LAMINATED TRANSMISSION LINKS. II 1187 



and tlic asyuiploLic c\pre.s.sioii for tlie eigeu\'aluc ol' liie mode wiiicli ia 

 essentiall}^ confined to $o < ^ ^ 1 is 



1 - ^^[^ '"^1/(1 -V)d 



+ / 



C 21?' 



L2(l - ^o) (1 - ^o)^ 



(544) 



y (i-"^o)cJ- 



It is clear that if ^o < \ the latter mode lias the smaller attenuation 

 const ant, while if ^,i > | the former mode has the smaller attenuation 

 constant. 



It is not difficuh, although the details will be omitted here, to investi- 

 gate the behavior of the e'gen values for small T and to show that no 

 matter whether $o < | or ^o > h, the eigenvalue which starts from tt 

 at C = tends to the asymptotic value which has the smaller real part, 

 so that this eigenvalue, whether its asymptotic form be given by (543) 

 or (544), may be called Ai . It appears that if ^o < 1) then Im Ai is 

 positive for positive C , while if |o > I; then Im Ai is negative for posi- 

 tive C. 



An interesting mathematical phenomenon appears when ^o = hj so 

 that the discontinuity in /(|) is exactly at the center of the stack. In 

 this case, when C is small Ai and Ao are both real, Ai being somewhat 

 greater than tt" and A2 somewhat less than 47r". For a certain value of 

 (' the two eigenvalues coincide; this value is approximately 



C = 17.9, Ai = A2 = 25.6. (545) 



For larger \alues of (', Ai and A2 are complex conjugates (it seems to be 

 immaterial which is which) whose asymptotic forms are gi\'en by (543) 

 and (544) with ^0 = i 



Approximate values of Ai and A2 were found for the symmetric case, 

 ^0 = 0.5, and for one unsymmetric case, ^0 = 0.6, on the Laboratories' 

 general piu'pose analog computer for ^ C ^ 100, and were refined 

 afterward by desk computation, using a method of successive approxi- 

 mations to solve ecjuation (542). The real and imaginary parts of Ai/tt' 

 and Ao/x' are plotted in Fig. 22 for the symmetric case, where it 

 should be noted that different vertical scales are used for Re A/tt and 

 Im A/V". The corresponding eigenf unctions lOi(^) and wt{^) are shown 

 in Fig. 23 for C = 0, C = 17.9, which corresponds to etiual eigenvalues, 

 and C = 100. It will be recalled that w{^) is equal to Hx{y), and the other 

 field components can be derived from H^ by equations (519) if desired. 

 Fig. 24 shows plots of Ai/x^ and A2/7r^ for the unsymmetric case 

 ^0 = 0.6. 



