# Quasi-Static Solution for ELF/SLF near-Zone Field of a Vertical Magnetic Dipole near the Surface of a Lossy Half-Space

Authors: Lina He; Tong He; Kai Li
Journal: International Journal of Antennas and Propagation
 (2020)

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## Abstract

Dipole antennas over the boundary between two different media have been widely used in the fields of geophysics exploration, oceanography, and submerged communication. In this paper, an analytical method is proposed to analyse the near-zone field at the extremely low frequency (ELF)/super low frequency (SLF) range due to a vertical magnetic dipole (VMD). For the lack of feasible analytical techniques to derive the components exactly, two reasonable assumptions are introduced depending on the quasi-static definition and the equivalent infinitesimal theory. Final expressions of the electromagnetic field components are in terms of exponential functions. By comparisons with direct numerical solutions and exact results in a special case, the correctness and effectiveness of the proposed quasi-static approximation are demonstrated. Simulations show that the smallest validity limit always occurs for componentH2z, and the value of k2ρ should be no greater than 0.6 in order to keep a good consistency.


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## Body

## 1. Introduction

The propagation properties of extremely low frequency (ELF: 3–30 Hz) and super low frequency (SLF: 30–300 Hz) waves over the boundary between two different media have been extensively investigated for many years because of their useful applications in geophysics exploration, oceanography, overwater or underwater communication, and so on. For example, the US system Seafarer, operated at the frequency of 78 Hz [1], while the Russian one, called ZEVS, operated at the frequency of 82 Hz [2]. These systems have been used to communicate from a fixed station on the sea surface with a submarine traveling close to the ocean floor since 1990s [3, 4]. The original expression of this problem can date back to 1909 by Sommerfeld in his classic literature [5]. Subsequently, Norton proposed simple formulas and graphs in 1936 [6]. Following their works, many other progresses were made on numerical [7, 8] and analytical solutions [9–15] at a large propagation distance of kρ≫1, where k represents the wave number and ρ is the propagation distance.Evidently, in practical physical models such as a dipole source on the interface between two different media like air and Earth or sea water and rock, the electric propagation distance for the ELF/SLF ranges is usually very small, namely,kρ≪1. Sommerfeld integrals are used for field components which can be expressed in terms of the derivatives of integrals including the Bessel function. Due to the divergence terms in Sommerfeld integrals, the investigation of ELF/SLF near-zone field is still not well developed yet. In the recent work by Xu et al. [16], a new quasi-static technique is used for analyzing the near-zone field radiated by a horizontal electric dipole near the sea-rock boundary. This new development rekindled our interest in investigating the magnetic-dipole excitation problem.Several investigations of simplifying formulas had been made by Banos and Wesley [17]. They obtained approximation solutions suitable for both near and far fields in terms of Hertz potentials. Based on their efforts, Durrani [18, 19] addressed the sufficiently accurate but relatively simple field components in the sea produced by vertical and horizontal dipoles (both electric and magnetic types) under reasonable limitations. Also of interest is the satisfactory communication range between air and sea. Later on, in a series of papers by Bannister [20, 21], previous theoretical work like [22] was well summarized, and all field components for these four different types of dipoles located on the boundary were presented over the quasi-static range. In recent years, some near-field research has gradually been carried out both analytically [23–25] and numerically [26]. Particularly in 2010, Parise [27] established the exact closed-form expressions of the electromagnetic field components excited by a vertical magnetic dipole lying on the surface of a flat and homogeneous lossy half-space, but unfortunately Hρ is still in terms of Bessel functions, and these formulas are valid only when both of the dipole source and the observation point are embedded on the boundary. In many practical applications, both the dipole source and the receiving point may be located near the boundary. However, the corresponding study on this near-field radiation of a vertical magnetic dipole is still in the dark. Since it is a formidable task to calculate it accurately by existing analytical methods, in this paper, we will attempt to extend the work by Xu et al. [16] and evaluate the integrals in purpose of deriving new approximation formulas.Note that the propagation distance of interest is up to dozens of kilometers, in this case, the propagation path via the reflection of the ionosphere is much longer than that along the Earth’s surface. Hence, the effect of the ionosphere can be neglected and a simple half-space flat model is established. At ELF/SLF ranges, media such as sea water/ground/Earth are always regarded as highly lossy ones, while the air is lossless. So, the ratio of the wave numbers is far smaller than 1, i.e.,kair/kmedia≪1. Furthermore, usually the heights of both the dipole source and the receiving point are much smaller than the horizontal distance, i.e., d≪ρ and z≪ρ. The following are some concrete evaluations and simplifications. Comparisons with direct numerical simulations and exact solutions are also carried out from both the amplitude and phase angle aspects eventually. The time dependence eiωt is assumed and suppressed throughout the analysis.


## 2. Formulations

The geometry and notations of the physical model are shown in Figure1. It consists of a loop antenna (viewed as a vertical magnetic dipole) situated at a distance d, above a half-infinite conducting space. Cartesian and cylindrical coordinates are introduced with z directed upwards through the dipole, and the radius ρ is taken horizontally in the x-y plane on the surface. The air characterized by the parameters ε2, μ2, and σ2 occupies the upper space z≥0, and the lossy medium (like seawater or Earth) characterized by the parameters ε1, μ1, and σ1 occupies the lower space z≤0. It is assumed that both regions are nonmagnetic so that μ1=μ2=μ0.Figure 1
Excitation of an ELF/SLF VMD above a lossy half-space.


