420 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1954 



It turns out that the results for the parabolic cylinder are closely 

 related to those obtained by Sommerfeld for the diffraction of a plane 

 wave by a perfectly conducting half-plane. In fact, the two fields are 

 surprisingly similar in the region of the shadow boundary. More pre- 

 cisely, the fields are similar for values of the angle 4/, defined in Fig. 2.3, 

 such that (roughly) 



-11/3 



111 

 xj/ m radians | < - 



wavelength 



4 |_radius of curvature of cylinder at its crest. 



where the coefficient j/^ has been selected somewhat arbitrarily. For 

 larger values of | i/' | the difference between the fields becomes pro- 

 nounced. As we go deeper and deeper into the shadow, i.e. as \f/ becomes 

 more and more negative, the field behind a cylinder ultimately decreases 

 exponentially with xp. On the other hand, the field behind a half-plane de- 

 creases roughly as 1/| lA I- Since the exponential function decreases more 

 rapidly than does 1/| lA |j the shadow behind a hill is darker than the one 

 behind a half -plane. High in the illuminated region the field consists of 

 the incident wave plus the wave reflected from the illuminated portion 

 of the cylinder. For the half-plane this reflected wave is negligibly small 

 until \l/ reaches 180°. 



First we shall review the situation pictured in Fig. 2.1. An incident 

 wave comes in from the left and strikes a perfectly conducting vertical 

 half -plane which casts a shadow as shown. The electric and magnetic 

 intensities are proportional to exp (io)t) where t is the time and co is 

 the radian frequency. The unit of length is chosen so that X, the wave- 

 length, is equal to 27r. This is done in order to simplify the expressions 

 we have to deal with. For example, a plane wave of unit amplitude 

 traveling in the positive x direction, as shown in Fig. 2.1, is represented 

 by exp ( — ix). 



Sommerfeld's exact expressions, for the special case of horizontal 

 incidence sho^vn in Fig. 2.1, may be written as 



(hp) E = (e-'^ + S,) + ^2(0), (2.1) 



(vp) H = (e-" -f Si) -f ^3(0), (2.2) 



where (2.1) holds when the electric intensity E is parallel to the edge, 

 and (2.2) when the magnetic intensity H is parallel to the edge. From 



8 Math. Annalen, 47, p. 317, 1896. Sommerfeld's results have been described in 

 a number of texts on optics. The book, Huygens' Principle by Baker and Copson 

 (Oxford 2nd edition, 1950) deals with this and many similar problems. See also 

 Chap. 20 of Frank-von Mises, Die Differential -zind Integralgleichungen der Me- 

 chanik und Physik, 2nd edition, Braunschweig: F. Vieweg and Sohn (1935). 



