1. Introduction
Over the past few decades, the study of electromagnetic wave (EMW) propagation in layered media has attracted considerable attention [
1,
2,
3], particularly in the context of EMW propagation across the sea-air interface. In comparison to techniques employed in acoustics [
4,
5] and optics [
6,
7], as well as other fields of ocean engineering, EMWs exhibit unique and valuable properties that render them a prime choice for applications in ocean communication engineering. Despite their limited transmission range owing to high attenuation, EMWs possess several advantageous features. Firstly, they can seamlessly traverse the “sea-air” interface, thereby extending the transmission range for both aerial and seabed paths. Secondly, electromagnetic transmission is resilient in the face of tidal waves and turbulence resulting from human activities. Thirdly, EMWs can operate effectively in turbid water conditions, among other advantages. Therefore, the utilization of EMWs in seawater environments has garnered significant interest in recent years [
8,
9,
10].
The problem of dipole radiation in multilayered media was first introduced by Sommerfeld in 1909 [
11]. Since then, this research area has witnessed substantial interest and notable achievements, finding applications in various engineering domains, including geophysical exploration [
12], submarine communication [
13], underwater navigation [
14], and submarine detection and communication [
15,
16]. Much of the subsequent work builds upon Sommerfeld’s integral [
11], which serves as the theoretical foundation for the propagation and scattering of EMW. However, in this context, calculating the electromagnetic field (EMF) necessitates solving the challenging Sommerfeld integral due to its singularity and high oscillation near the integration path. Margetis [
17] and Banos [
18] significantly advanced the theory of dipole sources embedded in conductive half-spaces, obtaining approximate solutions for different distances from the source. Numerous other scholars have also made substantial contributions to the analysis of EMF in layered conductive media [
19,
20,
21,
22]. Among them, the description and explanation of the physical process of EMW propagation across the sea–air interface is a topic of interest. In a study by Bush et al. (2012) [
23], the propagation path of EMWs between transmitting and receiving antennas located in seawater was investigated. The study proposed that the propagation path consists of direct wave, reflected wave, and lateral wave propagation, as shown in
Figure 1, and provided the analytical expression for the related components.
Bishay (2013) [
24] utilized complex image theory [
25] and the Hankel transform (integral over the radial distance
r) [
26] to calculate the far field of the electric field disturbance caused by a vertical electric dipole (VMD) in seawater. A good approximate solution for the far field was obtained. Wang (2015) [
27] derived the analytical expression of the EMF generated by a dipole antenna in the three-layer medium consisting of air, seawater, and seabed. Numerical calculations were performed using the fast Fourier transform (FFT) method to obtain the variation and spatial distribution of the EMF in the air with distance. The study confirmed that underwater EMWs primarily rely on air distance for long-distance transmission. Wang and Li (2017) [
28] comprehensively analyzed the airborne EMF generated by a horizontal electric dipole (HED) in shallow seawater through theoretical analysis and experimental verification. The study proved that low frequency and interface effects are crucial for achieving long-distance transmission. In the extremely low frequency (ELF) band, EMWs can travel over 3 km on the sea surface with reasonable transmission power. Subsequently, Xu (2018) [
29] investigated the near-field propagation of ELF waves excited by HED near the boundary between seawater and seafloor. The study proposed the use of the MacLaurin expansion method to derive an analytical solution for the near-field of the seafloor under quasi-static approximation conditions. Peng (2018) [
30] discussed the calculation of the electric field radiated by a horizontal magnetic dipole (HMD) antenna into a lossy half-space, simulating it as a two-layer medium. The study derived a set of EMF expressions composed of Sommerfeld-type integrals under quasi-static magnetic field conditions. Shoeiba (2020) [
31] considered the EMF generated by VMD under seawater in a planar three-layer conductive media (air, seawater, and seabed) and theoretically demonstrated that underwater VMD performs better when EMWs propagates through the boundary. The experiment verified that the depth and frequency of the transmitter significantly affect wave propagation in each region, with increased transmitter depth leading to a rapid reduction in the radiation field in the air. In the range of 50 m, changes in receiver height have a negligible influence on-field intensity, particularly in the far field. Zeng (2021) [
32] studied the modal theory of ELF wave propagation in layered marine lithospheric waveguides using surface impedance boundary conditions. The study derived the modal equations for transverse magnetic polarization guiding modes and transverse electric polarization guiding modes and obtained analytical expressions for propagatable modal parameters such as phase velocity, decay rate, excitation factor, and field strength under different components. Based on the conditional assumption of
γ1γ0 >>
k12 (where
γ is the propagation constant and
k is the wave number, subscripts 0 and 1 represent seawater and air, respectively), Xu (2021) [
33] mainly derived the complete and effective solution for the electromagnetic near field generated by VMD on the sea surface in the air and seawater region under cylindrical coordinates. Hu (2023) [
34] also utilized the McLaughlin expansion method to solve the near-field analytical solution under the quasi-static approximation condition at the uniformly infinite sea-air boundary. The study provided the spatial radiation distribution changes of the EMF but did not analyze the EMF propagation characteristics in different environments at the sea-air boundary.
