# Towards the Feasibility of Long Range Wireless Power Transfer over an Ocean Surface
## 2 Methodology and Materials
### Theory  
The problem as stated in the objective of this work can be thought of as a surface electromagnetic wave propagating along an interface between a low-loss medium and a high-loss medium (see Figure 1). The low-loss medium is either air or a non-polar substance. The high-loss medium is a polar substance. Non-polar substances include oil or petroleum chemicals. Polar substances include water. In this work, the high-voltage surface wave is a wave bound to a water surface, which can be water exposed to air or an interface between the rubber in the rubber tube and the water filling this tube. The electromagnetic wave capable of propagating on a flat interface is known to be transverse magnetic wave with the electric field component being normal to the interface. Since the rubber tube has a concave interior wall, there is a likelihood of transverse electric waves as well. However, our analysis is based on an assumption that the effects of this transverse electric waves are insignificant. Under our assumption, the model given in Figure 1 remains appropriate for our theoretical prediction. Figure 1: (**a**) Cross-sectional view of an interface between a lossless dielectric material and a lossy dielectric material. (**b**) Excitation of a surface electromagnetic wave on a monopole antenna partially submerged in a seawater, by a 100 kV pulsating source at 44 MHz. (**c**) Simulated surface electromagnetic wave as a result of the monopole antenna setup in (**b**). (**d**) Simulated S1,1 parameters of the setup in (**b**). Suppose the permittivities's for medium 1 and medium 2 are respectively \(\varepsilon_{1}\) and \(\varepsilon_{2}\). The permeabilities for medium 1 and medium 2 are respectively \(\mu_{1}\) and \(\mu_{2}\). The wavenumbers for medium 1 and medium 2 are respectively \(k_{1}\) and \(k_{2}\). \(\Gamma_{2}\) and \(\omega_{p,2}\) are respectively the damping coefficient and the plasma frequency of medium 2. \(\varepsilon_{1}\) and \(k_{1}\) can be straightforwardly written as:  
\[\varepsilon_{1}=\varepsilon_{1}^{\prime}+j\varepsilon_{1}^{{}^{\prime\prime} }=(\varepsilon_{\infty,1})-j\frac{\delta_{1}}{\omega} \tag{1}\]  
\[k_{1}=k_{1}^{\prime}+jk_{1}^{{}^{\prime\prime}} \tag{2}\]  
Medium 2 is a seawater. Seawater is a pure water with added moving Na\({}^{+}\) and Cl\({}^{-}\) ions. Pure water is an insulator that acts like a capacitor with no initial charge, whilst the moving Na\({}^{+}\) and Cl\({}^{-}\) ions contribute to the conductivity of the seawater. This conductivity is the key to the feasibility of wireless power transfer. There are many ways to model the permittivity of seawater. One of the easiest approaches is to treat medium 2 as a capacitor and an inductor connected in parallel, with the capacitor formed by the pure water and the inductor formed by the Na\({}^{+}\) and Cl\({}^{-}\) ions. The relative permittivity of the pure water can be easily found using the [PERSON]'s model:  
\[\varepsilon_{w}=\varepsilon_{w}^{\prime}+je_{w}^{{}^{\prime\prime}}=\left( \varepsilon_{\infty,w}+\frac{(\varepsilon_{s,w}-\varepsilon_{\infty,w})}{1+ \omega^{2}\tau^{2}}\right)+j\frac{(\varepsilon_{s,w}-\varepsilon_{\infty,w}) \omega\tau}{1+\omega^{2}\tau^{2}} \tag{3}\]  
where \(\varepsilon_{s,w}\) is the dielectric constant at DC, \(\varepsilon_{\infty,w}\) is the dielectric constant at infinite frequency, and \(\tau\) is the relaxation time which is also the inverse of water molecules' collision frequency. The model to be used for analysis is a Drude-based model, which was originally intended for modeling the permittivity of metals only.