www.astesj.com 82

**Singular Integral Equations in Electromagnetic Waves Reflection Modeling **

*A. S. Ilinskiy1 _{, T. N. Galishnikova}*,1*

*1 _{ Faculty of Computational Mathematics and Cybernetics, Lomonosov Moscow State University, 119991, Russia }*

A R T I C L E I N F O A B S T R A C T

*Article history: *

*Received: 01 March, 2017 *
*Accepted: 29 March, 2017 *
*Online: 11 April, 2017 *

*The processes of reflection of three-dimensional electromagnetic waves by locally irregular *
*media interfaces are investigated. The problem under study is mathematically reduced to *
*the solution of a boundary value problem for the Maxwell equations in an infinite space *
*with an irregular boundary. In order to develop a numerical algorithm, the potential theory *
*and a special Green’s function are applied to reduce the addressed boundary problem to *
*an equivalent system of two hypersingular integral equations. This system is solved with *
*the use of the approximation and collocation method. Special attention is focused on *
*calculation of the kernels of these equations. Results of simulation of the currents induced *
*on the irregularity and reflected field patterns in the resonance frequency range are *
*presented. *

*Keywords: *

*Reflection of electromagnetic *
*waves *

*Inhomogeneous medium *
*Singular integral equations*

**1.** **Introduction**

This paper is an extension of the work originally presented at the URSI Commission B International Symposium on Electromagnetic Theory (EMTS 2016) [1]. In this article we model the scattering of three-dimensional electromagnetic waves on a finite impedance section of a wavy surface separating two media. The scattering of electromagnetic waves by a wavy interface between two different media is an important problem for various applications. In this article we investigate the incidence of an arbitrary plane wave on a local inhomogeneity of the interface assuming sufficiently high conductivity of the underlying medium. The mathematical formulation of such problems reduces to that of solving a system of Maxwell equations in non-regular infinite regions. Depending on the polarization of the incident field, the boundary-value problems are reduced to independent systems of hypersingular integral equations solved by specially developed numerical algorithms utilizing a singularity isolating algorithm.

**2.** **Mathematical model and numerical algorithm**

*2.1.Mathematical formulation of wave scattering problems *

Let us assume that an interface between two media is situated
in Cartesian coordinates *x, y, z.* The interface consists of two
half planes

###

that are specified by the equation*y*0 and joined by an impedance cylindrical wavy section

*S*that stretches along the

*z*axis. The entire wavy section is located in the half-space

. 0

*y* The surface S is specified by the equation *y=f(x)*,

,

0*x**a* *f*(0) *f*(*a*)0, where *a*is the *x* length of irregularity
*S*. Let

*a*

###

### 2

###

### .

The reflecting surface*S*

###

###

does not depend on*z*coordinate. Denote by

*D*

_{1}the region over interface

*S*

###

###

where the 3D incident electromagnetic field propagates. This
region *D*_{1} is characterized by permittivity _{1}, permeability _{1},
and the *k*_{1} wave number 1 1

2 2

1

*k* , where

###

is the circular frequency (Figure 1). The time dependence is exp###

*i*

*t*

###

.Figure 1:Geometry of perfecting surface

### ASTESJ

ISSN: 2415-6698

*_{Corresponding Author: T. N. Galishnikova, Faculty CMC,}_{Lomonosov}_{Moscow }

State University, 119991, Russia.* *Email: tgalish@cs.msu.su

*www.astesj.com *

*Special Issue on Recent Advances in Engineering Systems *

*A. S. Ilinskiyet. al. / Advances in Science, Technology and Engineering Systems Journal Vol. 2, No. 3, 82-87 (2017) *

The field of the plane wave

),
exp(
)
,
,
(
),
exp(
)
,
,
(
0
0
0
0
0
0
0
0
0
0
*z*
*i*
*y*
*i*
*x*
*i*
*H*
*z*
*y*
*x*
*H*
*z*
*i*
*y*
*i*
*x*
*i*
*E*
*z*
*y*
*x*
*E*
0
0
1

0 sin

###

sin###

###

*k*

###

_{0}

*k*

_{1}cos

###

_{0}sin

###

_{0}, and

_{0}

*k*

_{1}cos

_{0}

is incident on the surface

*S*

###

###

in region*D*1. Here,

###

0 is theangle between axis *z *and the direction of the incident wave
propagation. When

###

_{0}

###

/2, we obtain a plane reflection two-dimensional problem. If _{0}0, we have the case of the normal incidence in the plane problem.

