# Electromagnetic Field Variation of ELF Near-Region Excited by HED in a Homogeneous Half-Space Model
## 3 Analytical Approximate Expression of the Field Component
### Quasistatic Approximate Propagation Conditions
#### 3.1.3 Approximation \(d<<\rho\) and \(z<<\rho\)  
Referring to (1), under the quasistatic assumption, the slot position of the Sommerfeld integral fulcrum \(k_{2}\) is close to the zero point of the imaginary axis. Because the ELF near-field conditions in seawater meet the relationship equations \(d<<\rho\) and \(z<<\rho\), the multivalue function of the Sommerfeld integral in seawater can be approximated as follows:  
\[\frac{k_{1}^{2}\gamma_{2}-k_{2}^{2}\gamma_{1}}{k_{1}^{2}\gamma_{2}+k_{2}^{2} \gamma_{1}}=\frac{1-k_{2}^{2}\gamma_{1}/k_{1}^{4}\gamma_{2}}{1+k_{2}^{2}\gamma_ {1}/k_{1}^{4}\gamma_{2}}\approx 1 \tag{4}\]Substituting the Equations (2)-(4) into the Sommerfeld integral expressions, the Sommerfeld integral expression of each electromagnetic field component at any observation point \(P\) in the homogeneous seawater medium can be simplified to  
\[E_{1\rho}=-\frac{\omega\mu_{0}Idl}{4\pi k_{1}^{2}}\cos\varphi\bigg{[}k_{1}^{2}F_ {1}(k_{1},d_{0})-\frac{1}{2}F_{2}(k_{1},d_{0})+\frac{1}{2}G_{4}(k_{1},d_{1})- \frac{k_{1}^{2}}{2}H_{1}(d_{1})\bigg{]} \tag{5}\]  
\[E_{1\varphi}=\frac{\omega\mu_{0}Idl}{4\pi k_{1}^{2}}\sin\varphi\bigg{[}k_{1}^{2 }F_{1}(k_{1},d_{0})-\frac{1}{2}F_{2}(k_{1},d_{0})+\frac{1}{2}G_{5}(k_{1},d_{1}) -\frac{k_{1}^{2}}{2}H_{2}(d_{1})\bigg{]} \tag{6}\]  
\[E_{1z}=\frac{j\omega\mu_{0}Idl}{4\pi k_{1}^{2}}\cos\varphi[\pm F_{3}(k_{1},d_{ 0})+G_{1}(k_{1},d_{1})] \tag{7}\]  
\[B_{1\rho}=-\frac{\mu_{0}Idl}{4\pi}\sin\varphi\bigg{[}\pm F_{4}(k_{1},d_{0})+ \frac{1}{2}G_{2}(k_{1},d_{1})-\frac{1}{2}H_{3}(d_{1})\bigg{]} \tag{8}\]  
\[B_{1\varphi}=-\frac{\mu_{0}Idl}{4\pi}\cos\varphi\bigg{[}\pm F_{4}(k_{1},d_{0}) +\frac{1}{2}G_{3}(k_{1},d_{1})-\frac{1}{2}H_{4}(d_{1})\bigg{]} \tag{9}\]  
\[B_{1z}=\frac{j\mu_{0}Idl}{4\pi}\sin\varphi[F_{5}(k_{1},d_{0})-H_{5}(d_{1})] \tag{10}\]  
where \(F_{1}\) (\(k_{1}\), \(d_{0}\))-\(F_{5}\) (\(k_{1}\), \(d_{0}\)) represents the direct wave from the field source point (0, 0, \(d\)) to any near-field observation point \(P\)(\(\rho\), \(\varphi\), \(z\)) in the lower half-space, \(G_{1}\) (\(k_{1}\), \(d_{1}\))-\(G_{5}\) (\(k_{1}\), \(d_{1}\)) represents the reflected wave from the field source image point (0, 0, \(-d\)) to any near-field observation point (\(\rho\), \(\varphi\), \(z\)) in the lower half-space, and \(H_{1}\) (\(d_{1}\))-\(H_{5}\) (\(d_{1}\)) represents the side wave affected by the interface of the two media. \(\varphi\)_is_ azimuth coordinates. \(r_{0}\) is the geometric distance from the source point to the receiving point of the seawater medium, \(r_{1}\) is the geometric distance from the image point of the field source to the receiving point of the air medium, \(d_{0}\) is the vertical height between the source and receiving points of the electric dipole in the seawater medium, and \(d_{1}\) is the vertical height between the image point of the field source and receiving point of the air medium. \(r_{0}\), \(r_{1}\) and \(d_{0}\), \(d_{1}\) are defined as follows:  
\[d_{0}=|z-d|,\;d_{1}=z+d \tag{11}\]  
\[r_{0}=\sqrt{d_{0}^{2}+\rho^{2}},\;r_{1}=\sqrt{d_{1}^{2}+\rho^{2}} \tag{12}\]