418 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1954 



As an application of our results, a brief study is made of the hypo- 

 thesis that very short radio waves are transmitted far beyond the hori- 

 zon by successive diffractions over hills or ridges. The ridges are as- 

 sumed to be of equal height, to l)e 40 miles apart, and to have a radius 

 of curvature of 100 meters at their crests. At the frequencies considered 

 (30 and 300 mc) and at the small angles of diffraction required to go 

 from one crest to the next, the ridges act like knife-edges. 



At 30 mc and a distance of 280 miles the calculated field is in fair 

 agreement with the observed field . At 300 mc and at the same distance 

 the calculated field is about 50 db below the observed field. This sug- 

 gests that the 30 mc long distance propagation may possibly be ex- 

 plained by successive diffractions. The discrepancy at 300 mc may be 

 due to any one of a numl^er of reasons. For example, it may be due to 

 the effect of the non-uniformity of the atmosphere, or to the roughness 

 of our approximations (for one thing, we neglect reflections from the 

 ground between the ridges). 



The first theoretical work on the diffraction of plane electromagnetic 

 waves by a parabolic cylinder apparently was done by P. S. Epstein.^ 

 His work makes use of a series of parabolic cylinder functions. When the 

 cylinder is large many terms are required for computation. By "large" 

 we mean that the radius of curvature at the vertex of the cylinder is 

 large compared to the wa^'elength of the radio wave. 



An entirely different approach was used by V. Fock*'^ in 1946. In the 

 first paper Fock sketches the derivation of an integral for the current 

 density on a large paraboloid of revolution. In the second paper he re- 

 derives this integral by considering the form assumed by the field equa- 

 tions when a plane wave strikes a gently curved conductor at grazing 

 incidence. His result gives the change in current density on a large and 

 highly conducting parabolic cylinder as we go from the illuminated 

 region into the shadow. 



In 1950 K. Artmann^ examined the diffraction field far behind a large 

 circular cylinder. He showed that, for small angles of diffraction, the 



2 A summary of experimental data is given by K. Bullington, Radio Trans- 

 mission beyond the Horizon in the 40-400 Megacycle Band, Proc. I.R.E., 41, pp. 

 132-135, 1953. 



' Dissertation, Munich (1914). A more accessible account of this work is given 

 in the Encyklopaedie der Matli. Wiss. 5, Part 3 (1909-1926) Phys. p. 511. See also 

 H. Bateman, Partial Differential Equations of Math. Phys., (Cambridge Univ. 

 Press 1932) p. 488._ 



* The Distribution of Currents Induced by a Plane Wave on the Surface of a 

 Conductor, J. Phys. (U.S.S.R.), 10, pp. 130-136, 1946. 



* The Field of a Plane Wave Near the Surface of a Conducting Body, J. Phys. 

 (U.S.S.R.), 10, pp. 399-409, 1946. 



* Beugung polarisierten Lichtes an Blenden endlicher Dicke im Gebiet der 

 Schattengrenze, Zeitschr. fur Phys. 127, pp. 468-494, 1950. 



