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.

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 [

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 [

In Chapter 15 of the monograph [

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.

The relevant geometry and Cartesian coordinate system are illustrated in Figure

Vertical dipole at height

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 [

It is assumed that the intermediate layer is satisfied by the condition that

With (

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

With the frequency-domain formulas for the field components and

It is seen that the integrals

Next, it is necessary to evaluate the integrals

In above formulas, the Heaviside unit step function

With the substitutions of (

From above derivations and analysis, it is seen that both the trapped-surface-wave terms and the lateral-wave terms in (

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

Magnitudes of

With

Magnitudes of the trapped-surface-wave term

Comparisons between the trapped-surface-wave pulses

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 lateral-wave pulses decrease with the amplitude factors

T. He is a coauthor who improved the English writing of this paper.

The authors declare that they have no competing interests.

This study has been supported by the National Natural Science Foundation of China under Grant no. 61271086 and no. 60971057. The authors are grateful to all referees for their constructive comments and suggestions in improving the quality of this paper. The authors thank all associate editors very much for their help and encouragements. The authors also wish to thank all reviewers for their insightful suggestions and comments.