In general, these limited studies underscore the analytical challenges posed by the Sommerfeld integral in the context of near-field wave propagation, particularly due to the presence of interfaces. In situations where an ELF wave propagates along the boundary of two media with a propagation distance of
kρ << 1, the Fourier–Bessel integral term in the Sommerfeld integral expression exhibits a divergence near the pole of
kρ → 0 at the seawater–air interface. For example, when EMWs propagate at the “sea-air” field interface, the ratio of ELF electromagnetic wave numbers denoted as
k1/
k0 tends toward infinitesimal values. Margetis’s integral expression [
17] can calculate the near-region EMF when
k1ρ << 1. However, when an ELF wave propagates along the sea-air interface, the dielectric constant of seawater can be approximated as purely imaginary (
εsea ≈
jσω), and |
k1/
k0| << 1 is not met. Consequently, Margetis’s method is unsuitable for near-region propagation issues. Conversely, numerical calculation methods [
35,
36] offer limited resolution, unable to provide a practical means of evaluating radiation intensity near the source and presenting difficulties or impossibilities in addressing near-field regions.
ELF, with an operating frequency range of 3–30 Hz, boasts an exceptionally long wavelength, approaching the circumference of the Earth—approximately three-quarters of it. When the field point and the source point are not significantly distant, the EMF propagation problem in the near-field of the HED becomes a pertinent subject of study. This paper delves into the EMF propagation in the near-field excited by an ELF HED at the sea-air interface, utilizing the HED model. To address this issue, we propose the use of the Sommerfeld numerical integral calculation method under quasi-static approximation conditions, specifically
ω → 0,
k1ρ << 1,
k1 <<
k0. Building upon this, we simplify the approximate numerical results of the Bessel-Fourier infinite integral term in the Sommerfeld integral expressions, leading to derived approximate integral expressions for EMF propagation in the near-region of seawater. To ascertain the validity of our proposed method, we compare its calculation results with those of Margetis [
17] and Pan [
22]. Furthermore, through simulation, we calculate and analyze the EMF propagation characteristics at the sea–air interface across different frequencies, dipole source heights, and observation point heights. These findings hold practical value in the realms of underwater target detection, underwater communication, and underwater navigation.
2. EMF Propagation Model in ELF at the Sea-Air Boundary
The model for the propagation of ELF EMF in the sea-air boundary, excited by a HED, is illustrated in
Figure 2. We employ cylindrical coordinates as the reference system. Let us consider the HED positioned at the air-sea interface (
z = 0), with the dipole directed downward along the positive
z-axis and parallel to the
x-axis (
φ = 0°). The distance from the
xoy plane, representing the sea surface, is denoted as “
d” meters, and the electric moment is represented by “
Idl”. The observation point, labeled as
P(
ρ,
φ,
z), is situated at a height “
z” meters above the sea surface.
