We consider the problem of radiation from a vertical short (Hertzian) dipole above flat lossy ground, which represents the wellknown "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.

In this paper we reconsider the problem of radiation from a vertical short (Hertzian) dipole above flat lossy ground, which represents the well-known in the literature 'Sommerfeld radiation problem'. Particularly, we expand on the problem's solution in the spectral domain, which ends up into simple one dimensional (1-D) integral expressions for the received EM field and represent the exact EM solution to the aforementioned problem. The advantage of the derived expressions is based on the fact that they can be analytically evaluated through the use of the Stationary Phase Method (SPM), which however is valid in the high frequency regime. To our knowledge, the literature lacks specifying the exact frequency range over which the SPM method is applicable. Hence, in this paper numerical integration on the above mentioned integral expressions is applied and the results are compared with those obtained through the SPM. These comparisons are then used as the basis of determining the frequency limits of applicability of the SPM solution. In fact it is shown that due to the specific peculiarities of the integrated expressions, which possess certain singularities, it is often preferable to use the SPM method as the best estimate for the received signal level, especially for most practical frequencies of interest in the area of wireless telecommunications. Additional practical implications that these findings suggest, as well as further research to be triggered, as a result of the overall progress that has been made by our research group so far on the specific subject, are provided as well.

In this paper we examine the problem of radiation from a vertical short (Hertzian) dipole above flat lossy ground, known in the literature as the 'Sommerfeld radiation problem'. Our formulation is in the spectral domain and ends up into simple one dimensional integral expressions for the received electromagnetic (EM) field, representing the exact solution of the problem. The problem can be solved analytically in an approximate sense in the high frequency regime using the Stationary Phase Method (SPM). In this paper the above spectral integrals for the received EM field are also mathematically represented as integrals over the 'grazing angle', a formulation that allows for a more accurate calculation since it avoids the singularities of the integrand expression. Also, a new SPM analytical solution, based on the above novel integral representation is obtained. Numerical comparisons between our SPM solution and the integral representations for the received EM field show that neither the horizontal Transmitter-Receiver distance, nor the frequency of operation are alone sufficient indicators regarding the most appropriate method to use (SPM or Numerical Integration). Instead, such a decision is to be based on their combined effect, given by their product k•r (electric distance).

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In this paper we consider the problem of radiation from a vertical short (Hertzian) dipole above flat ground with losses, which represents the well -known in the literature 'Sommerfeld radiation problem'. We end -up with a closedform analytical solution to the above problem for the received electric and magnetic field vectors above the ground in the far field area. The method of solution is formulated in the spectral domain, and by inverse three -dimensional Fourier transformation and subsequent application of the Stationary Phase Method (SPM) the final solutions in the physical space are derived. To our knowledge, the above closed -form solutions are novel in the literature for the Sommerfeld radiation problem.