### 2.1. Integral Expressions for the Field Excited by an ELF/SLF VMD

When its size is small enough compared with the propagation distance and free-space wavelength, it is a fairly common-sense idea to regard a small current loop as a vertical magnetic dipole (VMD) [28]. We identify IdS as the magnetic dipole moment, where I denotes the loop circulating current and dS represents the area of the loop. With a similar method by King [29], the integrals for the components of the electromagnetic field in Region 2 can be written as follows:(1)E2φρ,z=−ωμ0IdS2π∫0∞eiγ2z−d2γ2−eiγ2z+d2γ2+eiγ2z+dγ1+γ2J1λρλ2dλ,(2)H2ρρ,z=IdS2π∫0∞±eiγ2z−d2−eiγ2z+d2+γ2eiγ2z+dγ1+γ2J1λρλ2dλ,(3)H2zρ,z=iIdS2π∫0∞eiγ2z−d2γ2−eiγ2z+d2γ2+eiγ2z+dγ1+γ2J0λρλ3dλ.where(4)γn=kn2−λ2,n=1,2.It is noted that the upper sign “+” in (2) corresponds to z>d, and the lower sign “−” corresponds to 0<z<d, respectively. And the square root in (4) is taken in the first quadrant.The first two terms in formulas (1)–(3) stand for the direct wave and the ideal reflected wave. Meanwhile, the third term represents the lateral wave. Thus, these formulas can be rewritten in the following forms:(5)E2φρ,z=E2φdρ,z+E2φiρ,z+E2φlρ,z,H2ρρ,z=H2ρdρ,z+H2ρiρ,z+H2ρlρ,z,H2zρ,z=H2zdρ,z+H2ziρ,z+H2zlρ,z.In the above integrals, the direct-field and reflect-field terms can be derived via the method presented in the monograph by King et al. [15].The direct-field components are(6)E2φdρ,z=−ωμ0IdS4π∫0∞eiγ2z−dγ2J1λρλ2dλ=ωμ0IdS4πρk21r12+ik2r13eik2r1,H2ρdρ,z=±IdS4π∫0∞eiγ2z−dJ1λρλ2dλ=∓IdS4πρk2d1r1k2r12+3ir13−3k2r14eik2r1,H2zdρ,z=iIdS4π∫0∞eiγ2z−dγ2J0λρλ3dλ=−iIdS4πk2ik2r1−1r12−ik2r13eik2r1+k2d12r12ik2r1−3r12−3ik2r13eik2r1.The reflect-field components are(7)E2φiρ,z=−ωμ0IdS4π∫0∞eiγ2z+dγ2J1λρλ2dλ=ωμ0IdS4πρk21r22+ik2r23eik2r2,H2ρiρ,z=IdS4π∫0∞eiγ2z+dJ1λρλ2dλ=−IdS4πρk2d2r2k2r22+3ir23−3k2r24eik2r2,H2ziρ,z=iIdS4π∫0∞eiγ2z+dγ2J0λρλ3dλ=−iIdS4πk2​ik2r2−1r22−ik2r23​eik2r2+​k2d22r22​ik2r2−3r22−3ik2r23​eik2r2.where(8)d1=z−d,d2=z+d,r1=ρ2+d12,r2=ρ2+d22.The rest lateral-wave terms are defined by(9)E2φlρ,z=−ωμ0IdS2π∫0∞1Meiγ2z+dJ1λρλ2dλ=−ωμ0IdS2π⋅F2φlρ,d2,(10)H2ρlρ,z=IdS2π∫0∞γ2Meiγ2z+dJ1λρλ2dλ=IdS2π⋅F2ρlρ,d2,(11)H2zlρ,z=iIdS2π∫0∞1Meiγ2z+dJ0λρλ3dλ=iIdS2π⋅F2zlρ,d2,where M=γ1+γ2. It should be pointed out that these formulas contain Bessel functions, whose integrands are highly oscillatory in the near-zone. Therefore, a novel approximation approach is required to be taken into consideration in the following subsection.


