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

Authors: D. Cheng; T. T. Gu; P. Cao; T. He; K. Li
Journal: International Journal of Antennas and Propagation
 (2015)

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## Abstract

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.


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## Body

## 1. 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 [3–5]. In the comments by Wait, it is claimed that the trapped-surface-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 [11–13].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 [15–23]. 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 [24–26].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 [8–13] 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.


## 2. Transient Field of Vertical Electric Dipole with Delta-Function Excitation

### 2.1. The Simplified Representations for the Field Components in Frequency Domain

The relevant geometry and Cartesian coordinate system are illustrated in Figure1, where a unit vertical electric dipole in z^ direction is located at (0,0,d) and surrounded by air. Region 0 (z≥0) is the upper half-space filled with air, the intermediate layer in Region 1 is characterized by the permeability μi and relative permittivity εi, and Region 2 (z≤-l) is the rest space representing a conducting dielectric base. It is assumed that both Region 0 and Region 1 are nonmagnetic and Region 2 is a perfect conductor. So that the wave numbers of the three regions are(1)k0=ωμ0ε0=ωc,k1=ωμ0ε0ε1=ωε1c,k2⟶∞.Figure 1
Vertical dipole at heightd in air near the surface of a planar perfect conductor coated with a dielectric layer.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 eiωt, these three components can be written in the following forms:(2)B0φρ,z=-μ04πeik0r1ρr1ik0r1-1r12-μ04πeik0r2ρr2ik0r2-1r22+k02μ012πρei3π/4∑jγ1λj∗tan⁡γ1λj∗lq′λj∗γ0λj∗λj∗3/2eiγ0λj∗z+d+iλj∗ρ-k03μ0k12-k022k121πk0ρtan⁡k12-k02leik0r2+π/2e-ip∗Fp∗,(3)E0ρρ,z=-ωμ04πk0eik0r1ρr1z-dr1ik0r1-3r12-3ik0r13-ωμ04πk0eik0r2ρr2z+dr2ik0r2-3r22-3ik0r23-ωμ012πρeiπ/4∑jγ1λj∗tan⁡γ1λj∗lq′λj∗λj∗3/2eiγ0λj∗z+d+iλj∗ρ+ωk02μ02πk121πk0ρk12-k02tan⁡k12-k02leik0r2πk0ρ+πk0k12-k02k12tan⁡k12-k02le-ip∗Fp∗,(4)E0zρ,z=ωμ04πk0eik0r1ik0r1-1r12-ik0r13-z-dr12ik0r1-3r12-3ik0r13+ωμ04πk0eik0r2ik0r2-1r22-ik0r23-z+dr22ik0r2-3r22-3ik0r23-ωμ012πρei3π/4∑jγ1λj∗tan⁡γ1λj∗lq′λj∗γ0λj∗λj∗5/2eiγ0λj∗z+d+iλj∗ρ+ωk02μ02k121πk0ρk12-k02tan⁡k12-k02leik0r2+π/2e-ip∗Fp∗.In above formulas, r1=ρ2+(z-d)2 and r2=ρ2+(z+d)2. qλ and q′λ are expressed as follows:(5)qλ=k12γ0-ik02γ1tan⁡γ1l,q′λ=-k12λγ0+ik02λγ1tan⁡γ1l+ik02λlsec2⁡γ1l,where γ0=k02-λj∗2, γ1=k12-λj∗2, and the pole λj∗ is determined by the roots of the following pole equation:(6)k12γ0-ik02γ1tan⁡γ1l=0.The Fresnel function Fp∗ is defined by(7)Fp∗=C2p∗+iS2p∗=121+i-∫0p∗eit2πtdt,where(8)p∗=k0ρ2z+dρ-ik0k12-k02k12tan⁡k12-k02l2.It is assumed that the intermediate layer is satisfied by the condition thatl′=k1l≤0.6. Nominally, the dielectric layer is usually electrically very thin. Thus, we have(9)tan⁡k12-k02l≈ωlcε1-1.With (7), by introducing the change of the variable Qj∗=λj∗/k0, it follows that(10)q′λj∗≈-ω2Qj∗c2ε11-Qj∗2-i2ωlc.With above approximations, these three components in frequency domain can be rewritten in the following simplified forms:(11)B0φρ,0;ω=-μ02πc2ρ′iωeiωρ′+μ02πc2ρ′2eiωρ′-μ012πρ′·ei3π/4·lc3·∑jQj∗1/2ε1-Qj∗2ω5/2eiωQj∗ρ′ε1-2ωl/cQj∗2-1-μ0ε1-1l2ε1cπρ′eiπ/2ω5/2eiωρ′e-ip∗Fp∗,E0ρρ,0;ω=-μ0ε12πρ′ei3π/4lc2∑jQj∗1/2ε1-Qj∗2ω5/2eiωQj∗ρ′ε/Qj∗2-1-2ωl/c+μ0ε1-1ω2l2ε1ρ′c2eiωρ′+μ0ε1-12l22ε1c3πρ′ω7/2eiωρ′e-ip∗Fp∗,E0zρ,0;ω=μ02πceiωρ′iωρ′-1ρ′2-iωρ′3+μ012πρ′ei3π/4·lc3∑jε1-Qj∗2Qj∗3/2·ω5/2eiωQj∗ρ′ε1-2ωl/cQj∗2-1+μ0ε1-1l2ε1c2πρ′eiπ/2ω5/2eiωρ′e-ip∗Fp∗.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 thatp∗ can be written in the following form:(12)p∗≈-ω3l2ρ′2c2·ε1-12ε12,where ρ′=ρ/c.


