ELEKTRONIKA IR ELEKTROTECHNIKA, ISSN 1392-1215, VOL. 22, NO. 2, 2016 1 Abstract—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). Index Terms—Sommerfeld radiation problem; spectral domain; grazing angle; stationary phase method; electric distance. I. INTRODUCTION The ‘Sommerfeld radiation problem’ is a well-known problem in the area of propagation of electromagnetic (EM) waves above flat and lossy ground with applications in the area of wireless and mobile telecommunications [1]–[10]. The original Sommerfeld solution to this problem is provided in the physical space by using the ‘Hertz Manuscript received 3 October, 2015; accepted 26 February, 2016. potentials’ and it does not end up with closed form analytical solutions. Subsequently, K. A. Norton [11] focused in the engineering application of the above problem and provided approximate solutions represented by rather long algebraic expressions, suitable for engineering use. In the above expressions, the so-called ‘attenuation coefficient’ for the propagating surface wave plays an important role. In this paper, the authors advance on previous research work of theirs, concerning the solution of Sommerfeld’s problem in the spectral domain. Namely, in [12] the fundamental integral representations for the received EM field were given. Furthermore, in [13], [14] the Stationary Phase Method was proposed (SPM, [15], [16]) and as a result novel, closed-form analytical expressions were derived, for use in the high frequency regime. Moreover, in this article the authors elaborate more on the integral expressions of [12] – [14]. Particularly, it is shown that an appropriate selection of the integration variable and subsequent use of the SPM method lead to useful insights regarding the propagation mechanism. The expressions obtained are also more suitable for calculation purposes through numerical integration (NI) techniques, since some inherent singularities in previously derived integral representations are now removed, as shown in Section IV. As stated in [17], [18] determining the necessary conditions for the applicability of the SPM method, an inherently high frequency technique, is essential. There, the issue was investigated for the practical case of transmitter – receiver pairs that are separated by a rather long distance, in which case it was apparent that for most frequencies of interest in the telecommunication area, frequency alone is a good criterion for the selection of the most appropriate method for EM field calculation at the receiver’s point. Radiation of a Vertical Dipole Antenna over Flat and Lossy Ground: Accurate Electromagnetic Field Calculation using the Spectral Domain Approach along with Redefined Integral Representations and corresponding Novel Analytical Solution Ariadni Chrysostomou 1 , Sotiris Bourgiotis 1 , Seil Sautbekov 2 , Konstantina Ioannidi 1 , Panayiotis Vassilios Frangos 1 1 School of Electrical and Computer Engineering, National Technical University of Athens, 9, Iroon Polytechniou Str., 157 73 Zografou, Athens, Greece 2 Eurasian National University, 5 Munaitpassov Str., Astana, Kazakshtan pfrangos@central.ntua.gr http://dx.doi.org/10.5755/j01.eie.22.2.14592 54