LAMINATED TRANSMISSION LINES. II 1185 



(531) 



w(^) = IfM, 

 m = ^(//), 

 A = -T'la'. 

 'riu'ii oil making; use of (529) wo get 



(fw/d^' + [A - iCmM^) = 0, (532) 



with llic houudaiy conditions 



^v(0) = uil) = 0. (533) 



Once A has been determined for a i)articiilar mode, the i)ropagation 

 constant 7 is obtained from (52G) and (531), namely 



7 = /coa/mo€o [1 + A/iooflogQa^y. (534) 



Assuming as usual that the attenuation per radian is small, ^ve find that 

 the attenuation and phase constants are given by 



a = Re 7 = Re ~ — / — r- 5 , (535) 



2VMo/eo goa 



. A , ^ 



13 = Im 7 = ojVmo Co + Im ,^— _ "2 • (o36) 



2 V Mo/ Co S^oa 



The eigenvalues A of the differential equation (532) with boundary 

 conditions (533) may be found analytically for some simple forms of 

 /(^) , or numerically using a differential analyzer for any given /(^) which 

 does not fluctuate too rapidly. When C = 0, as in the case of a pei-fectly 

 uniform stack, the eigenvalues ai'e obviously 



Ai = tt", Ao = 47r , • • • , (537) 



corresponding to the eigenfunctions 



u'l = sin x^, uh = sin 27r^, • • • . (538) 



As C varies continuously, we expect the eigenvalues and eigenfunctions 

 to vary continuously in a manner depending on /($). In the following 

 paragraphs we shall discuss the behavior of Ai , and sometimes also 

 A2 , as a function of C for various simple types of nonuniformity. 



