The radiation problem from a vertical short dipole antenna above flat and lossy ground. Novel formulation in the spectral domain with closed form analytical solution in the high frequency regime

In this paper we consider 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. 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 field in the physical space are derived as one dimensional integrals over the radial component of wavevector, in cylindrical coordinates. 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 in the literature space wave, i.e. line of sight plus reflected EM field, 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 in the high frequency regime are presented in this paper, in comparison with corresponding numerical results based on Norton solution of the problem, i.e. space and surface waves.


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Abstract-In this paper we consider the problem of radiation from a vertical short (Hertzian) dipole above flat lossy ground, which represents the wellknown in the literature 'Sommerfeld radiation problem'. The problem is formulated in a novel spectral domain approach, and by inverse threedimensional Fourier transformation the expressions for the received electric and magnetic (EM) field in the physical space are derived as onedimensional integrals over the radial component of wavevector, in cylindrical coordinates. Subsequent use of the Stationary Phase Method (SPM) in the high frequency regime yields closedform 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 socalled in the literature 'space wave' (line of sight plus reflected EM field) 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 in the high frequency regime are presented in this paper, in comparison with corresponding numerical results based on Norton's solution of the problem (space and surface waves).

I. INTRODUCTION
The so -called 'Sommerfeld radiation problem' is a well known problem in the area of propagation of electromagnetic (EM) waves above flat lossy ground for obvious important applications in the area of wireless telecommunications [1,2]. The classical Sommerfeld solution to this problem is provided in the physical space by using the so-called 'Hertz potentials' and it does not endup with closed form analytical solutions. K. A. Norton [3] concentrated in subsequent years more in the engineering application of the above problem with obvious application to wireless telecommunications, and he provided approximate Manuscript received (month) XX, 20XX; accepted (month) XX, 20XX. solutions to the above problem, which are represented by rather long algebraic expressions for engineering use, in which the so -called 'attenuation coefficient' for the propagating surface wave plays an important role.
In this paper the authors take advantage of previous research work of them for the EM radiation problem in free space [4] by using the spectral domain approach.
Furthermore, in Ref. [5] the authors provided the fundamental formulation for the problem considered here, that is the solution in spectral domain for the radiation from a dipole moment at a specific angular frequency (ω) in isotropic media with a flat infinite interface. At that paper, the authors endup with integral representations for the received electric and magnetic fields above or below the interface [Line of Site (LOS) plus scattered fieldstransmitted fields, respectively], where the integration takes place over the radial spectral coordinate k ρ , in cylindrical coordinates. Then, in the present paper the authors concentrate to the solution of the classical 'Sommerfeld radiation problem' described above in the high frequency regime, where the radiation from a vertical dipole moment at angular frequency ω takes place above flat lossy ground [this is equivalent to the radiation of a vertical small (Hertzian) dipole antenna above the flat lossy ground]. By using the Stationary Phase Method (SPM method [6], i.e. high frequency approximation) integration over the radial spectral coordinate k ρ is performed and closedform analytical solutions for the received electric and magnetic (EM) fields are derived. These expressions coincide with the expressions corresponding to the EM field reflected from ground and originating from the image point, thus showing that this represents the total solution to the classical Sommerfeld radiation problem in the high frequency regime, where the surface wave can be ignored. Furthermore, the above derivations prove the accuracy of the expressions derived in this paper for the received EM field in spectral domain in all aspects. Finally, numerical results in the high frequency regime are presented in this paper, in comparison The radiation problem from a vertical short dipole antenna above flat and lossy ground : novel formulation in the spectral domain with closedform analytical solution in the high frequency regime

