Radio-Wave Propagation and Electromagnetic Surface Waves.

1. A Discrepancy and Its Resolution.-The problem of the propagation of radio-waves along the surface of the (plane) earth was first treated in a celebrated paper by A. Sommerfeld (1909).1 Let the (r, (p)-plane of a cylindric system of coordinates coincide with the plane of the earth surface and let the z-axis point vertically upward. Sommerfeld considered a vertically oscillating dipole (radio-antenna) in the air, close to the origin (z = 0, r = 0) and asked what secondary waves were produced by the discontinuity due to the presence of the partially conducting earth. His investigation led to the result that the Hertzian vector II describing the electromagnetic field can be divided, in either medium (air and earth), into two parts: H = Q + P, where the part Q has the character of space-waves, which at large distances from the origin are proportional to R-1, if R = (r2 + Z2)/2. On the other hand, P is a surface-wave: at large distances it becomes proportional to r-' /2 and is restricted to the vicinity of the earth surface. Ten years later (1919) the problem was re-:xamined by H. Weyl2 who used a somewhat different mathematical approach. He obtained a solution which was identical with Sommerfeld's space waves Q but which did not contain the surface wave P. The reason for this discrepancy has never been satisfactorily explained. Since Weyl's method seemed mathematically simpler his result was favored by public opinion in numerous papers by other authors. Finally, in 1935, Sommerfeld himself conceded that the surface wave has no reality.3 Referring to F. Noether,4 he attributed this to an inaccuracy in the evaluation of his general solution. This evaluation consisted in carrying out a contour integration in which a pole of the integrand yielded the surface-wave P and two branch-cuts accounted for the space-wave Q. According to Noether's explanation the pole is so close to one of the branch-cuts that the method of integration used by Sommerfeld, possibly, was not sufficiently reliable. The question has not only historical interest but retains even now some actuality because Sommerfeld's method was recently applied by C. Y. Fu° to the analysis of the propagation of seismic waves. In this case, the singularities, instead of nearly coalescing, as in Sommerfeld's problem, are rather far apart so that the evaluation does not present any difficulties. If Noether's suggestion is the complete explanation for the absence …