DIFFRACTION" OF HADIO WAVKS HV A I'AKAHOLIC CYLIXDKH l")! 



may also lie anywhere in < iv < oc , but in (5.18) it is resliictiMl lo 

 I < w < =o except when ^ > r? (roughly the shadow region) in which 

 case w may be unity. In (5.18) n = n', is the sth zero of 'ir„(2o). Tlie 

 zeros of 'Wn(zo) interlace those of Wn(zo) shown in Fig. 5.1. 



When h = 770/2 = the parabolic cylinder degenerates into a half- 

 plane antl our solutions reduce to Sommerfeld's expressions for waves 

 diffracted by a half -plane. It may be shown that if 



To = V cos - + ^ sm - = (2/-) sni r^— h t 1 , 



we have 



S-iiO) = -{i/tY- exp [ix sm 9 + ly cos ^j / c~"' d(, 



SM = -82(0). 



(5.19) 



(5.20) 



When this expression for ^2(0) is added to (5.6) we obtain Sommerfeld's 

 result for the case of horizontal polarization. 



One may verify that the series (5.12) leads to Sommerfeld's result as 

 z'o approaches zero. By neglecting 0(2^) terms in (9.3) and setting n^ = 

 — 2.S- + p, for the sth zero of Wn(zo) we may obtain the following rela- 

 tions which are needed in the course of the verification 



V, = -4:izoTis -f l/2)/7rr(s) + • • • , 



dWn{zo)/dn at ns = r(s)/4 + • • • , 



Vniz'o) at n, = -2izoT{s -f l/2)/7r + • • • , (^.21) 



r Vnjz'o) 1 ^ ^^^ 



\_s\mrndW„{zo)/dnjn=n, 



6. SURFACE CURRENTS ON THE CYLINDER 



As shown in Fig. 6.1, the surface current J on the perfectly conducting 

 cylinder t? = tjo is parallel to the crest of the cylinder (and to the elec- 

 tric intensity E) when the incident wave is hoi-izontally polarized. We 

 ha\e from Maxwell's equations in parabolic coordinates 



J = [-//,],.,o = (rTo)-^ (2r)-"'- [dE/drJl,^,,. (6.1) 



Here Ht is the component of magnetic intensity in the ^-direction. 

 fo is the intrinsic impedance of free space given by fn = (Mn/en)''" = como 

 where the second part of the equation follows from 27r/X = w(jtX(,eo)''^ 