II. GEOMETRY OF THE PROBLEM
The geometry of the problem is given in Fig. 1. Here a Hertzian (small) dipole with dipole moment p directed parallel to positive xaxis, at altitude x 0 above the infinite, flat and lossy ground, radiates timeharmonic electromagnetic (EM) waves at angular frequency ω=2πf [exp(-iωt) time dependence is assumed in this paper]. Here the relative complex permittivity of the ground (medium 2) is ε΄ r =ε΄/ε 0 =ε r +ix , where x=σ/ωε 0 =18×10 9 σ/f , σ being the ground conductivity, f the frequency of radiation and ε 0 =8.854 ×10 -12 F/m is the absolute permittivity in vacuum or air. Then the wavenumbers of propagation of EM waves in air and lossy ground, respectively, are given by the following equations:  [8] and [9], we find the following expressions for the received EM fields [in the higher halfspace, x>0, i.e. Line of Sight (LOS) field and field scattered from the ground, for receiver positions above it], as given below : Similarly, the expressions for the received EM field in the lower halfspace (x<0), i.e. EM field transmitted below the ground, are given by the follwing expressions (see [5] and [7] for mathematical details) : In eqs. (3)  , where φ is the angle defined by the image point of the radiating dipole, the observation point and the horizontal line drawn from the above mentioned image point, and cosφ is given by eq. (14) above (i.e. angle φ is the wellknown in the literature 'grazing angle' [10]), as shown in Fig. 2, below.
where angle φ has been defined just above, and It can be easily derived from eqs. (17) and (18) above that : where ζ is the free space impedance. The above expressions (17) and (18) are the classical expressions for the EM field reflected from the lossy ground and originating from the image point, as shown in Fig. 2. Then, by using the newly derived in this paper expressions for the received EM field in spectral domain, eqs. (3) -(6) above, and by applying the SPM method, i.e.
in the high frequency regime, the classical 'space wave' in region x>0 [10] was derived. This result has the following two interesting consequencies : 1. The 'space wave' [10], which corresponds to the coherent summation (interference) of the reflected fields, eqs. (17) and (18) above, and the Lineof -Sight (LOS) fields (the latter not included in these equations) is the solution to the Sommerfeld radiation problem in the high frequency regime, where the so -called 'surface wave' can be ignored [3], [10]. 2. The validity of expressions (3) and (4) above for the scattered EM field above the lossy and flat ground, derived in a novel way in this paper, has been confirmed in all aspects in the high frequency regime. Then it appears that expressions (3) -(6) above represent a very convenient starting point for further research with respect to the calculation of the received EM field for any frequency of the radiating dipole (i.e. including also low frequency effects), either above or below the ground, in an exact analytical way using the 'residue theorem' [11], or in a numerical way (i.e. through numerical integration techniques).

V. ΝUMERICAL RESULTS IN THE HIGH FREQUENCY REGIME -COMPARISON WITH NORTON'S RESULTS FOR SPACE AND SURFACE WAVES
In this Section indicative numerical results are provided for the electric field (magnitude) at the receiver point as a function of the horizontal distance (ρ) between transmitting Hertzian dipole and receiver position. These numerical results include the electric field scattered from the ground, magnitude of eq. (17), above, the Lineof -Sight (LOS) field, the so -called 'space wave' [which is just the coherent summation (interference) of the two fields mentioned above] and, finally, the so -called 'surface wave', according to Norton [3,10]. Furthermore, these numerical results are provided for frequency of radiating dipole f=80 MHz (Fig.  3) or f=30 MHz (Fig. 4). Note that at the higher frequency of formulation [3,10] is rather negligible, as compared to the 'space wave', while it becomes rather more important at the lower frequency of 30 MHz (Fig. 4). Our proposed SPM method of Section IV (which is inherently a 'high frequency method') ignores this surface wave contribution in the high frequency regime.
Moreover, note that the problem parameters in Figs. 3 and 4 are selected as following : height of transmitting dipole x 0 =60m, height of observation point (receiver position) x=15m, current of the radiating Hertzian dipole I=1A, length of the Hertzian dipole 2h=0.1m (much smaller than the wavelength λ=c/f in both cases), relative dielectric constant of ground ε r =20 and ground conductivity σ=0.01 S/m. Finally, note that the relation between current I and dipole moment p is given by : I(2h)=iωp , where ω=2πf and i is the unit imaginary number. Fig. 3. Electric fields at observation point as a function of horizontal distance ρ between transmitting Hertzain dipole and observation point, for frequency f=80 MHz.. Here the various components of received electric field are shown as following : Lineof -Sight (LOS) field (circle), field scattered from ground (asterisk), 'space wave' (square) and 'surface wave' (diamond). Note that in this case Norton's 'surface wave' is rather negligible as compared to the corresponding 'space wave' [10]. Fig. 4. Similarly with Fig. 3 above, except that here the frequency of radiating Hertzian dipole is now equal to 30 MHz (lower frequency). In this case of lower frequency the 'surface wave' can not be considered negligible, as compared to the 'space wave' [10].

VI. CONCLUSIONS -FUTURE RESEARCH
In this paper we formulated the radiation problem from a vertical short (Hertzian) dipole above flat and lossy ground in the spectral domain, which resulted in an easy to manipulate integral expression for the received EM field above or below the ground. Subsequently, by applying the Stationary Phase Methοd (SPM) in the high frequency regime, the classical solution for the 'space wave' was rederived in a new fashion, thus showing that this is the dominant solution in this high frequency regime. Finally, numerical results in this high frequency limit were obtained, and they were compared to Norton's results for 'space' and 'surface' waves [3], [10].
Corresponding research in the near future by our research group will concentrate to the calculation of the received EM field below the ground at the high frequency regime (by using again the SPM method). Furthermore, and probably most important, to the calculation of the received EM field, above or below the ground, for any frequency of the radiating dipole, in an exact and analytical manner [11] or in a numerical way (i.e. through the use of numerical integration techniques). Moreover, we intend also to investigate the formulation of the same radiation problem in spectral domain, but now in the case of a horizontal radiating Hertzian dipole above flat and lossy ground. Finally, further investigations will be performed in the case of rough (and not flat) ground, in the case of curvature of the earth's surface for large distance communication applications etc.