### 2.2. Approximated Formulas for Transient Field

With the frequency-domain formulas for the field components andρ′ replacing by ρ/c, the analytical formulas for the time-dependent components can be obtained by using Fourier transforms. Thus, the three time-dependent field components can be written in the following forms:(13)B0φρ,0;t=12πRe⁡∫∞∞eiωtB0φρ,0;ωdω=μ02πc2ρ′I1+μ02πc2ρ′2I2+μ0c212πρ′∑jQj∗1/2ε1ε1-Qj∗2ε1-1I3+μ02c2πρ′I4,E0φρ,0;t=12πRe⁡∫∞∞eiωtE0φρ,0;ωdω=-μ0c12πρ′·∑jQj∗1/2Qj∗2-1ε1ε1-Qj∗2ε1-1I3-μ02πcρ′I5-μ02cπρ′I6,E0zρ,0;t=-μ02πcρ′I1-μ02πcρ′2I2-μ02πcρ′3eiωρ′I7-μ0c12πρ′·∑jQj∗3/2ε1-Qj∗2ε1ε1-1I3-μ02cπρ′I4,where(14)I1=1πRe⁡∫0∞-iωe-iωt-ρ′dω,I2=1πRe⁡∫0∞e-iωt-ρ′dω,I3=1πRe⁡∫0∞ei3π/4-iω3/2ε∗ε1+2ε1ε∗/ε1-11-Qj∗2eiωQj∗ρ′e-iωtdω,I4=1πRe⁡∫0∞ω3/2ε∗e-ip∗Fp∗e-iωt-ρ′dω,I5=1πRe⁡∫0∞-iωε∗e-iωt-ρ′dω,I6=1πRe⁡∫0∞ω3/2ε∗2e-ip∗Fp∗e-iωt-ρ′dω,I7=1πRe⁡∫0∞1-iωe-iωt-ρ′dω.It is seen that the integralsI1, I2, I5, and I7 have been evaluated in [2, 11]. We write(15)I1=δ′t-ρ′;I2=δt-ρ′,I5=ε1-1lcε1δ′′t-ρ′;I7=Ut-ρ′.Next, it is necessary to evaluate the integralsI3, I4, and I6. With complex derivations, the evaluations of the integrals I3, I4, and I6 can be obtained. We write(16)I3=ε1-12ε1Qj∗2-1·1π-b5/2C2cos⁡bt-Qj∗ρ′+S2sin⁡bt-Qj∗ρ′-b2t-Qj∗ρ′-3/2-πb5/2·cos⁡bt-Qj∗ρ′+3π4Ut-Qj∗ρ′,I4=-1πρ′δ′t-ρ′-cε12ρ′l′2ε1-12t-ρ′t2Ut-ρ′,I6=-1πρ′ε1-1lcε1δ′′t-ρ′-cε1ρ′l′ε1-11tUt-ρ′.In above formulas, the Heaviside unit step functionU(x) is defined by(17)Ux=0x<01x≥0.The Fresnel cosine integral C2 and sine integral S2 are written in the following forms:(18)Cx=∫0xcos⁡ttdt,Sx=∫0xsin⁡ttdt,where x=b(t-Qj∗ρ′). The variable b is(19)b=cε12lQj∗2-1.With the substitutions of (15) and (16) into (13) and considering τ=t/ρ′, the approximate formulas of the time-dependent field components can be derived readily. We write(20)B0φρ,0;t=μ02πc2ρ′2δt-ρ′+μ02πρ′2l2·ε12ε1-12t-ρ′Ut-ρ′+μ02πc212ρ′·∑jQj∗1/2ε1-Qj∗2Qj∗2-1-b5/2C2cos⁡bt-Qj∗ρ′+S2sin⁡bt-Qj∗ρ′-b2bt-Qj∗ρ′-3/2-πb5/2cos⁡bt-Qj∗ρ′+3π4Ut-Qj∗ρ′,(21)E0ρρ,0;t=-μ0c2πρ3l′·ε13/2τε1-1Ut-ρ′-μ02πc12ρ′·∑jQj∗1/2ε1-Qj∗2-b5/2C2cos⁡bρcτ-Qj∗+S2sin⁡bρcτ-Qj∗-b2ρcτ-Qj∗-3/2-πb5/2·cos⁡bρcτ-Qj∗+3π4,(22)E0zρ,0;t=-μ0c2πcρ′2δt-ρ′-μ02πcρ′3Ut-ρ′-μ02πc12ρ′·∑jQj∗3/2ε1-Qj∗2Qj∗2-1·-b5/2C2cos⁡bt-Qj∗ρ′+S2sin⁡bt-Qj∗ρ′-b2t-Qj∗ρ′-3/2-πb5/2·cos⁡bt-Qj∗ρ′+3π4Ut-Qj∗ρ′-2μ02cπl2ρ′2·c2ε12ε1-12t-ρ′Ut-ρ′.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 ρ-1/2 and ρ-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.


