450 



THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1954 



as given by the left hand side of (5.8), to he the impressed field in the 

 equation 



E = impressed field + reflected field 



From the form of (5.8) and the discussion of expression (4.6) (given 

 between equations (4.14) and (4.15)) we are led to assume the reflected 

 field to be an outgoing wave of the form 



e sec - 



2i 



im 



r(n + 1) 



sm Trn 



V n{z)W r,{^z')a{n) dn 



where ain) must be chosen so as to make E vanish on the surface of the 

 cylinder. This gives a{n) = F„(2o)/^n(2o) and leads directly to the 

 expression (5.11) for E. 



When the incident wave is vertically polarized, integrals for H may 

 be obtained from the series of Section 4 in much the same manner as 

 were the integrals for E. The analogues of the earlier results are 



H = exp( — /.r sin 6 + iy cos 6) 



iy ^ 



- 14!^ f /l^y r(^i±l) uMW.iz'YU.U) dn/'W,Xzo), 

 2i Jli \ 2 / sm irn 



H = exp {-2X sin 6 + iy cos d) -{- Si + S^ih), 



e sec 



Szih) = 



2i 



LM 



iwV r(/i + 1) 



sm irn 



U„{z)Wn{z') 



'Vniz'o) dn/'Wn{Zo), 



(5.14) 



(5.15) 

 (5.16) 



H = 



e sec - 

 2i 



iw\ T{n + 1) 



r iwvn 



Jl. \2 s 



sui irn 



Un{z)Wn(z') 



H = -xc'^sec (0/2)1; 



_'W,XZ0) Wniz')_ 



■{iw/2)''T{n + l)Un(z)Wniz'yVn{zo 

 sin irnd'Wn{zo)/dn 



dn, 



(5.17) 



(5.18) 



In these formulas 'Un{z), etc. are defined by (4.19); w, z, z' by (4.13), 

 z'o by (4.17). In (5.14) w is restricted to < w < 1. In (5.15) S^ih) is 

 given by (5.16), and w may be anywhere in < w < co . In (5.17) w 