In region

*D*

_{1}we seek for a solution to the homogeneous system of Maxwell equations

,
0
)
,
,
(
)
,
,
(
,
0
)
,
,
(
)
,
,
(
1
1
*z*
*y*
*x*
*H*
*i*
*z*
*y*
*x*
*E*
*curl*
*z*
*y*
*x*
*E*
*i*
*z*
*y*
*x*
*H*
*curl*

which satisfies on the plane section the condition of perfect conduction

###

*n*

*E*(

*x*,

*y*,

*z*)

###

0,Leontovich condition on local section S with a finite conductance

###

*n*

*E*(

*x*,

*y*,

*z*)

###

*W*

_{2}

###

*n*

###

*n*

*H*(

*x*,

*y*,

*z*)

###

###

, ./ _{2}

2

2

*W* Here, _{2} and _{2} are the characteristics of the
section *S*, *n*

###

*n*,

_{x}*n*,0

_{y}###

is the outward normal to region*D*

_{1}. The reflected field satisfies the radiation conditions at infinity.

We look for the total electromagnetic field in *D*1 that has the

same dependence on *z* as the incident field, that is

###

*x*,

*y*,

*z*

###

*E*

*x*,

*y*exp

###

*i*

_{0}

*z*

###

,*E* *H*

###

*x*,

*y*,

*z*

###

*H*

###

*x*,

*y*exp

###

*i*

_{0}

*z*

###

. Then, the boundary condition on*S*can be represented as follows:

###

###

###

###

_{}

_{}

###

###

*y*

*x*

*E*

*i*

*n*

*y*

*x*

*H*

*i*

*k*

*W*

*y*

*x*

*H*

*n*

*y*

*x*

*E*

*i*

*y*

*x*

*H*

*i*

*k*

*W*

*y*

*x*

*E*

*z*

*z*

*z*

*z*

*z*

*z*, , 1 , , , , 0 1 2 0 2 1 2 1 0 2 0 2 1 2

### ,

, / //*n**n _{x}*

*x*

*n*

_{y}*y*

/

###

*n*/

_{x}*y*

*n*/

_{y}*x*are

the normal and tangent derivatives, respectively. Accordingly, boundary conditions on plane section

###

take the form###

, 0,###

, 0.
*n*
*y*
*x*
*H*
*y*
*x*

*Ez* *z*

We denote *u*

###

*x*,

*y*

*Ez*

###

*x*,

*y*,

*v*

###

*x*

### ,

*y*

###

*H*

*z*

###

*x*

### ,

*y*

### ,

. 2 0 2 1 2 1 *k* Then the considered boundary value problem can

be reduced to solving in region *D*1 of the 2D homogeneous

Helmholtz wave equation

0
)
,
(
)
,
(
2
1
2
2
2
2
*y*
*x*
*v*
*y*
*x*
*u*
*y*
*x*

with appropriate boundary conditions on

###

and*S*

.
*2.2.Reduction of the boundary problem for Helmholtz equations *
*to the system of singular equations in case of E-polarization *

In order to obtain a system of hypersingular integral equations
equivalent to the boundary value problem for Helmholtz equation,
we introduce Green’s function *gE*

###

*M*,

*P*

###

, satisfying in region1

*D* the inhomogeneous Helmholtz equation

###

,###

2###

,###

,2
1
2
2
2
2
*P*
*M*
*P*
*M*
*g*
*y*
*x*

*E* _{}_{}

_{}

the boundary condition for perfect conductor on half-plane

###

###

*M*,

*P*

###

0,*gE*

that have the form

###

###

###

###

###

ˆ###

.2 ) ,

( _{0}(1) _{1} _{M}_{,}_{P}_{0}(1) _{1} _{M}_{,}_{P}_{ˆ}
*E*
*r*
*H*
*r*
*H*
*i*
*P*
*M*

*g*

Here, *H*_{0}(1)(*x*) is the zero-order Hankel function of the first kind;

###

###

2###

2,,*P* *M* *P* *M* *P*

*M* *x* *x* *y* *y*

*r* where

###

*xM*,

*P*,

*yM*,

*P*

###

arecoordinates of *M* and *P* points;

*P*
*M*

*r*ˆ _{,}ˆ

###

*x*

_{M}*x*

_{P}###

2 *y*

_{M}*y*

_{P}###

2,

###

*x*

*P*

### ,

###

*y*

*P*

###

are the coordinatesof the *P*ˆ point.