### 2.2. Simplifications of the Lateral-Wave Terms under the Equivalent Infinitesimal Theory

For the beginning of simplification, we should figure out the definition of “quasi-static” or “near-field” involved in this paper. It is defined as the situation where the distance from the source to the observation point is far less than a wavelengthkρ≪1 [15, 18]. Typically, if the frequency remains invariable, distances satisfying this condition contain the “near-field”; otherwise, if the distance is fixed, qualified frequencies form the “quasi-static” state.It is readily understood from the definition, when the “quasi-static” assumption is considered,ω⟶0 and k2ρ≪1 hold. Under this circumstance, contribution of the Sommerfeld integrals in (9)–(11) is mainly from the integration path where λ≫k2. Therefore, the parameter γ2 can be expressed as follows:(12)γ2=k22−λ2≈iλ.Normally, the medium in Region 1 is highly conductive when compared with air in Region 2, namely,k2≪k1. Also, d and z are usually far less than the horizontal distance ρ, i.e., d≪ρ and z≪ρ. Thus, we have(13)iγ1−γ2−k1dm≈ik12−λ2−iλ−k1dm≪1,where m=1,2. It is seen from (13) that the limit of iγ1−γ2−k1dm can be approximated as zero. The equivalent infinitesimal theory, a special case of Taylor’s formula, could be subsequently applied here as follows:(14)eiγ1−γ2−k1dm−1≈iγ1−γ2−k1dm.Then, by some deformation, a useful alternative form ofγ1−γ2 is correspondingly obtained, so is the variable M. We write(15)γ1−γ2=i1−eiγ1−γ2−k1dmdm+k1,(16)1M=1γ1+γ2=γ1−γ2γ1+γ2γ1−γ2=i−eiγ1−γ2−k1dm+1−ik1dmk12−k22dm.Hence, by substituting (16) into (9), we have(17)F2φlρ,d2=∫0∞i−eiγ1−γ2−k1z+d+1−ik1z+dk12−k22z+deiγ2z+dJ1λρλ2dλ=−ik12−k22d2e−ik1d2∫0∞eiγ1−k1d2J1λρλ2dλ+ik1d2−1∫0∞eiγ2d2J1λρλ2dλ.Similarly, (10) is equal to(18)F2zlρ,d2=−ik12−k22d2e−ik1d2∫0∞eiγ1−k1d2J0λρλ3dλ+ik1d2−1∫0∞eiγ2d2J0λρλ3dλ.with ∫0∞γ2eiγ2xJ1λPλ2dλ=−i∂/∂x∫0∞eiγ2xJ1λρλ2dλ, the remaining term F2ρlρ,d2 in (10) can be evaluated as follows:(19)F2ρlρ,d2=∫0∞γ2Meiγ2z+dJ1λPλ2dλ≈∂∂d2∫0∞−eiγ1−γ2−k1z+d+1−ik1z+dk12−k22z+deiγ2z+dJ1λρλ2dλ=1k12−k22∂∂d2∫0∞−eiγ1−k1d2+1−ik1d2eiγ2d2d2J1λρλ2dλ.Finally, with the Appendix in [15], the final analytical results for F2φl, F2ρl, F2zl can be obtained readily. We write(20)F2φlρ,d2≈ρk1r2k12−k22k1r22+3ir23−3k1r24eik1r2−d2+ρk2ik1d2−1r2k12−k22k2r22+3ir23−3k2r24eik2r2,(21)F2ρlρ,d2=−iρk12−k22×k12r2k1r22+3ir23−3k1r24eik1r2−d2−k1d2r22k12r22+6ik1r23−15r24−15ik1r25eik1r2−d2−k1k2k2r23+3ir24−3k2r25eik2r2+k2d2r22k22r22+6ik2r23−15r24−15ik2r25eik2r2−k1k2d22r22ik22r22−6k2r23−15ir24+15k2r25eik2r2,(22)F2zlρ,d2≈k1k22−k12k12r22+4ik1r23−9r24−9ik1r25eik1r2−d2+k2ik1d2−1k22−k12×k22r22+4ik2r23−9r24−9ik2r25eik2r2.So far, we have provided the process of calculating the field components for a VMD source over the near-field range in Region 2 in detail. The complete approximation solutions for the field components in Region 1 can be evaluated with a similar method as well.


### 2.3. Solutions for the near-Zone Field withz=0 and d=0

The complete quasi-static field components are derived readily with (9)–(11) and (20)–(22). For comparison purpose, in the following, we will attempt to reduce these formulas to the special case when both the dipole source and the receiving point are embedded on the boundary. For brevity and clarity, only H2z’s derivation procedure is given as a demonstration. The magnetic component H2z can be expressed as(23)H2z=IdS4πk2k2r1+ir12−1k2r13eik2r1−d12r12k2r1+3ir12−3k2r13eik2r1−k2r2+ir22−1k2r23−d22r22k2r2+3ir22−3k2r23eik2r2+2ik1k2k12−k22k12r22+4ik1r23−9r24−9ik1r25eik1r2−d2+2i−2k1d2k12−k22k22r22+4ik2r23−9r24−9ik2r25eik2r2.Sincek2ρ≪1 is assumed, with Maclaurin series, we have(24)eik2ρ≈1+ik2ρ−k2ρ22.By substituting (24) into (23) and setting z=0, d=0, it converts to:(25)H2z≈12πik1ρ2−4ρ3−9ik1ρ4−9k12ρ5eik1ρ+k22k121ρ3+9ρ5.Now, we have derived the analytical expressions whenz=d=0. To understand this method better, computations and discussions under several different conditions will be carried out in the following section.


