The Radiation Problem from a Vertical Hertzian Dipole Antenna above Flat and Lossy Ground: Novel Formulation in the Spectral Domain with Closed-Form Analytical Solution in the High Frequency Regime

Abstract

We consider the problem of radiation from a vertical short (Hertzian) dipole above flat lossy ground, which represents the well-known “Sommerfeld radiation problem” in the literature. The problem is formulated in a novel spectral domain approach, and by inverse three-dimensional Fourier transformation the expressions for the received electric and magnetic (EM) field in the physical space are derived as one-dimensional integrals over the radial component of wavevector, in cylindrical coordinates. This formulation appears to have inherent advantages over the classical formulation by Sommerfeld, performed in the spatial domain, since it avoids the use of the so-called Hertz potential and its subsequent differentiation for the calculation of the received EM field. Subsequent use of the stationary phase method in the high frequency regime yields closed-form analytical solutions for the received EM field vectors, which coincide with the corresponding reflected EM field originating from the image point. In this way, we conclude that the so-called “space wave” in the literature represents the total solution of the Sommerfeld problem in the high frequency regime, in which case the surface wave can be ignored. Finally, numerical results are presented, in comparison with corresponding numerical results based on Norton’s solution of the problem.