LAMINATED TRANSMISSION LINES. II 1 1 SO 



^'- ^ t < i^„ , 



w(^) = A sin [A + iCmon, 



w(^) = B cos [A - iC/2(l - $n)]-(| - ^), ko < ^ ^ h 



(547) 



Tlic ro(|iiiroments that w and dw/d^ must i)o contiiiiious at ^ = |^o 

 lead t(» the (Miuations 



A sin i[A + /CV2^o]-st,) = B cos i[A - /6V2(1 - ^o)]-{l - ^o) , 

 .1[A + iCmo\- cos i[A + /C/2^o]-s^o 



= B[\ - /C/2(l - ^o)]- sin i[A - 7-C/2(l - ^0)]^! - t„), 



(548) 



which will be consistent if tlie t'oHowiug characteristic ecination is 

 satisfied: 



tan 



|[A + iC/2^,]ko ^ cot |[A - iC /2(l - ^o)]^(l - ^0) ,^^g^ 

 [A + iC/2^o\^ [A - tC/2(l - ^0)]^ 



The roots in A of equation (549) are the eigenvahies corresiioncHnj;- to 

 the even modes of the symmetrical structure. 



When C = 0, the roots of (549) are A = tt', 9x", • • • .It appears tliat 

 for C > we have Re Ai > tt^ and Im Ai > 0. For large C the asymi)- 

 totio expression for Ai turns out to be 



C = 100 C = 100 



(a) (b) 



Fig. 23 — Real and iniaginary parts of the first two eigenfunctions, wi = wi +ivi 

 and wi = U2 + ivi , for the nonuniform stack of Fig. 22. 



