**2. Time-harmonic EM model**

When deriving the wave equation for electromagnetic wave propagation through a vacuum or a dielectric medium such as soil or pure water, the free charge density *ρ <sup>f</sup>* ¼ *0* and the free current density *J <sup>f</sup>* ¼ *0*. However, in the case of conductors like seawater or metals, we cannot control the flow of charges and, in general, *J <sup>f</sup>* is certainly not equal to zero. With this, Maxwell's equations for linear media assume the form [11].

$$\begin{cases} \begin{array}{ll} (i) & \nabla \cdot E = \frac{1}{\varepsilon} \rho\_f \\\\ (ii) & \nabla \cdot B = \mathbf{0} \end{array} & (iv) \quad \nabla \mathbf{x} E = \cdot \frac{\partial B}{\partial t} \\\\ (i\dot{i}) & \nabla \cdot B = \mathbf{0} \end{cases} \tag{1}$$

Apply the curl to (iii) and (iv), and we obtain modified wave equations for the electric field *E* and the magnetic field *B*:

$$
\nabla^2 u = \mu \varepsilon \frac{\partial^2 u}{\partial t^2} + \mu \sigma \frac{\partial u}{\partial t} \tag{2}
$$

where *u* represents the scalar component of the electric or magnetic field, *ε* is the permittivity of the medium *μ* ¼ *μrμ0*, is the magnetic permeability (*μ<sup>r</sup>* is the relative permeability of the soil *<sup>μ</sup><sup>r</sup>* <sup>¼</sup> *<sup>1</sup>*, for non-magnetic soil, *<sup>μ</sup><sup>0</sup>* <sup>¼</sup> *<sup>4</sup>π*x*10*‐*<sup>7</sup> N=A<sup>2</sup>* is the magnetic constant in vacuum), and *σ* is the electric conductivity of the soil medium. Eq. (2) admits the plane wave solution

$$u(\mathbf{x},t) = ue^{i(k\mathbf{x}\cdot\mathbf{a}t)} \tag{3}$$

*Signal Propagation in Soil Medium: A Two Dimensional Finite Element Procedure DOI: http://dx.doi.org/10.5772/intechopen.99333*

where *ω* ¼ *2πf* is the angular frequency, *t* is the time, and *k* is the complex wavenumber or propagation constant, which can be derived from Eq. (3). The boundary value problem (BVP) used to solve the time-harmonic electromagnetic problem in 2-D, can be expressed in its generic form as

$$-\frac{\partial}{\partial \mathbf{x}} \left( p\_x \frac{\partial u}{\partial t} \right) - \frac{\partial}{\partial \mathbf{y}} \left( p\_y \frac{\partial u}{\partial \mathbf{y}} \right) + qu = f \qquad \text{for } (\mathbf{x}, \mathbf{y}) \in \Omega \tag{4}$$

*u* ¼ *u0* on Γ<sup>D</sup> : Dirichlet Boundary Condition (5)

(7)

$$\left(p\_x \frac{\partial u}{\partial t} \hat{a}\_x + p\_y \frac{\partial u}{\partial y} \hat{a}\_y\right) \cdot \hat{n} = \beta \qquad \text{on } \Gamma\_\mathcal{N} \text{ : Neumann Boundary Condition (6)}$$

$$\left(p\_x \frac{\partial u}{\partial t} \hat{a}\_x + p\_y \frac{\partial u}{\partial y} \hat{a}\_y\right) \cdot \hat{n} + au = \beta \qquad \text{on } \Gamma\_\mathcal{M} \text{ : Mixed Boundary Condition}$$

where *u x*ð Þ , *y* is the unknown function to be determined, and *px*ð Þ *x*, *y py*ð Þ *x*, *y q x*ð Þ , *y f x*ð Þ , *y* are given functions. *u*<sup>0</sup> *α*, and *β* are given functions in boundary conditions (BCs); *Γ<sup>D</sup> Γ<sup>N</sup>* and *Γ<sup>M</sup>* refers to boundaries where Dirichlet, Neumann, and mixed BCs are imposed, respectively; *n*^ ¼ *a*^*xnx* þ *a*^*yny* is the unit vector normal to the boundary in the outward direction.

The weak form is used to provide the finite element solution. The weak form is written as [12–14].

$$\iint\limits\_{\Omega} \left[ p\_x \frac{\partial u}{\partial x} \frac{\partial \boldsymbol{\nu}}{\partial x} + p\_y \frac{\partial u}{\partial y} \frac{\partial \boldsymbol{\nu}}{\partial y} \right] ds - \oint\_I \boldsymbol{\nu} f \cdot \left[ p\_x \frac{\partial u}{\partial y} n\_x + p\_y \frac{\partial u}{\partial y} n\_y \right] dl + \left[ \int\_\Omega q \boldsymbol{\nu} u \, ds - \left[ \int\_\Omega \boldsymbol{\nu} f \, ds = 0 \right] \right] \tag{8}$$

where *ψ*ð Þ *x*, *y* is the weight function.

The weak form is applied in each element domain, and element matrices are formed by expressing the unknown function as a weighted sum of nodal shape functions. The sum of line integrals of two neighboring elements cancels out while combining the element matrices. Therefore, the line integral can be omitted for interior elements and should be considered only for elements adjacent to the boundary. Due to its special form, the line integral makes easier the imposition of mixed types of BCs. The mesh generation will be discussed first, and the shape functions will be given, and then the finite element solution of the time-harmonic problem of electromagnetic scattering in a dielectric medium (soil) will be presented.
