Asymptotic Theory for Dipole Radiation

The basic theory for dipole radiation in the presence of a two-layer half-space is outlined with special reference to its use as a model for studying radio propagation through and over heavily vegetated terrain. The source dipole may be located above or below the top surface of the slab. The dipole orientation is either vertical or horizontal. The asymptotic derivations for the field expressions are carried out without making the usual assumption that the refractive index of the uppermost layer is large compared with unity. The final results exhibit the expected inverse square dependence of the field on horizontal range. IF IhTRODUCTION OR SHORT-RANGE communication (i.e., < 50 km), the earth’s boundary and its immediate environment determine the total path loss between transmitter and receiver. The influence of the local environment is particularly noticeable in the case of terrain that is heavily obscured by dense vegatation. There have been some attempts to describe this situation in terms of a single half-space model with semiempirical modifications [I], [2] introduced to account for the inhomogeneities. In this paper we discuss the theory for an idealized slab model that has already shown some success [3]-[6] in predicting received field strengths for propagation over terrain covered with dense vegetation. This same model has also been applied to electromagnetic investigations of other environments, such as in the polar ice caps and in deep basement rock layers [7]-[9]. Manuscript received May 10, 1967. The author is with the Institute for Telecommunication Sciences and Aeronomy, Environmental Science Services Administration, Boulder, Colo. 80302. He was formerly with the Division of Engineering and Applied Physics, Harvard University, Cambridge, Mass. VERTICAL DPOLE EXCITATION We consider &st; a vertical electric diple located at a height h over a two-layer conducting half-space. The situation is illustrated in Fig. 1 where we have used a cylindrical coordinate system ( p , 4, z) with the dipole of moment Ids located at z=R on the axis. It is assumed that free space, with properties eo and po, occupies the region z>O, while the two-layer medium occupies z z> D, the electrical properties are cl, el, and p1, and for D > z> m , the electrical properties are uz, ez, and p2 . The electromagnetic fields at the point P, in the free-space region, may be derived from a Hertz vector that has only a z component noP. A straightforward analysis [8] gives the following exact result, for a time factor exp ( iwt) , I d s [ e--il:oR no,=___-R + F ] (1) 4m’eow where Ro= [ (~ -hy+p~] l ;~ , R= [(~+h>~+p2]~~~, ko=(poeo)l% = w / c , and F is a contour integral. The latter is given by