. The Bell System technical journal . of the incident wave are related byHo = Eo/^o = 1/fo since £0 = 1. In rational il//v*S units fo = 1207r derivative in () may be obtained by differentiating expression() for E and then setting ry = rjo. Use of the Wronskian () forVn{zo), Wn{zo) then takes () into = M, L dn sni irn {iwrUrXz)/Wn{Zo) () where iv = tan (6/2), Lo is shown in Fig. , 2 and Zo are given by ()and (), and Mo = {i/STrY^^e sec - ,r = {^ + r,l)/2. () In this Section r will be restricted to mean a radius vector drawn to thetrace of the cylinder
. The Bell System technical journal . of the incident wave are related byHo = Eo/^o = 1/fo since £0 = 1. In rational il//v*S units fo = 1207r derivative in () may be obtained by differentiating expression() for E and then setting ry = rjo. Use of the Wronskian () forVn{zo), Wn{zo) then takes () into = M, L dn sni irn {iwrUrXz)/Wn{Zo) () where iv = tan (6/2), Lo is shown in Fig. , 2 and Zo are given by ()and (), and Mo = {i/STrY^^e sec - ,r = {^ + r,l)/2. () In this Section r will be restricted to mean a radius vector drawn to thetrace of the cylinder on the (.r, y) plane of Fig. Closing L2 on the right and on the left gives the two series = 20/0 Z i-ilVrUn{z)/Wn{Zo), 0 < IV < 1, n=0 (iwyUniz) UJ = -2/7ril/o2] sin irn dWn{zo)/dn ()1 < 10 < oo, () where ris is the sth zero of Wn(zo) regarded as a function of the half-plane case 770 = 0, and () gives WniO) = -r/2r(l + n/2). In this case the series () may be expressed as an infinite integral. Fig. — Relationship between surface current density J and electromagneticfield when incident wave is horizontally polarized. E and / are normal to theplane of the paper. DIFFRACTION OF RADIO WAVES BY A rARAHOLIC fVLINDKH 153 when the integral for r(l + n/2) is inserted and the sum () [,the sum for the generating function of U„(z)] used. Tntogratiug part ofthe result gi\es a fo-/ = {2/iirry - cos - (()- 2/t sin ^ c^ / exp (-//) dt . ■i ^5 sin («/2) In () /• is the distance along the half-plane as measuied from theedge: r = ^~/2 = \ y \. Positive values of ^ correspond to the shadowedside of the half-plane and negative values to the illuminated side. Withthis interpretation () agrees with the current density obtained fromSommerfelds expression for the field. Although () has been derived from () on the assumption that0 < u? < 1 it also holds for 0 < w < oo as may be shown by analyticcontinuation. Again,
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