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THE BELL SYSTEM TECHNICAL JOURNAL, MARCH l()o4 



and X = 27r. The Eo and the Ho of the incident wave are related by 

 Ho = Eo/^o = 1/fo since £"0 = 1. In rational il//v*S units fo = 1207r ohms. 

 The derivative in (6.1) may be obtained by differentiating expression 

 (5.11) for E and then setting ry = rjo. Use of the Wronskian (9.9) for 

 Vn{zo), Wn{zo) then takes (6.1) into 



fo./ = M, 



L 



dn 



sni irn 



{iwrUrXz)/Wn{Zo) 



(6.2) 



where iv = tan (6/2), Lo is shown in Fig. 5.1, 2 and Zo are given by (4.13) 

 and (4.17), and 



Mo = {i/STrY^^e "' sec - , 

 r = {^' + r,l)/'2. 



(6.3) 



In this Section r will be restricted to mean a radius vector drawn to the 

 trace of the cylinder on the (.r, y) plane of Fig. 6.1. 



Closing L2 on the right and on the left gives the two series 



ro./ = 20/0 Z i-ilVrUn{z)/Wn{Zo), < IV < 1, 



n=0 



(iwyUniz) 



UJ = -2/7ril/o2] 



sin irn dWn{zo)/dn 



(6.4) 

 1 < 10 < oo, (6.5) 



where ris is the sth zero of Wn(zo) regarded as a function of n. 

 For the half-plane case 770 = 0, and (9.4) gives 



WniO) = -r/2r(l + n/2). 



In this case the series (6.4) may be expressed as an infinite integral 



Fig. 6.1 — Relationship between surface current density J and electromagnetic 

 field when incident wave is horizontally polarized. E and / are normal to the 

 plane of the paper. 