## 2.1. Integral Expressions for the Field Excited by an ELF/SLF VMD

When its size is small enough compared with the propagation distance and free-space wavelength, it is a fairly common-sense idea to regard a small current loop as a vertical magnetic dipole (VMD) [28]. We identify IdS as the magnetic dipole moment, where I denotes the loop circulating current and dS represents the area of the loop. With a similar method by King [29], the integrals for the components of the electromagnetic field in Region 2 can be written as follows:(1)E2φρ,z=−ωμ0IdS2π∫0∞eiγ2z−d2γ2−eiγ2z+d2γ2+eiγ2z+dγ1+γ2J1λρλ2dλ,(2)H2ρρ,z=IdS2π∫0∞±eiγ2z−d2−eiγ2z+d2+γ2eiγ2z+dγ1+γ2J1λρλ2dλ,(3)H2zρ,z=iIdS2π∫0∞eiγ2z−d2γ2−eiγ2z+d2γ2+eiγ2z+dγ1+γ2J0λρλ3dλ.where(4)γn=kn2−λ2,n=1,2.It is noted that the upper sign “+” in (2) corresponds to z>d, and the lower sign “−” corresponds to 0<z<d, respectively. And the square root in (4) is taken in the first quadrant.The first two terms in formulas (1)–(3) stand for the direct wave and the ideal reflected wave. Meanwhile, the third term represents the lateral wave. Thus, these formulas can be rewritten in the following forms:(5)E2φρ,z=E2φdρ,z+E2φiρ,z+E2φlρ,z,H2ρρ,z=H2ρdρ,z+H2ρiρ,z+H2ρlρ,z,H2zρ,z=H2zdρ,z+H2ziρ,z+H2zlρ,z.In the above integrals, the direct-field and reflect-field terms can be derived via the method presented in the monograph by King et al. [15].The direct-field components are(6)E2φdρ,z=−ωμ0IdS4π∫0∞eiγ2z−dγ2J1λρλ2dλ=ωμ0IdS4πρk21r12+ik2r13eik2r1,H2ρdρ,z=±IdS4π∫0∞eiγ2z−dJ1λρλ2dλ=∓IdS4πρk2d1r1k2r12+3ir13−3k2r14eik2r1,H2zdρ,z=iIdS4π∫0∞eiγ2z−dγ2J0λρλ3dλ=−iIdS4πk2ik2r1−1r12−ik2r13eik2r1+k2d12r12ik2r1−3r12−3ik2r13eik2r1.The reflect-field components are(7)E2φiρ,z=−ωμ0IdS4π∫0∞eiγ2z+dγ2J1λρλ2dλ=ωμ0IdS4πρk21r22+ik2r23eik2r2,H2ρiρ,z=IdS4π∫0∞eiγ2z+dJ1λρλ2dλ=−IdS4πρk2d2r2k2r22+3ir23−3k2r24eik2r2,H2ziρ,z=iIdS4π∫0∞eiγ2z+dγ2J0λρλ3dλ=−iIdS4πk2​ik2r2−1r22−ik2r23​eik2r2+​k2d22r22​ik2r2−3r22−3ik2r23​eik2r2.where(8)d1=z−d,d2=z+d,r1=ρ2+d12,r2=ρ2+d22.The rest lateral-wave terms are defined by(9)E2φlρ,z=−ωμ0IdS2π∫0∞1Meiγ2z+dJ1λρλ2dλ=−ωμ0IdS2π⋅F2φlρ,d2,(10)H2ρlρ,z=IdS2π∫0∞γ2Meiγ2z+dJ1λρλ2dλ=IdS2π⋅F2ρlρ,d2,(11)H2zlρ,z=iIdS2π∫0∞1Meiγ2z+dJ0λρλ3dλ=iIdS2π⋅F2zlρ,d2,where M=γ1+γ2. It should be pointed out that these formulas contain Bessel functions, whose integrands are highly oscillatory in the near-zone. Therefore, a novel approximation approach is required to be taken into consideration in the following subsection.


## 2.2. Simplifications of the Lateral-Wave Terms under the Equivalent Infinitesimal Theory

For the beginning of simplification, we should figure out the definition of “quasi-static” or “near-field” involved in this paper. It is defined as the situation where the distance from the source to the observation point is far less than a wavelengthkρ≪1 [15, 18]. Typically, if the frequency remains invariable, distances satisfying this condition contain the “near-field”; otherwise, if the distance is fixed, qualified frequencies form the “quasi-static” state.It is readily understood from the definition, when the “quasi-static” assumption is considered,ω⟶0 and k2ρ≪1 hold. Under this circumstance, contribution of the Sommerfeld integrals in (9)–(11) is mainly from the integration path where λ≫k2. Therefore, the parameter γ2 can be expressed as follows:(12)γ2=k22−λ2≈iλ.Normally, the medium in Region 1 is highly conductive when compared with air in Region 2, namely,k2≪k1. Also, d and z are usually far less than the horizontal distance ρ, i.e., d≪ρ and z≪ρ. Thus, we have(13)iγ1−γ2−k1dm≈ik12−λ2−iλ−k1dm≪1,where m=1,2. It is seen from (13) that the limit of iγ1−γ2−k1dm can be approximated as zero. The equivalent infinitesimal theory, a special case of Taylor’s formula, could be subsequently applied here as follows:(14)eiγ1−γ2−k1dm−1≈iγ1−γ2−k1dm.Then, by some deformation, a useful alternative form ofγ1−γ2 is correspondingly obtained, so is the variable M. We write(15)γ1−γ2=i1−eiγ1−γ2−k1dmdm+k1,(16)1M=1γ1+γ2=γ1−γ2γ1+γ2γ1−γ2=i−eiγ1−γ2−k1dm+1−ik1dmk12−k22dm.Hence, by substituting (16) into (9), we have(17)F2φlρ,d2=∫0∞i−eiγ1−γ2−k1z+d+1−ik1z+dk12−k22z+deiγ2z+dJ1λρλ2dλ=−ik12−k22d2e−ik1d2∫0∞eiγ1−k1d2J1λρλ2dλ+ik1d2−1∫0∞eiγ2d2J1λρλ2dλ.Similarly, (10) is equal to(18)F2zlρ,d2=−ik12−k22d2e−ik1d2∫0∞eiγ1−k1d2J0λρλ3dλ+ik1d2−1∫0∞eiγ2d2J0λρλ3dλ.with ∫0∞γ2eiγ2xJ1λPλ2dλ=−i∂/∂x∫0∞eiγ2xJ1λρλ2dλ, the remaining term F2ρlρ,d2 in (10) can be evaluated as follows:(19)F2ρlρ,d2=∫0∞γ2Meiγ2z+dJ1λPλ2dλ≈∂∂d2∫0∞−eiγ1−γ2−k1z+d+1−ik1z+dk12−k22z+deiγ2z+dJ1λρλ2dλ=1k12−k22∂∂d2∫0∞−eiγ1−k1d2+1−ik1d2eiγ2d2d2J1λρλ2dλ.Finally, with the Appendix in [15], the final analytical results for F2φl, F2ρl, F2zl can be obtained readily. We write(20)F2φlρ,d2≈ρk1r2k12−k22k1r22+3ir23−3k1r24eik1r2−d2+ρk2ik1d2−1r2k12−k22k2r22+3ir23−3k2r24eik2r2,(21)F2ρlρ,d2=−iρk12−k22×k12r2k1r22+3ir23−3k1r24eik1r2−d2−k1d2r22k12r22+6ik1r23−15r24−15ik1r25eik1r2−d2−k1k2k2r23+3ir24−3k2r25eik2r2+k2d2r22k22r22+6ik2r23−15r24−15ik2r25eik2r2−k1k2d22r22ik22r22−6k2r23−15ir24+15k2r25eik2r2,(22)F2zlρ,d2≈k1k22−k12k12r22+4ik1r23−9r24−9ik1r25eik1r2−d2+k2ik1d2−1k22−k12×k22r22+4ik2r23−9r24−9ik2r25eik2r2.So far, we have provided the process of calculating the field components for a VMD source over the near-field range in Region 2 in detail. The complete approximation solutions for the field components in Region 1 can be evaluated with a similar method as well.