Applying in region

*D*

_{1}the Green’s formula successively to functions

*u*

###

*M*,

*gE*

###

*M*,

*P*

###

, taking into account the radiation conditions and the properties of surface potentials, we obtain a system of two integral equations for unknown functions###

*P*

*n*

*u*

/ and *v*

###

*P*/ on finite section

*S*for

*E-*polarization case:

*M*

*M*

*M*

*v*

*W*

*i*

*n*

*M*

*u*

*W*

*W*

*ik*

###

###

###

###

) ( 2 ) (2 _{1}2

2 0 2 1 1 2 1

###

_{P}*S*

*P*

*E*

*E*

*n*

*P*

*u*

*n*

*P*

*M*

*g*

*W*

*W*

*ik*

*P*

*M*

*g* ( , ) ( , ) ( )

2 1 2 1 1 2 1

###

###

### (1)

), ( ) ( ) , ( 0 2 2

0 *v* *P* _{ds}_{u}_{M}

*A. S. Ilinskiyet. al. / Advances in Science, Technology and Engineering Systems Journal Vol. 2, No. 3, 82-87 (2017) *

###

_{}

###

_{P}*S* *M* *P*

*E*

*M*
*E*

*M* *n*

*P*
*u*
*n*
*n*

*P*
*M*
*g*
*W*

*W*
*ik*
*n*

*P*
*M*
*g*
*n*

*M*

*u* ( , ) ( , ) ( )

2 1 2

1 2

2 1 1

2 1

###

###

, , ) ( )

( ) ,

( 0

2

2 1 2

0 _{M}_{S}

*n*
*M*
*u*
*ds*
*P*
*v*
*n*
*n*

*P*
*M*
*g*
*W*
*i*

*M*
*E*

*P*
*P*
*P*
*M*
*E*

### (2)

where *u*_{0}*E*(*x*,*y*)exp

###

*i*

_{0}

*x*

*i*

_{0}

*y*

###

exp###

*i*

_{0}

*x*

*i*

_{0}

*y*

###

. In the case of the*H-*polarization we can obtain a system of two integral equations for unknown functions

*v*

###

*P*and

*u*

###

*P*/ on finite section

*S.*

*2.3.The numerical algorithm for the solution of singular *
*integral equations for the E-polarization *

The systems of integral equations are singular and can be solved through reducing them to the complex system of linear algebraic equation by means of the collocation and approximation method.

To this end, we divide the interval

###

0,*a*, on which the equation

*y*

*f*(

*x*) of the irregular cylindrical surface

*S*

is
defined, by points *x*

_{i}*ia*/

*N*,

*x*

_{0}0,

*x*

_{N}*a*, into

*N*

equal
parts. The points dividing the curve *S*

have the coordinates
###

*x*,

_{i}*f*(

*x*)

_{i}###

. Then we replace the integrals entering into the systems of equations (1), (2) with the sums of the integrals overthe subintervals and set ,

2
1
*Mi*

*M* *i*1,...,*N*,
2
1
*i*

*M* are

collocation points. The sought solution on the section

### ,

1

, *j*

*jP*

*P*

*S*

, 1 ,...,

0

*N*

*j* is approximated by a zero-order spline. As a
result, we obtain a complex system of linear algebraic equations.
When calculating the integrals over a section

### ,

1

, *j*

*jP*

*P*

*S*

in which
the integrand depends on either *gE*

###

*M*,

*P*

###

or###

,###

,*P*

*E*

*n*
*P*
*M*
*g*

we

replace the integration with respect to

*s*

with the integration with
respect to *x*

by setting *ds* 1

_{P}###

*f*

###

*x*

###

2*dx*.

