## 3 The Electric Fields in the Medium
### Wave Propagation in Medium  
The Lossless medium has zero conductivity so that the electromagnetic wave does not lose power. To analyze wave propagation in a Lossless medium, it is necessary to consider the relationship between the electric field vector and the magnetic field vector [11]. Based on [PERSON]'s equations, the phasor form of [PERSON]'s law and the [PERSON]'s law can be determined as follows:  
\[\
abla\times\hat{E}=-j\omega\mu_{0}\mu_{r}\hat{H} \tag{4a}\] \[\
abla\times\hat{H}=\hat{J}+j\omega\varepsilon_{0}\epsilon_{r}\hat{E}=\sigma \hat{E}+j\omega\varepsilon_{0}\epsilon_{r}\hat{E} \tag{4b}\]  
where \(\hat{E}\) and \(\hat{H}\) represent the electric and magnetic fields, respectively, and \(\omega\) is \(2\pi f\). The permeability in free space is \(\mu_{0}=4\pi\times 10^{-7}(\mathrm{H/m})\) and the permittivity is \(\varepsilon_{0}\cong(1/36\pi)\times 10^{-9}\ (\mathrm{F/m})\). The electrical characteristics of a medium can be expressed using the relative permeability \(\mu\), and relative permittivity \(\varepsilon_{r}\). Assuming that the direction of the electric field is the x-direction, the direction of the magnetic field is the y-direction, and the direction of electromagnetic waves is the z-direction; the left side of Equation (4a,b) are expanded in the rectangular coordinate system as follows:  
\[\
abla\times\hat{E}=\left(\frac{\partial\hat{E}_{z}}{\partial y}-\frac{ \partial\hat{E}_{y}}{\partial z}\right)\mathbf{a}_{x}+\left(\frac{\partial\hat{E} _{x}}{\partial z}-\frac{\partial\hat{E}_{z}}{\partial x}\right)\mathbf{a}_{y}+ \left(\frac{\partial\hat{E}_{y}}{\partial x}-\frac{\partial\hat{E}_{x}}{ \partial y}\right)\mathbf{a}_{z}=\frac{\partial\hat{E}_{x}}{\partial z}\mathbf{a}_{y} \tag{5a}\]  
Figure 5: Fabricated underwater stacked CPT system. (**a**) Stacked CPT coupler. (**b**) Experimental setup. \[\
abla\times\hat{H}=\left(\frac{\partial\hat{H}_{z}}{\partial y}-\frac{\partial\hat{ H}_{y}}{\partial z}\right)\mathbf{a}_{x}+\left(\frac{\partial\hat{H}_{x}}{\partial z }-\frac{\partial\hat{H}_{z}}{\partial x}\right)\mathbf{a}_{y}+\left(\frac{\partial \hat{H}_{y}}{\partial x}-\frac{\partial\hat{H}_{x}}{\partial y}\right)\mathbf{a}_{z} =-\frac{\partial\hat{H}_{y}}{\partial z}\mathbf{a}_{x} \tag{5b}\]  
Since the directions of the electric and magnetic fields are the x-axis and y-axis, the \(\hat{E}_{y}\), \(\hat{E}_{z}\), \(\hat{H}_{x}\), \(\hat{H}_{z}\) must be zero. The \(\frac{\partial\hat{E}_{x}}{\partial y}\) and \(\frac{\partial\hat{H}_{y}}{\partial x}\) of the \(\mathbf{a}_{z}\) component also becomes zero because the \(\hat{E}_{x}(z)\) and \(\hat{H}_{y}(z)\) components of the uniform plane wave are constant regardless of the x-axis and y-axis. Substituting the results of Equation (5a,b) into Equation (4a,b) and applying the characteristics of a Lossless medium (\(\sigma=0\)), the equation below can be obtained. \[\frac{\partial\hat{E}_{x}}{\partial z}=-j\omega\mu_{0}\mu_{r}\hat{H}_{y} \tag{6a}\] \[\frac{\partial\hat{H}_{y}}{\partial z}=-j\omega\varepsilon_{0}\varepsilon_{r} \hat{E}_{x} \tag{6b}\]  
The equation consisting of only \(\hat{E}_{x}\) or \(\hat{H}_{y}\) can be obtained by differentiating one equation with respect to z and then substituting it into the other equation. Under the condition of \(\sigma\
eq 0\), Equation (6a,b) can be rewritten as follows:  
\[\frac{\partial\hat{E}_{x}}{\partial z}=-j\omega\mu_{0}\mu_{r}\hat{H}_{y} \tag{7a}\] \[\frac{\partial\hat{H}_{y}}{\partial z}=-\sigma\hat{E}_{x}-j\omega\varepsilon_{ 0}\varepsilon_{r}\hat{E}_{x} \tag{7b}\]  
Since [PERSON]'s law does not include a term for conductivity, Equation (7a) is identical to Equation (6a).