As depicted in
Figure 2, the upper half-space is filled with air, referred to as region 1 (
z < 0), with respective electrical parameters denoted as
k1,
μ1,
σ1, and
ε1. The lower half-space is occupied by seawater, termed region 0 (
z ≥ 0), with electrical parameters labeled as
k0,
μ0,
σ0, and
ε0. We define “
ρ” as the propagation distance between the projection point of the electric dipole source and the observation point, “
r0” as the geometric distance from the electric dipole source point to the observation point in seawater, and “
r1” as the geometric distance from the mirrored electric dipole source point to the observation point in air. For ELF electric dipole excitation varying with the time-harmonic factor
e−jωt, the EMF it generates in seawater region 0 (
z ≥ 0) can be expressed as an integral [
11], as comprehensively demonstrated in
Appendix A. In this paper, we use the electric field component
E0ρ and the magnetic field component
B0ρ from [
11] for illustrative purposes.
where
Here, γ0 represents the propagation parameter in region 0, γ1 represents the propagation parameter in region 1, ω is the angular frequency, λ is the wavelength, c is the speed of light in free space, and j is the imaginary unit. Jn(λρ) is a Bessel function of order n, with n = 0, 1, 2.
4. Simulation and Analysis of Near-Region Fields
To delve deeper into the characteristics of ELF near-field distribution by an HED near the “sea-air” interface, we maintain the aforementioned relative dielectric constants for seawater and air within the cylindrical coordinate system, as shown in
Figure 2. For illustration, we employ the EMF components
E0ρ and
B0ρ. Our methodology, as outlined in this paper, is used to simulate and compute the amplitude variation of these components concerning propagation distance (
ρ) across different frequencies, dipole heights, and observation point heights. The results are graphically presented in
Figure 4,
Figure 5,
Figure 6 and
Figure 7.
In
Figure 4a,b, the field’s source and observation points are situated at distances of
d = 10 m and
z = 100 m from the sea surface, respectively. The figures illustrate the EMF components
E0ρ and
B0ρ’s trends as they relate to propagation distance (
ρ) at various operating frequencies. These graphs reveal that both the electric field component
E0ρ and magnetic field component
B0ρ decrease as horizontal distance in the
ρ direction increases. When
ρ < 50 m, the amplitude of EMF intensity experiences the most significant changes and decays rapidly. Conversely, when
ρ > 300 m, the electromagnetic field intensity exhibits less substantial decay, showing a slow declining trend with an attenuation amplitude of approximately 40 dB. Additionally, it is evident that, at the same observation point, the amplitudes of EMF components
E0ρ and
B0ρ are more sensitive to changes in frequency and decrease as operating frequency increases. Notably, the reduction in amplitude is more pronounced in the high-frequency range within the ELF band, emphasizing the preference for low-frequency EMWs in seawater applications.
Figure 5a,b illustrate the variations in the electric field component
E0ρ and magnetic field component
B0ρ concerning propagation distance (
ρ) at an operating frequency of 8 Hz. The observation point’s height is adjusted to align with the field source height at
d = 10 m, 50 m, 100 m, and 150 m, respectively. As depicted in
Figure 5, the dipole’s height significantly influences EMW propagation. A higher dipole source height corresponds to a more substantial attenuation in the electric field component
E0ρ and magnetic field component
B0ρ. For propagation distances (
ρ) less than 200 m, both
E0ρ and
B0ρ experience significant decreases with increasing horizontal distance. However, beyond 200 m, the field intensity amplitude stabilizes. Remarkably, near the interface, the electric field component
E0ρ and magnetic field component
B0ρ exhibit the least decline in intensity, with an attenuation amplitude of approximately 60 dB. This suggests that the attenuation rate of HED EMF components in the sea–air half-space with horizontal distance is significantly lower than in a pure seawater medium.
We further investigated the effects of varying the field source point’s height on the amplitude of
E0ρ at observation points near the interface.