_{P}Note that the terms that enter the kernels of integral equations
and depend on *gE*(*M*,*P*) have a logarithmic singularity when the
arguments coincide. Therefore, when developing the numerical
algorithm, this singularity is separated in an explicit form and the
integrals containing it are calculated analytically.

The normal derivatives of the function *gE*(*M*,*P*) have no
singularity when the arguments are equal, and

), ( 2 1 ) , (

lim 0 *M*

*n*
*P*
*M*
*g*

*p*
*E*

*M*
*P*

where

###

_{0}(

*M*)is the curvature of the contour

*S*

at the point *M*

.
The kernels of the system depending on

###

###

*P*

*M*

*E*

*n*
*n*

*P*
*M*
*g*

2 ,

have a strong singularity. In order to calculate the matrix elements that are integrals of the functions with a hypersingularity, it is necessary to develop a numerical algorithm. To this end, we preserve the integration over the arc

*s*

in these integrals. We defined the
surface *S*

in the form *x*

*x*

###

*s*,

*y*

*y*

###

*s*. The outward normal to the domain

*D*

_{1}we write in the form

*n*

*y*

_{s}*i*

*x*

_{s}*j*, where

*i*

###

and

###

*j*

are the *x*

and*y*

unit vectors. Note that if the boundary of
*S*is specified in the form

*y*

*f*

###

*x*, the relationship between the directional cosines

*n*

_{x}### ;

*n*

*of the outer normal*

_{y}*n* to

*D*

_{1}is given by

###

###

###

21 *f* *x*

*n*

*y _{S}*

* ,*

_{x}*x*

_{S}*n*1

_{y}###

*f*

###

*x*

###

2 ,###

###

###

###

21

/ *f* *x*

*x*
*f*

*n _{x}* ,

*n*1/ 1

_{y}###

*f*

###

*x*

###

2 .We have

###

###

###

1 ,

,

2 / 1 2

*j*
*P*
*j*
*P*

*S*

*P*
*i*

*E*

*P*
*M*

*ds*
*P*
*M*
*g*
*n*
*n*

(3)

###

###

###

1 ,

2 / 1 ,

1 ) 1 ( 0 2

2
*j*
*P*
*j*
*P*

*i*

*S*

*P*
*P*
*M*
*P*

*M*

*ds*
*r*

*H*
*n*
*n*

*i* _{}

###

###

###

1 ,

2 /

1 .

ˆ

2 1 ,

) 1 ( 0 2

*j*
*P*
*j*
*P*

*i*

*S*

*P*
*P*
*M*
*P*

*M*

*ds*
*r*

*H*
*n*
*n*

*i* _{}

To develop a special computing algorithm, every integrands is represented by a sum of two terms: the first one is the total differential of some function has been calculated analytically; the function in the second term has a logarithmic singularity and is integrated by separating this singularity in the explicit form, thereafter the corresponding integrals are calculated analytically.

In the first term /*nM* let us consider the component /*xM*
(without the multiplier

*i*

###

### /

### 2

): _{}

###

*M*

*P*

*P*

*S* *M* *P*

*ds*
*r*

*H*
*n*

*x* *i*

*j*
*P*
*j*
*P*

, 1 1 0 2

2 1

1 ,

_{}

###

*M*

*P*

*P*

*P*

*S* *P*

*ds*
*r*

*H*
*x*

*n* *i*

*j*
*P*
*j*
*P*

, 1 1 0

2 1

1 ,

###

_{}

1 ,

2 1,

1 1 0

*j*
*P*
*j*
*P*

*i*

*S*

*P*
*M*
*P*

*r*
*H*
*y*

*A. S. Ilinskiyet. al. / Advances in Science, Technology and Engineering Systems Journal Vol. 2, No. 3, 82-87 (2017) *
_{} _{} _{}

