## 2 The Plane \(E\)-Polarized EM Wave Scattering by a Coplanar Strip System
### The Integral Equations Technique  
According to classical integral equations technique, the unknown function of the mixed plane wave scattering boundary-value problem will be represented in form of the sum: \(v=v_{0}+v_{s}\). The known function \(v_{0}=\exp(ik\overset{\rightarrow}{q}\cdot\overset{\rightarrow}{r})\), where \(\overset{\rightarrow}{r}=(x,z)\) is the radius vector of the Cartesian coordinate system, the vector \(\overset{\rightarrow}{q}=(q_{1},q_{3})\) points the direction of exciting wave. The first addend presents the initial plane \(E\)-polarized EM wave. The second addend, dealing with scattering process of initial wave with coplanar system, has to be written in standard integral form (see, for example, [8]):  
\[v_{s}(x,z)=-\frac{i}{4}\sum_{n=1}^{N}\int_{-a_{n}}^{a_{n}}[\psi_{n}(x^{\prime} )H_{0}^{(1)}(kR_{n})+\varphi_{n}(x^{\prime})\frac{\partial}{\partial z}H_{0}^{ (1)}(kR_{n})]dx^{\prime} \tag{4}\]  
Here \(N\) is the number of strips, the function \(H_{0}^{(1)}(z)\) is the first kind Hankel function of order zero and the root of its argument \(R_{n}=\sqrt{\left(x_{n0}+x^{\prime}-x\right)^{2}+\left(z_{n0}-z\right)^{2}}\) (see Figure 1). New unknown functions of one variable \(\psi_{n}(x)\) and \(\varphi_{n}(x)\), determined in intervals, are expressed by the following formulas:  
\[ \begin{array}{l}\psi_{n}(x)=\frac{\partial}{\partial z}v(x_{n0}+x,z_{n0}+0) -\frac{\partial}{\partial z}v(x_{n0}+x,z_{n0}-0),\\ \varphi_{n}(x)=v(x_{n0}+x,z_{n0}+0)-v(x_{n0}+x,z_{n0}-0)\end{array}  \tag{5}\]  
Using the boundary condition (2) in the interval \(S_{n}\), it is easy to obtain the relationship between these unknown functions \(\varphi_{n}(x)=i\mu\cdot\psi_{n}(x)/k\). This important relationship will be helpful to simplify integrals under sum in (4) and that formula can be simplified to the form:  
\[v_{s}=-\frac{i}{4}\sum\limits_{n=1}^{N}\int_{-a_{n}}^{a_{n}}\psi_{n}(x)\cdot(1+i \frac{\mu}{k}\frac{\partial}{\partial z})H_{0}^{(1)}(kR_{n})dx \tag{6}\]  
Due to the first kind Hankel function properties, the function \(v_{s}=v_{s}(x,z)\) satisfies both the Equation (1) and the radiation condition at infinity (3) [8]. The main step of integral equations technique's realization will be reducing of the two-dimensional exterior boundary-value problem to a one-dimensional one. For this purpose we ought to use the boundary impedance condition (2) upon the interval \(S_{m}\):  
\[\lim\limits_{z\to z_{m0}}(ik+\mu\frac{\partial}{\partial z})\cdot v_{s}(x_{m},z)=-ik(q_{2}\mu+1)v_{0}(x_{m},z_{m0}). \tag{7}\]  
Substitution into which of the expression (6) yields the system of coupled integral-differential equations:  
\[ \begin{array}{c}\sum\limits_{n=1}^{N}\int_{-a_{n}}^{a_{n}}\psi_{n}(x^{\prime })\lim\limits_{z\to z_{m0}}(k^{2}+\mu^{2}\frac{\partial^{2}}{\partial z^{2}}) H_{0}^{(1)}(k\sqrt{\left(x_{m0}+x^{\prime}-x_{m}\right)^{2}+\left(z_{m0}-z \right)^{2}})dx^{\prime}=\\ =4 ik^{2}(q_{2}\mu+1)\cdot v_{0}(x_{m},z_{m0}),\quad m=1,2,\ldots,N.\end{array}  \tag{8}\]  
To take the limits in kernel functions we ought to use the relationship:  
\[\frac{\partial^{2}}{\partial z^{2}}H_{0}^{(1)}(kR_{n})=-(\frac{\partial^{2}}{ \partial x^{2}}+k^{2})H_{0}^{(1)}(kR_{n}).\]  
It is true for \((x,z)\
otin S_{n}\). That relationship allows separate differential and integral parts of the coupled integral-differential equations' system (8) as well.