432 



THE BELL SYSTEM TECHXICAL JOFRXAL, MARCH 1954 



2.03 e 



we see from (2.13) that 



|^(r) + 1/r, 



I ^.(r) + 1/r I ~ 3.42 e^ 



SO that the absolute value of the field is 



(2.25) 



E 

 H 



S-l/2 ,1/3 



1/3, 



{2Trp)~"' K" 2.03 exp (2.025 K"^p) 



(2.26) 



(hp) 



(vp) I i^ I ~ (27rp)"''' h}'^ 3.42 exp (0.882 h}"^p). 



where the angle ^ is negative. Thus, as Artmann has pointed out, the 

 field decreases exponentially as we go into the shadow. The larger h is, 

 the more rapid is the decrease. This supports the statement made earlier 

 that it is darker behind a large cylinder than behind a half -plane. 



Comparison of the expressions for the current density and field 

 strength for the shadow regions shows that there is a simple approxi- 

 mate relation between them. Near the crest of the cylinder, where x is 

 small, the radius of curvature is nearly 2h. Hence the tangent to the 

 parabola drawn from the point P (located at (p, i/') deep in the shadow) 

 touches the parabola at T where x is approximately —2h\f/. This is 

 shown in Fig. 2.6. Replacing x by —2h\l/ in the expressions (2.19) shows 

 that the current density at T is proportional to the field at P as given 

 by (2.26). It follows that 



(hp) I E/^oJ 1 ~ 1.41 h'" {2irpr"' 



(2.27) 



(vp) \H/J,\ - 1.87/i'''(27rp)"''-. 



This leads us to picture the field at P as being produced principally 

 by the surface currents around T. The effect of the stronger currents 



(A^) 



Fig. 2.6 — The field strength at point P (deep in the shadow) specified by the 

 polar coordinates (p, 4^) is nearly proportional to the current density at the tangent 

 point T specified by the rectangular coordinates (x, y). 