## 2.1. The Simplified Representations for the Field Components in Frequency Domain

The relevant geometry and Cartesian coordinate system are illustrated in Figure1, where a unit vertical electric dipole in z^ direction is located at (0,0,d) and surrounded by air. Region 0 (z≥0) is the upper half-space filled with air, the intermediate layer in Region 1 is characterized by the permeability μi and relative permittivity εi, and Region 2 (z≤-l) is the rest space representing a conducting dielectric base. It is assumed that both Region 0 and Region 1 are nonmagnetic and Region 2 is a perfect conductor. So that the wave numbers of the three regions are(1)k0=ωμ0ε0=ωc,k1=ωμ0ε0ε1=ωε1c,k2⟶∞.Figure 1
Vertical dipole at heightd in air near the surface of a planar perfect conductor coated with a dielectric layer.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 eiωt, these three components can be written in the following forms:(2)B0φρ,z=-μ04πeik0r1ρr1ik0r1-1r12-μ04πeik0r2ρr2ik0r2-1r22+k02μ012πρei3π/4∑jγ1λj∗tan⁡γ1λj∗lq′λj∗γ0λj∗λj∗3/2eiγ0λj∗z+d+iλj∗ρ-k03μ0k12-k022k121πk0ρtan⁡k12-k02leik0r2+π/2e-ip∗Fp∗,(3)E0ρρ,z=-ωμ04πk0eik0r1ρr1z-dr1ik0r1-3r12-3ik0r13-ωμ04πk0eik0r2ρr2z+dr2ik0r2-3r22-3ik0r23-ωμ012πρeiπ/4∑jγ1λj∗tan⁡γ1λj∗lq′λj∗λj∗3/2eiγ0λj∗z+d+iλj∗ρ+ωk02μ02πk121πk0ρk12-k02tan⁡k12-k02leik0r2πk0ρ+πk0k12-k02k12tan⁡k12-k02le-ip∗Fp∗,(4)E0zρ,z=ωμ04πk0eik0r1ik0r1-1r12-ik0r13-z-dr12ik0r1-3r12-3ik0r13+ωμ04πk0eik0r2ik0r2-1r22-ik0r23-z+dr22ik0r2-3r22-3ik0r23-ωμ012πρei3π/4∑jγ1λj∗tan⁡γ1λj∗lq′λj∗γ0λj∗λj∗5/2eiγ0λj∗z+d+iλj∗ρ+ωk02μ02k121πk0ρk12-k02tan⁡k12-k02leik0r2+π/2e-ip∗Fp∗.In above formulas, r1=ρ2+(z-d)2 and r2=ρ2+(z+d)2. qλ and q′λ are expressed as follows:(5)qλ=k12γ0-ik02γ1tan⁡γ1l,q′λ=-k12λγ0+ik02λγ1tan⁡γ1l+ik02λlsec2⁡γ1l,where γ0=k02-λj∗2, γ1=k12-λj∗2, and the pole λj∗ is determined by the roots of the following pole equation:(6)k12γ0-ik02γ1tan⁡γ1l=0.The Fresnel function Fp∗ is defined by(7)Fp∗=C2p∗+iS2p∗=121+i-∫0p∗eit2πtdt,where(8)p∗=k0ρ2z+dρ-ik0k12-k02k12tan⁡k12-k02l2.It is assumed that the intermediate layer is satisfied by the condition thatl′=k1l≤0.6. Nominally, the dielectric layer is usually electrically very thin. Thus, we have(9)tan⁡k12-k02l≈ωlcε1-1.With (7), by introducing the change of the variable Qj∗=λj∗/k0, it follows that(10)q′λj∗≈-ω2Qj∗c2ε11-Qj∗2-i2ωlc.With above approximations, these three components in frequency domain can be rewritten in the following simplified forms:(11)B0φρ,0;ω=-μ02πc2ρ′iωeiωρ′+μ02πc2ρ′2eiωρ′-μ012πρ′·ei3π/4·lc3·∑jQj∗1/2ε1-Qj∗2ω5/2eiωQj∗ρ′ε1-2ωl/cQj∗2-1-μ0ε1-1l2ε1cπρ′eiπ/2ω5/2eiωρ′e-ip∗Fp∗,E0ρρ,0;ω=-μ0ε12πρ′ei3π/4lc2∑jQj∗1/2ε1-Qj∗2ω5/2eiωQj∗ρ′ε/Qj∗2-1-2ωl/c+μ0ε1-1ω2l2ε1ρ′c2eiωρ′+μ0ε1-12l22ε1c3πρ′ω7/2eiωρ′e-ip∗Fp∗,E0zρ,0;ω=μ02πceiωρ′iωρ′-1ρ′2-iωρ′3+μ012πρ′ei3π/4·lc3∑jε1-Qj∗2Qj∗3/2·ω5/2eiωQj∗ρ′ε1-2ωl/cQj∗2-1+μ0ε1-1l2ε1c2πρ′eiπ/2ω5/2eiωρ′e-ip∗Fp∗.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 thatp∗ can be written in the following form:(12)p∗≈-ω3l2ρ′2c2·ε1-12ε12,where ρ′=ρ/c.