Figure 6a,b depict the amplitude changes in the EMF component
E0ρ as the height of the field source point varies from 0 m to 300 m, with propagation distances (
ρ) set at 500 m and 1000 m and an operating frequency of 8 Hz. The source point’s height significantly affects the amplitude of
E0ρ at observation points near the interface. Higher field source points correspond to faster attenuation rates in
E0ρ amplitude. However, as the source point’s height increases, the attenuation rate of
E0ρ amplitude decreases. Notably, at a depth of
z = 150 m, the change in electric field component
E0ρ becomes less pronounced, but the attenuation value reaches its zenith. Moreover, when
ρ = 500 m and 1000 m, and the field source point’s height is 100 m and 50 m, respectively, a distinct critical point emerges. The attenuation of
E0ρ remains essentially unchanged when the source height exceeds 120 m. These observations suggest that as propagation distance (
ρ) increases, the attenuation amplitude of the electric field intensity in seawater becomes more pronounced. Specifically, when
ρ = 500 m, the
E0ρ component is effective within a dipole height of
d = 50 m, while for
ρ = 1000 m, the
E0ρ component is effective within a dipole height of
d = 20 m, offering the most favorable field intensity effects. Nonetheless, with the observation point’s depth (
z) increasing, the field intensity undergoes a shift: the field intensity values gradually decrease. A critical point is reached at
z = 150 m, and the field strength component remains nearly constant at a dipole height of
d = 100 m.
As illustrated in
Figure 7a,b, the magnetic field component
B0ρ varies with the observation point’s height, exhibiting an attenuation pattern closely resembling that of the electric field intensity
E0ρ. The variation trend of the
B0ρ component at the observation point mirrors that of the
E0ρ component. The attenuation rate is significantly influenced by the field source point’s proximity to the interface. The horizontal distance between the observation point and the field source point plays a critical role in influencing the
B0ρ component. Notably, when the propagation distance
ρ = 500 m, the
B0ρ component experiences its steepest decline at a dipole height of
d = 50 m, followed by a more gradual descent. After this initial decline, magnetic field intensity
B0ρ demonstrates reduced sensitivity to changes in observation point height once
d > 200 m. Conversely, when the propagation distance
ρ = 1000 m, the
B0ρ component exhibits pronounced changes at a dipole height of
d = 20 m, followed by an attenuation amplitude that remains nearly constant at a dipole height of
d = 200 m. These findings underscore the substantial impact of propagation distance (
ρ) on ELF antennas in seawater.
5. Conclusions
In this study, we established a model for the propagation of EMWs by HED at the boundary between seawater and air. By leveraging the quasi-static assumption, where ω → 0, k1ρ << 1, and k1 << k0, we derived integral expressions for Sommerfeld EMF in the near region induced by ELF HED in seawater. These expressions simplified the Bessel–Fourier integral terms within Sommerfeld’s integral expressions, ultimately yielding the final expressions for EMF in the near region of seawater. We comprehensively analyzed the EMF propagation characteristics of an ELF HED source in seawater, taking into account different frequencies, dipole source heights, and observation point heights. Our results indicated that the electric and magnetic fields were strongest at the interface between seawater and air, with attenuation amplitudes increasing alongside higher dipole source heights, observation point heights, and propagation distances. From our simulation results, we conclude that the optimal conditions for EMF intensity occur at the seawater–air interface when the propagation distance is ρ = 500 m and 1000 m, the dipole source height is d = 50 m and 30 m, and the observation point depth is z = 150 m and 100 m, respectively, within the ELF band. Beyond these conditions, the amplitude of EMF intensity becomes less responsive to changes. These findings underscore that lateral waves represent the primary propagation mode of ELF HED at the seawater–air interface. The practical applications of these conclusions hold significance in underwater communication, underwater target location, and underwater navigation. We observed consistency between our calculated results and those of Pan and Margetis, validating the accuracy of our proposed method. Notably, our method excels in predicting near-field behavior in seawater but exhibits less precision in the far-field context.