(292) 



LAMI.VATKD TRANSMISSION' LINES. IT 1127 



to the conduction currents in the laminated medium. This is equivalent 

 to assuming that the bounchuy impedances Za(y) and Zi,(y) arc effec- 

 tively infinite, so that equation (279) reduces to the simple form 



Equation (290) may l)e converted to one involving ordinary Bessel 

 and Xeumaini functions by the substitution 



r' = -x', r, - ix. (291) 



Tiicn since 



K„(r,p) = wr'"^'' [./„(xp) - iXnixp)], 



the e(iuation may easily l)e transformed into 



Ji(xa)Xi(xh) - ./i(x6).Vi(xa) - 0. (293) 



For any given value of the ratio a/b, equation (293) has an infinite 

 number of real roots in x- The lowest root xi has been tabulated^* as 

 a function of h/a, and may be written in the form 



X, = #^, (294) 



where fi{a/b) is a monotone decreasing function of a/b which is equal 

 to 1.2197 when a/b = and to 1 when a/b = 1. Hence the attenuation 

 and phase constants of the principal mode in a coaxial Clogston 2 with 

 infinitesimally thin laminae and high-impedance walls are 



rflia/b) , . 



a = j= , (295) 



2VM/e g(b - ay 

 /3 = coV^e , (296) 



and again to this approximation tiicre is neither amplitude nor phase 

 distortion. 



Comparing equations (288) and (295), we see that the attenuation 

 constant of the principal mode in a coaxial Clogston 2 with infinitesimally 

 thin laminae (that is, the low-frequency attenuation constant if the 

 laminae are of finite thickness) is equal to the attenuation constant of 



** E. Jahrike and F. Emde, Tables of Functions, fourth ed., Dover, New York, 

 1945, pp. 204-207. What we call irji{a/h) is tabulated by Jahnke and Emde, p. 

 205, as {k — l)x/^', where k = b/n, while our/i(o/6) is plotted as 1 -(- a on p. 207. 