## 2.3. Solutions for the near-Zone Field withz=0 and d=0

The complete quasi-static field components are derived readily with (9)–(11) and (20)–(22). For comparison purpose, in the following, we will attempt to reduce these formulas to the special case when both the dipole source and the receiving point are embedded on the boundary. For brevity and clarity, only H2z’s derivation procedure is given as a demonstration. The magnetic component H2z can be expressed as(23)H2z=IdS4πk2k2r1+ir12−1k2r13eik2r1−d12r12k2r1+3ir12−3k2r13eik2r1−k2r2+ir22−1k2r23−d22r22k2r2+3ir22−3k2r23eik2r2+2ik1k2k12−k22k12r22+4ik1r23−9r24−9ik1r25eik1r2−d2+2i−2k1d2k12−k22k22r22+4ik2r23−9r24−9ik2r25eik2r2.Sincek2ρ≪1 is assumed, with Maclaurin series, we have(24)eik2ρ≈1+ik2ρ−k2ρ22.By substituting (24) into (23) and setting z=0, d=0, it converts to:(25)H2z≈12πik1ρ2−4ρ3−9ik1ρ4−9k12ρ5eik1ρ+k22k121ρ3+9ρ5.Now, we have derived the analytical expressions whenz=d=0. To understand this method better, computations and discussions under several different conditions will be carried out in the following section.


## 3. Computation and Discussion

### 3.1. Comparison with Numerical Solutions Whenz≠0 and d≠0

It is shown in literature studies [7, 8] that, by some proper numerical approaches, one can solve Sommerfeld integrals directly. Therefore, direct numerical simulation is employed and then compared with the analytical result obtained in this paper. We assume that Region 1 is seawater characterized by the permeability μ0, relative permittivity εr1=80, and conductivity σ1=4 S/m. Region 2 is air characterized by the permeability μ0 and the permittivity ε0. For a general case, the dipole source and receiving point are placed at the height d=1m and z=5m, respectively. The amplitudes and phase angles of the three nonzero field components with respect to the propagation distance ρ are evaluated in Figures 2–4 at operating frequencies f=3Hz and 300 Hz.Figure 2
Amplitudes and phase angles of field componentH2z versus the propagation distance ρ with the proposed and numerical solutions. (a, c) f=3Hz. (b, d) f=300Hz.
(a)
(b)
(c)
(d)Figure 3
Amplitudes and phase angles of the field componentE2φ versus the propagation distance ρ with the proposed and numerical solutions. (a, c) f=3Hz. (b, d) f=300Hz.
(a)
(b)
(c)
(d)Figure 4
Amplitudes and phase angles of the field componentH2ρ versus the propagation distance ρ with the proposed and numerical solutions. (a, c) f=3 Hz. (b, d) f=300 Hz.
(a)
(b)
(c)
(d)It is seen that both the amplitudes and phase angles obtained by analytical and direct numerical methods have a consistent trend in general. Figures2(a) and 2(b) show that at the beginning there exists a relatively apparent difference between these two computed results. This is probably because when ρ is too small, d≪ρ and z≪ρ are not very restrictive. But when ρ>10m, two curves shed light on a good correspondence with each other. The magnitude of E2φ and H2ρ in Figures 3(a), 3(b), 4(a), and 4(b) also show similar trends with H2z. Obviously, in most cases of practical interest, ρ≫z and ρ≫d, so the results confirm the high reliability and validity of the proposed method. Furthermore, it is also noted that the operating frequency has little influence to the magnitude of H2z and H2ρ, but a larger frequency will lead to a smaller amplitude of E2φ.As illustrated in Figures2(c) and 2(d), initially the phase of H2z ascend sharply until reaching the value of π. It is noted that a higher frequency leads to a closer mutation point to the source. Eventually, for a sudden change to −π, as the distance further increases, the phase starts rising smoothly and stabilizes around −π/2. Higher frequency also causes a faster phase variation from –π to −π/2, which results in a shorter distance for reaching the final state. We can see that phase angles of the two methods are almost alike except with some latency. By observing other two components, though trajectories are not in full accord, similar conclusions can still be drawn.It should also be born in mind that with large distances,k2ρ is more close to “1,” which then also leads to a larger difference between the numerical and analytical results. In this paper, attention is restricted to the near field, and k2ρ∼1 is out of consideration. Computations show that different components have different validity limits of k2ρ for maintaining a consistency between the proposal and numerical solutions. For E2φ and H2ρ components, results of both the amplitude and phase obtained by the two methods remain the same even if k2ρ is approaching 1. However, for H2z, when the distance reaches 100 km (i.e., k2ρ≈0.6, as can be seen in Figure 2(b)), two curves start to deviate from each other.