## 2.2. Approximated Formulas for Transient Field

With the frequency-domain formulas for the field components andρ′ replacing by ρ/c, the analytical formulas for the time-dependent components can be obtained by using Fourier transforms. Thus, the three time-dependent field components can be written in the following forms:(13)B0φρ,0;t=12πRe⁡∫∞∞eiωtB0φρ,0;ωdω=μ02πc2ρ′I1+μ02πc2ρ′2I2+μ0c212πρ′∑jQj∗1/2ε1ε1-Qj∗2ε1-1I3+μ02c2πρ′I4,E0φρ,0;t=12πRe⁡∫∞∞eiωtE0φρ,0;ωdω=-μ0c12πρ′·∑jQj∗1/2Qj∗2-1ε1ε1-Qj∗2ε1-1I3-μ02πcρ′I5-μ02cπρ′I6,E0zρ,0;t=-μ02πcρ′I1-μ02πcρ′2I2-μ02πcρ′3eiωρ′I7-μ0c12πρ′·∑jQj∗3/2ε1-Qj∗2ε1ε1-1I3-μ02cπρ′I4,where(14)I1=1πRe⁡∫0∞-iωe-iωt-ρ′dω,I2=1πRe⁡∫0∞e-iωt-ρ′dω,I3=1πRe⁡∫0∞ei3π/4-iω3/2ε∗ε1+2ε1ε∗/ε1-11-Qj∗2eiωQj∗ρ′e-iωtdω,I4=1πRe⁡∫0∞ω3/2ε∗e-ip∗Fp∗e-iωt-ρ′dω,I5=1πRe⁡∫0∞-iωε∗e-iωt-ρ′dω,I6=1πRe⁡∫0∞ω3/2ε∗2e-ip∗Fp∗e-iωt-ρ′dω,I7=1πRe⁡∫0∞1-iωe-iωt-ρ′dω.It is seen that the integralsI1, I2, I5, and I7 have been evaluated in [2, 11]. We write(15)I1=δ′t-ρ′;I2=δt-ρ′,I5=ε1-1lcε1δ′′t-ρ′;I7=Ut-ρ′.Next, it is necessary to evaluate the integralsI3, I4, and I6. With complex derivations, the evaluations of the integrals I3, I4, and I6 can be obtained. We write(16)I3=ε1-12ε1Qj∗2-1·1π-b5/2C2cos⁡bt-Qj∗ρ′+S2sin⁡bt-Qj∗ρ′-b2t-Qj∗ρ′-3/2-πb5/2·cos⁡bt-Qj∗ρ′+3π4Ut-Qj∗ρ′,I4=-1πρ′δ′t-ρ′-cε12ρ′l′2ε1-12t-ρ′t2Ut-ρ′,I6=-1πρ′ε1-1lcε1δ′′t-ρ′-cε1ρ′l′ε1-11tUt-ρ′.In above formulas, the Heaviside unit step functionU(x) is defined by(17)Ux=0x<01x≥0.The Fresnel cosine integral C2 and sine integral S2 are written in the following forms:(18)Cx=∫0xcos⁡ttdt,Sx=∫0xsin⁡ttdt,where x=b(t-Qj∗ρ′). The variable b is(19)b=cε12lQj∗2-1.With the substitutions of (15) and (16) into (13) and considering τ=t/ρ′, the approximate formulas of the time-dependent field components can be derived readily. We write(20)B0φρ,0;t=μ02πc2ρ′2δt-ρ′+μ02πρ′2l2·ε12ε1-12t-ρ′Ut-ρ′+μ02πc212ρ′·∑jQj∗1/2ε1-Qj∗2Qj∗2-1-b5/2C2cos⁡bt-Qj∗ρ′+S2sin⁡bt-Qj∗ρ′-b2bt-Qj∗ρ′-3/2-πb5/2cos⁡bt-Qj∗ρ′+3π4Ut-Qj∗ρ′,(21)E0ρρ,0;t=-μ0c2πρ3l′·ε13/2τε1-1Ut-ρ′-μ02πc12ρ′·∑jQj∗1/2ε1-Qj∗2-b5/2C2cos⁡bρcτ-Qj∗+S2sin⁡bρcτ-Qj∗-b2ρcτ-Qj∗-3/2-πb5/2·cos⁡bρcτ-Qj∗+3π4,(22)E0zρ,0;t=-μ0c2πcρ′2δt-ρ′-μ02πcρ′3Ut-ρ′-μ02πc12ρ′·∑jQj∗3/2ε1-Qj∗2Qj∗2-1·-b5/2C2cos⁡bt-Qj∗ρ′+S2sin⁡bt-Qj∗ρ′-b2t-Qj∗ρ′-3/2-πb5/2·cos⁡bt-Qj∗ρ′+3π4Ut-Qj∗ρ′-2μ02cπl2ρ′2·c2ε12ε1-12t-ρ′Ut-ρ′.