444 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1954 



and the two other equations obtained when U is replaced by V and W. 

 The prime is placed in front of U instead of behind to avoid mistaking 

 'Un(z) for dU„(z)/dz. The function 'Un{z') makes its appearance when 

 dH/dt] is calculated for the boundary condition. Since iy = —{z'-\- z''^)/2 

 we have 



-e'Wniz') = i-'V 'Un(z'). (4.20) 



or] 



The analogues of (4.15) and (4.16) for vertical polarization are (as- 

 suming now that H for the incident wave is of unit amplitude). 



H = c'" sec (6/2) Z n!{-iw/2rUn(z) ^^ 2i) 



[Uniz') - W„{z7Un{zo)nV,Xzo)l 



= exp [ — ix sin d + iy cos d] 



(4.22) 

 - e^^sec {e/2) J2 n!(-iw/2rUn{z)Wdz'yUn{zo)/'Wnizo), 







and these series converge if \ iv \ < 1. 



If the parabolic cylinder is merely a good conductor, instead of being 

 perfect, the boundary conditions at r? == ijo are approximately E = 

 — ^Ht, Et = ^H. Here E^ and Ht denote the ^ components of the elec- 

 tric and magnetic intensities and f is the intrinsic impedance 



r = [io>ui/(g + ic,e)V" (4.23) 



of the cylinder material, f is assumed to be small compared to the 

 intrinsic impedance fo = (mo/^o)^'^ = co/xo (since X = 2x) of the external 

 medium. In these expressions m, e, y are the permeability, dielectric 

 constant, and conductivity of the cylinder; and mo and eo refer to the 

 external medium. 

 When we set 



a = l~"\e + Vir ^./^, (4.24) 



the boundary conditon for hp becomes aE — —dE/dz' at z' = Zq. When 

 a is assumed to be constant we obtain 



00 



E = e'-" sec(e/2) E n!{-iw/2)"U.Xz) 



(4.25) 



[Udz') - W,Xz')[aU,Xzo) + 'U,Xzo)]/[aW.Xzo) + 'W,Xzo)]] 



" Electromagnetic Waves, S. A. Schelkunoff, D. Von Nostrand Co., N. Y. (1943) 

 p. 89. See also G. A. Hufford, Quart. Appl. Math. 9, pp. 391-403, 1952, where ref- 

 erence is made to the work of Leontovich and Fock. 