### 3.2. Comparison with Exact Solutions Whenz=0 and d=0

When both the dipole source and receiving point are located at the interface of a half-space, the exact field components excited by a vertical magnetic dipole have been addressed by Parise already [27]. The exact solutions of H2z, E2φ, and H2ρ can be expressed in terms of(26)exactzexact=−IdS2πk22−k12Q^2ρ−Q^1ρ,exactφexact=−iIdS2πk22−k12Q2′ρ−Q1′ρ,exactρexact=IdSπρα2+β22K1αρI1βρ−αβK2αρI2βρ,where(27)Q^nρ=ikn3ρ3−4kn2ρ2−9iknρ+9eiknρρ5,Qn′ρ=kn2ρ2+3iknρ−3eiknρρ4.Note thatIn and Kn are the nth order modified Bessel functions of the first and second kind, respectively, where n=1,2.Since it is a special case of the dipole radiation problem discussed in this paper, we can actually better validate the manner if we plot how that field changes as a function of the distanceρ by our approximation method, the direct numerical calculation and this exact analytical solution. Comparisons are depicted in Figures 5–7. All the parameters involved are the same with those in Figures 2–4 except with different d and z.Figure 5
Amplitudes and phase angles of field componentH2z versus the propagation distance ρ with the proposed, exact, and numerical solutions. (a, c) f=3 Hz. (b, d) f=300 Hz.
(a)
(b)
(c)
(d)Figure 6
Amplitudes and phase angles of field componentE2φ versus the propagation distance ρ with the proposed, exact, and numerical solutions. (a, c) f=3 Hz. (b, d) f=300 Hz.
(a)
(b)
(c)
(d)Figure 7
Amplitudes and phase angles of field componentH2ρ versus the propagation distance ρ with the proposed, exact, and numerical solutions. (a, c) f=3 Hz. (b, d) f=300 Hz.
(a)
(b)
(c)
(d)Apparently, our analytical results not only coincide with the numerical simulations but also have an excellent agreement with the exact solutions, and curves in Figures5–6 bear a strong resemblance to those in Figures 2–3. It indicates that all methods are of good accuracy. Meanwhile, comparisons of H2ρ are not satisfactory as can be seen from Figure 7. On the whole, curves of the analytical and numerical solution are relatively consistent, but do not match up to the exact solution as the propagation distance ρ increases. Take f=3Hz as an example. If ρ reaches several hundred meters, the gap in magnitude between exact solutions and the other two methods will become larger. And the phase difference is always about π. This is probably because the exact solution in [27] is expressed in terms of the modified Bessel functions rather than concrete analytical ones.


### 3.3. Evaluations for These Analytical Formulas

For further investigation about the features of the proposed method, several other evaluations are performed. As a natural extension, just to make it clear and to see how the locations of the dipole source and receiving point influence the propagation strength of electromagnetic fields, we consider the following simulations presented in Figure8, with the relative permittivity εr1=80 and conductivity σ1=4 S/m in Region 1. The operating frequency is taken as f=10 Hz.Figure 8
Amplitude of field componentH2z versus the receiving height z: (a) ρ=50 m, (b) ρ=100 m, (c) ρ=500 m, and (d) ρ=1000 m.
(a)
(b)
(c)
(d)Figure8 illustrates the relationship between H2z and z with different heights of the source at a constant propagating distance, where all other parameters are held unchanged. The constant is chosen as 50 m, 100 m, 500 m, and 1000 m, respectively. It is seen that, with the increasing of the radial distance, H2z will exhibit quite different characteristics. When ρ is relatively small (see Figures 8(a) and 8(b)), the amplitude decreases at first and then increases gradually as the height of the observing point becomes higher. Typically, the minimum point of the curve will have a shift to a farther position from the origin if ρ is larger. However, for sufficiently large values of ρ (Figures 8(c) and 8(d)), the amplitude increases monotonically as z increases. We also know that, when the dipole source is closer to the interface, the attenuation will be larger. Perhaps it is due to the different degrees of the effect by the lossy medium in Region 1 at different heights.When the receiving point is at an arbitrary place on thex-z plane in Region 2, and the dipole source is located at d= 0 m, 50 m, 150 m, respectively, with the operating frequency taken as f=10 Hz, the strength distributions of the electric and magnetic components are revealed in Figures 9–10. We can actually see that both the electric and magnetic fields decay rapidly in all directions. As the position of the source is higher, the attenuation of the electromagnetic components will become slower. It is also noted that the distribution in Figures 9(c) and 10(c) is very likely to that in free space, which implies that when the height of the dipole is high enough away from the interface, the radiation result is nearly not affected by the sea water.Figure 9
Field distribution ofH2z in the x-z plane: (a) d=0 m, (b) d=50 m, and (c) d=150 m.
(a)
(b)
(c)Figure 10
Field distribution ofE2φ in the x-z plane: (a) d=0 m, (b) d=50 m, and (c) d=150 m.
(a)
(b)
(c)


