a cosine function across the stack. 



to -(-1 and back to —1. The eigenf unction of the lowest mode (m = 1) 

 is sei(r, t>), and the values of Ai may be obtained from Reference 29 for 

 ten equally spaced values of C out to C = 327r". Since our ^ is negative 

 imaginary, in the notation of this reference we have Ai = 4:t^^i . Ap- 

 proximate values of Ai/x obtained on the analog computer for C at 

 smaller intervals in the range ^ C ^ 100 are plotted in Fig. 29; 

 and the eigenfunctions are similar to those shown in Fig. 26 for the 

 symmetric rectangular step. The leading terms of the asymptotic formula 

 for Ai when C is large are as follows : 



Ai ^ [3.1416(7^ - 2.4674 - 0.9689C~' - • • • ] 



(569) 

 + i[C - 3.1416C' - 0.9689(r^ - • • • ]. 



V = 3. Now /(^) is a three-cycle cosine function and the lowest mode 

 corresponds to se'^ir, t?). Approximate values of Ai/tf" for ^ C ^ 100 

 were obtained on the analog computer and are plotted in Fig. 30; the 

 eigenfunctions are shown in Fig. 31 for C = and C = 100. 



v » 1. For a i/-cycle cosine variation, the lowest eigenfunction is 

 sei/^(r, t?), and for the lowest eigenvalue there is an approximate formula 

 given by McLachlan.^" Incidentally this formula predicts no imaginary 

 part for Ai if t? is purely imaginary and p> 1, which agrees approximately 

 with the results of our analog computations for i/ = 3; we found the 

 imaginary part of Ai to be only about 1 per cent of the real part even for 

 C = 100. If C is fixed, one expects that asv ^ oo the effects of the I'apid 

 fluctuations in/(^) will average out, so that Ai will ultimately approach 



'" Reference 28, p. 20, equation (6), where a corresponds to our \i , q to our 

 I?, and V to our l/v. McLachlan's formula was ostensibly derived for real q, but the 

 derivation appears equally valid for complex q. 



