428 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1<)54 



the order of h~^^'^ when 7 is of moderate size. It will be shown later that 

 the current density decreases exponentially as one moves into the 

 shadow, and that its rate of decrease is related to that of the field as 

 shown in Fig. 2.6. 



We now take up the detailed discussion of the expressions for the field 

 and the current density. It is convenient to consider the current density 

 first. When a plane wave strikes a perfectly conducthig plane, the sur- 

 face current is proportional to the tangential component of H in the 

 incident wave, and flows at right angles to it. In the rational MKS units 

 we are using, the surface curi-ent density is two times the incident 

 tangential H. When we consider the illuminated side of a large parabolic 

 cylinder and calculate the current density by doubling the tangential 

 component of the incident H we obtain the approximations 



(2.18) 



which hold when x is large and negative. When h is very large but x/2h 

 small these formulas agree with the leading terms of (2.17) which were 

 obtained from our integrals for the current density. 



Expressions for the current density deep in the shadow may be ob- 

 tained from the leading terms of the convergent series by letting 7 

 become large and positive. The exponential decrease is found to be 



I fo./ I ^ 1.43/i~'" exp (-I.013.i7r'''), 



(2.19) 



I J. I ;^ 1.83 exp (-0.44.t/i~'''), 



where the numbers appearing in these equations are associated with the 

 smallest zeros of Ai{ii) and Ai'iii), respectively. These formulas for a 

 large cylinder are roughly similar to those for propagation over a smooth 

 earth. The radius of curvature at the crest of the cylinder is 2h. Setting 

 this equal to the radius of the earth gives an exponential rate of decrease 

 for / and J„ which is the same as that over a smooth earth for the 

 two polarizations. Of course, the coefficients multiplying the exponential 

 functions are different. This agreement is not surprising since the Airy 

 integrals are closely related to the Hankel functions of order l^3 used in 

 the smooth earth theory. 



The surface current densities as a function of the distance h — y 

 below the crest for h = 1000 and for h — are shown in Fig. 2.4. The 

 equation of the cylinder shows that h —y = x /4/i so that, from (2.19), 

 fo/ and Jv decrease in proportion to h~^'^ exp [ — 2.02bh~^'^ (h — yY''] 

 and exp [— .88/r^'* (h — yf^j, respectively, far down in the shadow. 



