Electromagnetic Pulse of a Vertical Electric Dipole in the Presence of Three-Layered Region

Approximate formulas are obtained for the electromagnetic pulses due to a delta-function current in a vertical electric dipole on the planar surface of a perfect conductor coated by a dielectric layer. The new approximated formulas for the electromagnetic field in time domain are retreated analytically and some new results are obtained. Computations and discussions are carried out for the time-domain field components radiated by a vertical electric dipole in the presence of three-layered region. It is shown that the trapped-surface-wave terms should be included in the total transient field when both the vertical electric dipole and the observation point are on or near the planar surface of the dielectric-coated earth.


Introduction
It is known that the properties of the electromagnetic wave radiated by a vertical or horizontal electric dipole in the presence of layered region have been studied analytically by many researchers over a century.This problem has led to many published papers and was well summarized in two famous monographs [1,2].In a series of works by King et al., the approximated solutions are obtained for the electromagnetic field of a dipole in the presence of three-layered region [3][4][5].In the comments by Wait, it is claimed that the trappedsurface-wave term should be included in the solutions by King et al. [6].Lately, the old problem was revisited by many researchers, particularly including Mahmoud et al. [7], Collin [8,9], and Zhang and Pan [10].Subsequent work was also carried out and some new developments were made.The details are summarized in recent book by Li [11].It is concluded that the trapped surface wave can be excited efficiently by a dipole source in the presence of three-layered region [11][12][13].
Evidently, it is also necessary to study the properties and applications for the transient electromagnetic field of a dipole in a layered region.In 1950s, the first analytical solution for the transient electromagnetic pulse of a delta current in a vertical electric dipole on the interface between two different media was carried out by van der Pol [14].The subsequent contributions on the transient field radiated by a dipole source on or near the boundary of two different media were made by other pioneers [15][16][17][18][19][20][21][22][23].Some important work on the exact formulas for the transient field radiated by a vertical electric dipole with delta-function excitation on the boundary of two dielectrics was carried out by Wu and King [20].As the extensions of above works, the transient electromagnetic field of horizontal electric dipole on or near the boundary between two different media was reexamined analytically [24][25][26].
In Chapter 15 of the monograph [2], the propagation of the electromagnetic pulses radiated by a horizontal electric dipole with delta-function excitation in the presence of three-layered region was processed analytically.It is found that the trapped-surface-wave term was not included in the analytical formulas for the transient field components.The new developments on the analytical frequency-domain results for the trapped surface wave in [8][9][10][11][12][13] aroused interest in the study on the transient electromagnetic field due to a dipole in the presence of three-layered region.
In the present study, we are attempting to derive the analytical formulas for the transient pulses radiated by a vertical electric dipole with a delta-function excitation on the surface of a perfect conductor coated with a dielectric layer.

Transient Field of Vertical Electric Dipole
with Delta-Function Excitation The approximated formulas were derived for the electromagnetic field in the frequency domain excited by a vertical electric dipole in the presence of three-layered region [10,11].With the time dependence of   , these three components can be written in the following forms: In above formulas,  1 = √  2 + ( − ) 2 and  2 = √  2 + ( + ) 2 .() and   () are expressed as follows: where , and the pole  *  is determined by the roots of the following pole equation: The Fresnel function F( * ) is defined by where It is assumed that the intermediate layer is satisfied by the condition that   =  1  ≤ 0.6.Nominally, the dielectric layer is usually electrically very thin.Thus, we have With (7), by introducing the change of the variable  *  =  *  / 0 , it follows that With above approximations, these three components in frequency domain can be rewritten in the following simplified forms: In above three formulas, it is assumed that both the dipole and the observation point are on or near the surface of the coating conductor.It is noted that  * can be written in the following form: where   = /.

Approximated Formulas for Transient Field.
With the frequency-domain formulas for the field components and   replacing by /, the analytical formulas for the timedependent components can be obtained by using Fourier transforms.Thus, the three time-dependent field components can be written in the following forms: where It is seen that the integrals  1 ,  2 ,  5 , and  7 have been evaluated in [2,11].We write Next, it is necessary to evaluate the integrals  3 ,  4 , and  6 .With complex derivations, the evaluations of the integrals  3 ,  4 , and  6 can be obtained.We write In above formulas, the Heaviside unit step function () is defined by The Fresnel cosine integral  2 and sine integral  2 are written in the following forms: where  = ( −  *    ).The variable  is With the substitutions of ( 15) and ( 16) into ( 13) and considering  = /  , the approximate formulas of the timedependent field components can be derived readily.We write From above derivations and analysis, it is seen that both the trapped-surface-wave terms and the lateral-wave terms in ( 20)-( 22) converge with  −1/2 and  −2 , respectively.When both the vertical electric dipole and the observation point are on or near the planar surface of the dielectric-coated earth, the trapped-surface-wave terms are included in the total transient field components.

Computations and Discussions
For the asphalt-and cement-coated earth or ice-coated seawater at low frequencies, the earth or sea can be usually regarded as a perfect conductor, so that the region of interest is taken as a perfect conductor, a dielectric layer, and air above.With  = 10 km, the magnitudes of the component (2 0  1 ) 0 (, 0, ) due to a vertical electric dipole on the surface of a dielectric-coated conductor are computed at  1 = 2, 2.65, 2.85, and 3.65 and showed in Figure 2, respectively.It is noted that the properties of other two components  0 (, 0, ) and  0 (, 0, ) are similar to those of the component  0 (, 0, ).   the dielectric parameters and the thickness of the dielectric layer [11].Once the poles  *  are determined, the trappedsurface-wave terms can be computed readily.In Figure 4, the trapped-surface-wave terms   0 (, 0, ) are computed at  = 10 km and  = 20 km, respectively.

Conclusions
In this paper, the transient electromagnetic field components due to a delta-function current in vertical electric dipole on the surface of a dielectric-coated conductor are studied analytically.Particularly, the completed analytical formulas for the trapped-surface-wave terms are derived.It is concluded that both the trapped-surface-wave pulses and the lateralwave pulses decrease with the amplitude factors  −1/2 and  −2 , respectively.Furthermore, the present study can be extended to the case in which electromagnetic pulse is excited by a horizontal electric dipole in the presence of three-layered region.

2 InternationalFigure 1 :
Figure 1: Vertical dipole at height  in air near the surface of a planar perfect conductor coated with a dielectric layer.