The Horizontal Electric Dipole in a Conducting Half-Space, II

This report, Part II, constitutes the culmination of a research study which was described initially in a paper of the same Title, Part I, that appeared as SIO Reference 53-33, September 1953, and provides a further account of the mathematical theory involved in the determination of the electromagnetic field components generated by a horizontal electric dipole embedded in a conducting half-space separated from the non-conducting medium above by a horizontal plane. In particular, a detailed account is given of the computations involved for points of observation in the non-conducting medium when the depth of the source and the height of the point of observation are small in comparison with the horizontal range.The first part of this report is concerned mainly with the general evaluation of the fundamental integrals for both media by the double saddle point method of integration developed earlier, and the salient feature of the present analysis is the fact that the new asymptotic expansions are term-wise differentiable to any order with respect to three essential parameters: horizontal range, depth (or height) of dipole source, and height (or depth) of the point of observation. It is shown that this important achievement is a consequence of applying the saddle point method of integration to a more judicious choice of exponent with the result that the asymptotic expansions presented here are much simpler than those reported in Part I.The remainder of the report is concerned with the application of the new asymptotic expansions to the evaluation of the Cartesian components of the Hertzian vector and of the cylindrical components of the electromagnetic field vectors for points of observation in the non-conducting medium. Simplified approximations in which numerical substitutions can be readily made are presented for three distinct ranges corresponding to the asymptotic, the intermediate, and the near field; and, in each case, a detailed account is given of the power flow in the field. In addition, there is presented for the first time, for points of observation In the non-conducting medium, an approximation valid down to zero horizontal range, which is attained by equating to zero the propagation constant in the non-conducting medium. Numerical results are given in a manner similar to the numerical example presented in Part I