# Towards the Feasibility of Long Range Wireless Power Transfer over an Ocean Surface
## 2 Methodology and Materials
### Theory
To be able to apply this model to a seawater, we need to obtain the measured permittivity of the NaCl electrolytic solution first, followed by subtracting the influence due to the dielectric constant of the pure water, that is [28; 29; 30]:  
\[\varepsilon_{2,\text{Drude}}=\varepsilon_{2,\text{ Electrolyte}}-\varepsilon_{w} \tag{4}\]  
In so doing, the real part of the permittivity of NaCl as calculated using Equation (4) will be always negative at frequencies below the plasma frequencies. The Drude model of the medium 2 has the following form:  
\[\varepsilon_{2}=\varepsilon_{2}^{\prime}+j\varepsilon_{2}^{{}^{\prime\prime} }=\left(1-\frac{\omega_{p,2}{}^{2}}{\omega^{2}+\Gamma_{2}{}^{2}}\right)+j \frac{\Gamma_{2}{}^{2}}{\omega}\left(\frac{\omega_{p,2}{}^{2}}{\omega^{2}+ \Gamma_{2}{}^{2}}\right) \tag{5}\]  
In Equation (5), \(\omega_{p,2}\) is the plasma frequency given by \(\omega_{p,2}=Ne^{2}/(\varepsilon_{0}m)\) where \(N\) is the number of particles per unit volume, \(e\) is the electron charge, \(\varepsilon_{0}\) is the permittivity of vacuum, \(m\) is the mass of the conductive particle, and \(\Gamma_{2}\) is the damping factor of seawater. Likewise, the propagation vector will be obtained in the following form:  
\[k_{2}=k_{2}^{\prime}+jk_{2}^{{}^{\prime\prime}} \tag{6}\]  
Due to the conservation of light momentum, the wave numbers can be expressed as:  
\[k_{1}^{2}=\beta^{2}-k_{0}^{2}\varepsilon_{1} \tag{7}\]  
\[k_{2}^{2}=\beta^{2}-k_{0}^{2}\varepsilon_{2} \tag{8}\]  
Due to the continuity conditions for transverse magnetic mode, the following relation must be fulfilled in order to satisfy [PERSON]'s equations:  
\[\frac{k_{1}}{\varepsilon_{1}}+\frac{k_{2}}{\varepsilon_{2}}=0 \tag{9}\]For a surface electromagnetic wave to exist on the propagation interface between media 1 and 2, the relation in Equation (9) must be fulfilled. This means that both the real part and the imaginary part of Equation (9) must be zero. \[\varepsilon_{1}=\varepsilon_{1}^{\prime}+j\varepsilon_{1}^{{}^{\prime\prime} }=(\varepsilon_{\infty,1})-j\frac{\delta_{1}}{\omega} \tag{1}\]0  
and  
\[\varepsilon_{1}=\varepsilon_{1}^{\prime}+j\varepsilon_{1}^{{}^{\prime\prime} }=(\varepsilon_{\infty,1})-j\frac{\delta_{1}}{\omega} \tag{1}\]1  
\(k_{1}\) and \(k_{2}\) are assumed to have a positive real part in order to attain a damping condition, i.e., \(Re(k_{1})>0\) and \(Re(k_{2})>0\). This assumption is invalid if there exists a wave oscillation at the contact interface. By solving Equations (10) and (11), we obtain the following widely agreed conclusion [23; 24; 25]:  
\[\varepsilon_{1}=\varepsilon_{1}^{\prime}+j\varepsilon_{1}^{{}^{\prime\prime} }=(\varepsilon_{\infty,1})-j\frac{\delta_{1}}{\omega} \tag{1}\]2  
By multiplying the top and bottom of Equation (12) with the conjugate of \(\varepsilon_{2}\), we obtain the following very neat inequality:  
\[\varepsilon_{1}=\varepsilon_{1}^{\prime}+j\varepsilon_{1}^{{}^{\prime\prime} }=(\varepsilon_{\infty,1})-j\frac{\delta_{1}}{\omega} \tag{1}\]3  
Inequality (13) is based on an assumption that both media 1 and 2 have a non-zero imaginary part. By direct inspection of Equation (13), the most straightforward condition that satisfies (9) is for \(\varepsilon^{\prime}{}_{2}\) to be negative. \(\varepsilon^{\prime}{}_{2}\) is deduced from the measured dielectric constant of the liquid seawater using Equation (4).