DIFFRACTION OF KADIO WAVES BY A PARABOLIC CYLINDER 419 



diffraction pattern is shifted l)y an amount proportional to the J-^ power 

 of the radius of curvature. Whether the shift is towards the shadow or 

 in the opposite direction depends upon the polarization of the incident 

 wave. 



In this paper we derive some of the results given by Fock and Art- 

 mann l)y starting with Epstein's series. In addition we investigate the 

 diffracted field at a great distance behind the cylinder. The cylinder is 

 assumed to be a perfect conductor in all of our work except for a few 

 equations given near the ends of Sections 4, 6, and 7. The procedure is 

 similar to that used in the smooth-earth theory.^ The series is converted 

 into an integral and then the path of integration is deformed so as to 

 gain as much simplification as possible. As might be expected, the results 

 for a large parabolic cylinder are similar in some respects to those for a 

 smooth earth. Aluch of the work requires a knowledge of the behavior 

 of parabolic cylinder functions of large complex order. Although several 

 studies of this behavior have been published, the results are not in the 

 form required. For this reason, and for the sake of completeness, several 

 sections dealing with the properties of parabolic cylinder functions have 

 been included. 



Incidentally, W. Magnus^ has studied the field produced by a line 

 source located at the focus of a parabolic cylinder. However, his problem 

 is somewhat different from the one with which we are concerned. 



I am grateful to Prof. Erdelyi of the California Institute of Tech- 

 nology and to my colleagues at Bell Telephone Laboratories for helpful 

 discussions and references which resulted in a number of improvements 

 throughout the paper. I am also indebted to Miss Marian Darville for 

 performing the rather laborious computations upon which the various 

 curves and tables are based. 



2. DISCUSSION OF RESULTS 



Various expressions are given later for the electromagnetic field in 

 terms of parabolic cylinder functions. In this section we shall confine a 

 good share of our attention to the case in which the cylinder is very 

 large compared to a wavelength so that the cylinder functions may be 

 approximated by Airy integrals. As in the remainder of the paper, we 

 shall be concerned chiefly with the field behind the cylinder and the 

 current density on the cylinder. 



' By "Smooth-earth theory" we mean the formulas for the field produced by a 

 dipole near a large sphere. A complete discussion of the theory is given in the 

 book by H. Bremmer, Terrestrial Radio Waves (Elsevier, 1949). 



* Zur Theorie des zylindrisch-parabolischen Spiegels, Zeitschr. fiir Physik, 

 118, pp. 343-356, 1941. 