## 3.1. Comparison with Numerical Solutions Whenz≠0 and d≠0

It is shown in literature studies [7, 8] that, by some proper numerical approaches, one can solve Sommerfeld integrals directly. Therefore, direct numerical simulation is employed and then compared with the analytical result obtained in this paper. We assume that Region 1 is seawater characterized by the permeability μ0, relative permittivity εr1=80, and conductivity σ1=4 S/m. Region 2 is air characterized by the permeability μ0 and the permittivity ε0. For a general case, the dipole source and receiving point are placed at the height d=1m and z=5m, respectively. The amplitudes and phase angles of the three nonzero field components with respect to the propagation distance ρ are evaluated in Figures 2–4 at operating frequencies f=3Hz and 300 Hz.Figure 2
Amplitudes and phase angles of field componentH2z versus the propagation distance ρ with the proposed and numerical solutions. (a, c) f=3Hz. (b, d) f=300Hz.
(a)
(b)
(c)
(d)Figure 3
Amplitudes and phase angles of the field componentE2φ versus the propagation distance ρ with the proposed and numerical solutions. (a, c) f=3Hz. (b, d) f=300Hz.
(a)
(b)
(c)
(d)Figure 4
Amplitudes and phase angles of the field componentH2ρ versus the propagation distance ρ with the proposed and numerical solutions. (a, c) f=3 Hz. (b, d) f=300 Hz.
(a)
(b)
(c)
(d)It is seen that both the amplitudes and phase angles obtained by analytical and direct numerical methods have a consistent trend in general. Figures2(a) and 2(b) show that at the beginning there exists a relatively apparent difference between these two computed results. This is probably because when ρ is too small, d≪ρ and z≪ρ are not very restrictive. But when ρ>10m, two curves shed light on a good correspondence with each other. The magnitude of E2φ and H2ρ in Figures 3(a), 3(b), 4(a), and 4(b) also show similar trends with H2z. Obviously, in most cases of practical interest, ρ≫z and ρ≫d, so the results confirm the high reliability and validity of the proposed method. Furthermore, it is also noted that the operating frequency has little influence to the magnitude of H2z and H2ρ, but a larger frequency will lead to a smaller amplitude of E2φ.As illustrated in Figures2(c) and 2(d), initially the phase of H2z ascend sharply until reaching the value of π. It is noted that a higher frequency leads to a closer mutation point to the source. Eventually, for a sudden change to −π, as the distance further increases, the phase starts rising smoothly and stabilizes around −π/2. Higher frequency also causes a faster phase variation from –π to −π/2, which results in a shorter distance for reaching the final state. We can see that phase angles of the two methods are almost alike except with some latency. By observing other two components, though trajectories are not in full accord, similar conclusions can still be drawn.It should also be born in mind that with large distances,k2ρ is more close to “1,” which then also leads to a larger difference between the numerical and analytical results. In this paper, attention is restricted to the near field, and k2ρ∼1 is out of consideration. Computations show that different components have different validity limits of k2ρ for maintaining a consistency between the proposal and numerical solutions. For E2φ and H2ρ components, results of both the amplitude and phase obtained by the two methods remain the same even if k2ρ is approaching 1. However, for H2z, when the distance reaches 100 km (i.e., k2ρ≈0.6, as can be seen in Figure 2(b)), two curves start to deviate from each other.


