Scripps Institution of Oceanography

This report gives a thorough and complete account of the mathematical problems involved in the determination of the electromagnetic field components generated by a horizontal electric dipole embedded in a conducting half-space whose plane boundary is also horizontal. The problem is formulated by introducing the Hertzian vectors or polarization potentials and employing the technique of triple Fourier transforms in Cartesian coordinates, in configuration space as well as in transform space. Suitable integral representations are obtained for the components of the Hertzian vectors.

It is shown that this formulation is fundamental in the sense that it contains 'per se’ all other known formulations of the problem. Thus, by suitable transformations of the variable or variables of integration one readily obtains the formulations of Sommerfeld (1909), Weyl (1919) , Ott (1942) , etc. Further, by correctly specifying the original path of integration in Sommerfeld’s formulation of the problem and by carefully analyzing the class of permissible deformations of the original path, the whole moot question of poles and residues is clarified in a straightforward manner.

The report also presents the complete independent solution of the static problem and it is shown that all solutions for the alternating case converge uniformly to the static solutions as the frequency is made to vanish. Further, the static solution is applied to an extended source pointing out the way for a similar extension of the alternating dipolar solution.

The Cartesian components of the Hertzian vectors and the cylindrical components of the field vectors (E and H) are given, for both media, in terms of four fundamental integrals, which are expanded in asymptotic series by saddle point methods, two of these integrals belonging to the conducting medium and the other two to the free space above. It is shown, in the treatment of each of the four integrals mentioned, that there are two distinct asymptotic contributions arising from two saddle points and the notable feature of the results is that one of the saddle points yields a solution which is not exponentially attenuated in the horizontal direction in accordance with known experimental results. Thus, the possibility of large ranges of the field in the horizontal direction at depths which are not too great is clearly established.