DIFFRACTION OF RADIO WAVES BY A PAl{AHOLI(' CVLINDKH 443 



When the incident wave strikes the cylinder tlic loflected wave has 

 some of the characteristics of a wave spreadinji; radially outwards. Such 

 a wave contains the factor exp ( — ir) = exp [ — r(r + '7")/2] . Consideration 

 of the exponential factors in (4.6) and (4.11) suggests that the reflected 

 wave may be expressed as the sum of terms of the form (4.6). The co- 

 efficients in this series are to be determined so (hat E = or dll/d-q = 

 at the surface r? = rjo, the incident wave being represented by (4.12). 



For the case of horizontal polarization this procedure gives 



E = e'" sec (6/2) Z n!{- iw/2yUM [U,Xz') u .^^ 



(4.15; 



-Wn{z')USz,)/WrXz,)] 

 E = exp [— ix sin 6 -f iy cos d] 



- e'" sec (9/2) E n!{- iw/2yUn{z)Wr.{z')Ur.{z',)/Wr.{z,) ^^'^^^ 







for the complete field. These are special cases of Epstein's results. Here 

 ^■o is the value of z' which corresponds to the surface of the cylinder: 



z', = i~"\o = (2h/iy" (4.17) 



The entries for regions II and II' (these are regions in the w-plane 

 {m = n -\- 1) which, as Figs. 11.2 and 12.2 show, contain the large posi- 

 tive values of n) in Tables 12.2 and 12.4 of Section 12 may be used to 

 show that as n — > 00 



Vn(zo)/Wr.{zo) - ^-'" exp [2r,o{2iny'% 



Un{Zo)/M\{zo) = -1 - Vnizo)/Wn{Zo), 



Un{z)WnKz) 4[r(l + n/2)P 



{exp[- ^(.2n/iy"] + r" exp mn/iy^']}. 



(4.18) 



Since 



n!/\T(l + w/2)]''-2" (7rn/2)- 



-1/2 



the series in (4.15) and (4.16) converge if | w | = | tan 9/2 \ < 1. 



Series for H similar to those of (4.15) and (4.16) may ])e obtained for 

 the case of vertical polarization. The boundary condition at t? = 770 

 is now dll/dr] = 0. It is convenient to introduce the functions 'Un(z), 

 'Vn{z), 'Wn(z) defined by 



'Un(z) = -zUn{z) + dUn{z)/dZ (4.19) 