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 ρ-1/2 and ρ-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.


## 3. 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. Withρ=10 km, the magnitudes of the component (2πε0ε1)E0z(ρ,0,t) due to a vertical electric dipole on the surface of a dielectric-coated conductor are computed at ε1=2,2.65,2.85, and 3.65 and showed in Figure 2, respectively. It is noted that the properties of other two components E0ρ(ρ,0,t) and B0φ(ρ,0,t) are similar to those of the component E0z(ρ,0,t).Figure 2
Magnitudes of(2πε0ε1)E0z(ρ,0,t) due to a vertical electric dipole with delta-function excitation with ρ=10 km.Withρ=30 km and ε1=2,2.65,2.85, and 3.65, the magnitudes of the trapped-surface-wave terms B2φS in (20) are computed at Qj∗=1.2 and Qj∗=1.5 and shown in Figure 3, respectively. It is known that the trapped-surface-wave terms are contributed by the poles λj∗, which are determined by both the dielectric parameters and the thickness of the dielectric layer [11]. Once the poles λj∗ are determined, the trapped-surface-wave terms can be computed readily. In Figure 4, the trapped-surface-wave terms B0φS(ρ,0,t) are computed at ρ=10 km and ρ=20 km, respectively.Figure 3
Magnitudes of the trapped-surface-wave termE0zS(ρ,t) due to a vertical electric dipole with delta-function excitation with ρ=30 km and ε1=2.65,2.85, and 3.65.Figure 4
Comparisons between the trapped-surface-wave pulsesB0φS(ρ,0,t) at ρ=10 km and those at ρ=20 km.


## 4. 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 lateral-wave pulses decrease with the amplitude factorsρ-1/2 and ρ-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.



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