## 3.2. Comparison with Exact Solutions Whenz=0 and d=0

When both the dipole source and receiving point are located at the interface of a half-space, the exact field components excited by a vertical magnetic dipole have been addressed by Parise already [27]. The exact solutions of H2z, E2φ, and H2ρ can be expressed in terms of(26)exactzexact=−IdS2πk22−k12Q^2ρ−Q^1ρ,exactφexact=−iIdS2πk22−k12Q2′ρ−Q1′ρ,exactρexact=IdSπρα2+β22K1αρI1βρ−αβK2αρI2βρ,where(27)Q^nρ=ikn3ρ3−4kn2ρ2−9iknρ+9eiknρρ5,Qn′ρ=kn2ρ2+3iknρ−3eiknρρ4.Note thatIn and Kn are the nth order modified Bessel functions of the first and second kind, respectively, where n=1,2.Since it is a special case of the dipole radiation problem discussed in this paper, we can actually better validate the manner if we plot how that field changes as a function of the distanceρ by our approximation method, the direct numerical calculation and this exact analytical solution. Comparisons are depicted in Figures 5–7. All the parameters involved are the same with those in Figures 2–4 except with different d and z.Figure 5
Amplitudes and phase angles of field componentH2z versus the propagation distance ρ with the proposed, exact, and numerical solutions. (a, c) f=3 Hz. (b, d) f=300 Hz.
(a)
(b)
(c)
(d)Figure 6
Amplitudes and phase angles of field componentE2φ versus the propagation distance ρ with the proposed, exact, and numerical solutions. (a, c) f=3 Hz. (b, d) f=300 Hz.
(a)
(b)
(c)
(d)Figure 7
Amplitudes and phase angles of field componentH2ρ versus the propagation distance ρ with the proposed, exact, and numerical solutions. (a, c) f=3 Hz. (b, d) f=300 Hz.
(a)
(b)
(c)
(d)Apparently, our analytical results not only coincide with the numerical simulations but also have an excellent agreement with the exact solutions, and curves in Figures5–6 bear a strong resemblance to those in Figures 2–3. It indicates that all methods are of good accuracy. Meanwhile, comparisons of H2ρ are not satisfactory as can be seen from Figure 7. On the whole, curves of the analytical and numerical solution are relatively consistent, but do not match up to the exact solution as the propagation distance ρ increases. Take f=3Hz as an example. If ρ reaches several hundred meters, the gap in magnitude between exact solutions and the other two methods will become larger. And the phase difference is always about π. This is probably because the exact solution in [27] is expressed in terms of the modified Bessel functions rather than concrete analytical ones.


## 3.3. Evaluations for These Analytical Formulas

For further investigation about the features of the proposed method, several other evaluations are performed. As a natural extension, just to make it clear and to see how the locations of the dipole source and receiving point influence the propagation strength of electromagnetic fields, we consider the following simulations presented in Figure8, with the relative permittivity εr1=80 and conductivity σ1=4 S/m in Region 1. The operating frequency is taken as f=10 Hz.Figure 8
Amplitude of field componentH2z versus the receiving height z: (a) ρ=50 m, (b) ρ=100 m, (c) ρ=500 m, and (d) ρ=1000 m.
(a)
(b)
(c)
(d)Figure8 illustrates the relationship between H2z and z with different heights of the source at a constant propagating distance, where all other parameters are held unchanged. The constant is chosen as 50 m, 100 m, 500 m, and 1000 m, respectively. It is seen that, with the increasing of the radial distance, H2z will exhibit quite different characteristics. When ρ is relatively small (see Figures 8(a) and 8(b)), the amplitude decreases at first and then increases gradually as the height of the observing point becomes higher. Typically, the minimum point of the curve will have a shift to a farther position from the origin if ρ is larger. However, for sufficiently large values of ρ (Figures 8(c) and 8(d)), the amplitude increases monotonically as z increases. We also know that, when the dipole source is closer to the interface, the attenuation will be larger. Perhaps it is due to the different degrees of the effect by the lossy medium in Region 1 at different heights.When the receiving point is at an arbitrary place on thex-z plane in Region 2, and the dipole source is located at d= 0 m, 50 m, 150 m, respectively, with the operating frequency taken as f=10 Hz, the strength distributions of the electric and magnetic components are revealed in Figures 9–10. We can actually see that both the electric and magnetic fields decay rapidly in all directions. As the position of the source is higher, the attenuation of the electromagnetic components will become slower. It is also noted that the distribution in Figures 9(c) and 10(c) is very likely to that in free space, which implies that when the height of the dipole is high enough away from the interface, the radiation result is nearly not affected by the sea water.Figure 9
Field distribution ofH2z in the x-z plane: (a) d=0 m, (b) d=50 m, and (c) d=150 m.
(a)
(b)
(c)Figure 10
Field distribution ofE2φ in the x-z plane: (a) d=0 m, (b) d=50 m, and (c) d=150 m.
(a)
(b)
(c)


## 4. Conclusions

This paper deals with the integrals for the electromagnetic field of a vertical magnetic dipole lying above the boundary of a lossy half-space in order to derive analytical formulas for the near-zone field. On the basis of the feature of “quasi-static” state and the equivalent infinitesimal theory, we can induce two rational assumptions. One isγ2≈iλ, and the other is about the substitution of γ1−γ2. Comparisons with numerical solutions show that both the amplitude and phase are quite consistent with each other. Curves for contrast with the exact formulas are also drawn when the dipole source and observing point are both on the boundary, which is a further proof of the high accuracy and reliability of our manner. Simulations also show that different components have different validity limits of k2ρ for maintaining a consistency between the proposal and numerical solutions. The smallest validity limit always occurs for the component H2z, and the value of k2ρ should be no greater than 0.6 in order to keep a good consistency. Subsequent computations, which indicate that operating frequencies and the positions of the dipole source and the receiving point will all affect the field components, are also performed. In addition, it should be pointed out that these formulas are also suitable for other cases where the condition k1≪k2 is satisfied. And the media in two regions are interchangeable. Under this circumstance, γ1≈iλ is acquired, and the approximation in (13) changes to γ2−γ1−k2. Then, by a similar method presented in this paper, the final field expression can be obtained.



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*Source: 9710723-2020-03-20.xml*