## 2 The Plane \(E\)-Polarized EM Wave Scattering by a Coplanar Strip System
### Formulation of the Scattering Problem  
This subsection deals with reformulation of the plane EM wave scattering problem starting from [PERSON]'s equations in the vector differential equation form. Let a plane linearly polarized EM wave be incident upon a coplanar system, consisting of zero-thickness electrically conducting flat strips with parallel edges. The strips' edges will be oriented to the ordinate axis of a Cartesian frame. The transverse cross-section of that coplanar strip system represents an appropriate number of segments or intervals, depending on type of mathematical law. In mathematics, a segment is a closed interval, but an interval (in short) means open interval. It is important for the construction of a pre-fractal set (where segments are used) or a stage of terrible Cantor function, where intervals are used (see Section 4.1). Those intervals have to be placed in the \(xOz\) plane upon parallel straight lines. In Figure 1, one can see the simplest coplanar system, which consists of two unequal zero-thickness electrically conducting flat strips, illuminated by the plane \(E\)-polarized EM wave. The transverse cross-section of that simplest coplanar system consists of two intervals placed in the \(xOz\) plane upon parallel straight lines \(z=const\).  
Figure 1: System of parallel flat strips, illuminated by the plane \(E\)-polarized EM wave.  
Starting from the first two basic [PERSON]'s equations in the vector differential equation form, we obtain partial differential equation of the second order with respect to single second component of the electric field intensity vector \(E_{y}=v(x,z)\):  
\[(\
abla^{2}+k^{2})\cdot v(x,z)=0 \tag{1}\]  
Here the symbol \(\
abla^{2}\) denotes Laplacian operator and the letter \(k\) denotes the wave number. The magnetic field intensity will be of the form: \(\overset{\rightarrow}{H}=(-v^{\prime}z,0,v^{\prime}x):ik\chi\), where the letter \(i\) denotes the imaginary unit and the variable \(\chi\) is the wave resistance of a medium around the strips. The boundary conditions on the electrically conducting flat strips have the form: \(E_{y}=w\cdot H_{x}\), here \(w\) is the surface strip impedance [2,6]. So, the impedance boundary condition on intervals will be in more simple and convenient form:  
\[ikv+\mu\cdot v^{\prime}z=0 \tag{2}\]  
where the variable \(\mu=w/\chi\) is the dimensionless impedance parameter. Thus, we have the exterior mixed boundary-value problem for the Equation (1). This two-dimensional boundary-value problem has unique solution provided two types of conditions. The first one is the radiation condition at infinity:  
\[\lim_{r\rightarrow\infty}\sqrt{r}(v^{\prime}r-iv)=0, \tag{3}\]  
The second type of conditions deal with conditions at the strip boundary points in the Meixner form [6].  
Thus, the plane \(E\)-polarized EM wave scattering problem in three-dimensional space is correctly simplified to a two-dimensional boundary-value problem. It is a powerful aid to simplify and examine similar exterior boundary-value problems for elliptic partial equations of mathematical physics is a classical integral equations technique [2]. So, this technique modification will be used firstly for reformulation of that plane \(E\)-polarized EM wave scattering boundary-value problem.