## 3 Electromagnetic Modeling of Balise for Underwater Debris
### Magnetic Field Strength of Conductive Media  
Some studies on electromagnetic coupling in conductive media have been conducted using electromagnetic field analysis. Most of them are based on circular coil models, assuming the coils are fully immersed in water for calculations and analysis [28; 29]. However, the OAU and the balise are approximately rectangular, with the magnetic field primarily concentrated in the main lobe zone. Therefore, electromagnetic field analysis is performed for the main lobe zone in the presence of conductive media. As shown in Figure 4, the balise \(ABCD\) is composed of four finite-length conductors, with a cavity of height \(h_{0}\) above it. Regions 0 and 2 are free space, while region 1 is dielectric water. The electrical conductivity, dielectric constant, and magnetic permeability of each region are denoted by \(\sigma_{i}\), \(\epsilon_{i}\), and \(\mu_{i}\). The three regions are analyzed separately, assuming the medium in each region is linear, homogeneous, and isotropic. For region 0, \(z<0\), and the magnetic field coexists with the reflected magnetic field generated by eddy currents in this area. Assuming the current density of the balise is \(J\), the magnetic vector potential \(A_{0}\) at any point \((x,y,h)\) in the free space above it can be expressed as:  
\[A_{0}(x,y,h)=\frac{\mu_{0}}{4\pi}\int_{v}\frac{J(x^{\prime},y^{\prime},z^{ \prime})dv^{\prime}}{r} \tag{18}\]  
\[B_{0}=\
abla\times A_{0} \tag{19}\]where \(J\) represents the current density of the balise, \(dv^{\prime}\) represents the elementary volume of the source, and \(r\) is the distance between any point (\(x,y,h\)) and a point source (\(x^{\prime},y^{\prime},z^{\prime}\)) on the balise. The distance \(r\) can be expressed as:  
\[r=\sqrt{\left(x-x^{\prime}\right)^{2}+\left(y-y^{\prime}\right)^{2}+\left(h-z^ {\prime}\right)^{2}} \tag{20}\]  
Considering that no eddy currents exist in this region, the formula for calculating the reflected magnetic flux density Br is as follows:  
\[\
abla^{2}B_{r}=0 \tag{21}\]  
\[\
abla\times B_{r}=0 \tag{22}\]  
For region 1, 0 \(<z<h\), and the magnetic flux density \(B_{1}\) is given by:  
\[\
abla^{2}B_{1}-\left(\omega^{2}\mu_{1}\varepsilon_{1}-j\omega\mu_{1} \sigma_{1}\right)B_{1}=0 \tag{23}\]  
\[\
abla\cdot B_{1}=0 \tag{24}\]  
For region 2, \(h<z\), and the magnetic flux density \(B_{2}\) is given by:  
\[\
abla^{2}B_{2}=0 \tag{25}\]  
\[\
abla\cdot B_{2}=0 \tag{26}\]  
To solve the aforementioned equation, we introduce the method of double Fourier transform [30; 31] and its inverse transform as follows:  
\[b(\xi,\eta,z)=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}B(x,y,z)e^{-j(x \xi+y\eta)}dxdy \tag{27}\]  
\[B_{0}=\
abla\times A_{0} \tag{19}\]0  
where \(\xi\) and \(\eta\) are integration variables used in the double Fourier transform. The normal component of the magnetic flux density and the tangential component of the magnetic  
\[B_{0}=\
abla\times A_{0} \tag{19}\]1 Electromagnetic coupling for underwater debris. field strength remain continuous across the boundary interface. The following relationship holds true:  
\[\left\{ \begin{array}{ll}b_{iz}+b_{rz}=b_{12},&z=0;\\ \mu_{1}(b_{ix}+b_{rx})=\mu_{0}b_{1x},&z=0;\\ b_{1z}=b_{2z},&z=h;\\ \mu_{2}b_{1x}=\mu_{1}b_{2x},&z=h;\end{array} \right. \tag{29}\]  
Considering that the magnetic flux density at infinity is 0, the \(Z\)-axis component \(B_{2z}\) of the magnetic flux density from the balise to the OAU is:  
\[B_{2z}(x,y,z)=\frac{\mu_{0}I_{1}}{2\pi^{2}}\int_{-\infty}^{\infty}\int_{- \infty}^{\infty}\frac{k\delta\text{sin}(a_{\xi}^{\zeta})\text{sin}(b\eta)}{ \xi\eta}e^{-k(z+h_{0})}e^{-j(x_{\xi}^{\zeta}+y\eta)}d\xi d\eta \tag{30}\]  
In Equation (30), regions 0 and 2 comprise the same medium, with magnetic permeability \(\mu_{0}\).