###

1 , 2 1 , 1 1 0 2 1*j*

*P*

*j*

*P*

*P*

*i*

*S*

*P*

*S*

*P*

*M* *y* *ds*

*r*
*H*

###

###

###

###

###

*P*

*S*

*S*

*ds*

*y*

*P*

*M*

*i*

*j*

*P*

*j*

*P*

*P*1 , , 4

### (4)

_{}

_{}

*j* *i* *j*

*i* *M* *P*

*P*
*P*
*M*
*P*
*r*
*H*
*y*
*r*
*H*

*y* 1 ,

1
0
,
1
1
0
2
1
1
2
1
_{} _{} _{}

###

*P*

*S*

*S*

*P*

*M* *y* *ds*

*r*
*H*
*j*
*P*
*j*
*P*
*P*
*i*
1
,
2
1,
1
1
0
2
1

###

,###

. 4 1 ,*P*

*S*

*S*

*ds*

*y*

*P*

*M*

*i*

*j*

*P*

*j*

*P*

*P*

###

The points

*M*

and *P*

can coincide, hence, there is an integral of the
Dirac delta-function.
Similarly let us consider the component/*y _{M}* in the first
term /

*n*:

_{M} _{} _{}

###

*M*

*P*

*P*

*S* *M* *P*

*ds*
*r*
*H*
*n*
*y* *i*
*j*
*P*
*j*
*P*
,
1
1
0
2
2
1
1
,
_{}

###

*M*

*P*

*P*

*P*
*S* *P*
*ds*
*r*
*H*
*y*
*n* *i*
*j*
*P*
*j*
*P*
,
1
1
0
2
1
1
,

###

_{} 1 , 2 1, 1 1 0

*j*

*P*

*j*

*P*

*i*

*S*

*P*

*M*

*P*

*r*

*H*

*x*

*d*

_{}

_{}

_{}

###

*P*

*S*

*S*

*P*

*M* *x* *ds*

*r*
*H*
*j*
*P*
*j*
*P*
*P*
*i*
1
,
2
1 ,
1
1
0
2
1

###

###

###

###

###

1 , , 4*j*

*P*

*j*

*P*

*P*

*S*

*P*

*S*

*ds*

*x*

*P*

*M*

*i* (5)

_{}

_{}

*j* *i* *j*

*i* *M* *P*

*P*
*P*
*M*
*P*
*r*
*H*
*x*
*r*
*H*

*x* 1 ,

1
0
,
1
1
0
2
1
1
2
1
_{} _{}

###

*P*

*S*

*S*

*P*

*M* *x* *ds*

*r*
*H*
*j*
*P*
*j*
*P*
*P*
*i*
1
,
2
1 ,
1
1
0
2
1

###

,###

.4*i*

###

*M*

*P*

*x*

*ds*

Using (4)-(5) and the recurrence relations for Hankel functions

###

###

###

###

; , , 1 1 1 1 , 1 1 0*P*

*M*

*P*

*M*

*P*

*M*

*P*

*M*

*P*

*r*

*x*

*x*

*r*

*H*

*r*

*H*

*x*

_{}

_{}

_{}

###

###

###

###

, , , 1 1 1 1 , 1 1 0*P*

*M*

*P*

*M*

*P*

*M*

*P*

*M*

*P*

*r*

*y*

*y*

*r*

*H*

*r*

*H*

*y*

_{}

_{}

_{}

we obtain the final formula for the calculation of the second normal derivative of the Green function [2].

*2.4.Calculation of the reflected field pattern for the *
*E-polarization *

Using the solution to the system of integral equations (1)-( 2) and taking into account the boundary conditions on the section

*S*

, we can obtain the integral
representation for the field *u*

_{1}

###

*M*

*u*

###

*M*

*u*

_{0}

*E*

###

*M*

**:**

###

_{}

###

###

###

###

###

*S* *P* *P*

*E*
*E*
*n*
*P*
*u*
*n*
*P*
*M*
*g*
*W*
*W*
*k*
*i*
*P*
*M*
*g*
*M*

*u* , ,

2 1 1 2 1 2 1

1 _{} _{}

###

###

1 2

1 2

0 , *vP* _{d}_{s}_{,} _{M}_{D}

*n*
*P*
*M*
*g*
*W*
*i*
*P*
*P*
*P*
*E*

_{. }

(6)
We introduce spherical coordinate frame *r*,,, where angle

###

/2 /2###

is measured from the *y* axis and angle

###

0 /2###

is measured from the *z* axis. We seek for a
representation of the scattered field for *r** _{M}*

*r*

*,*

_{P}*k*

_{1}

*r*

*1. Following that, we introduce the notation*

_{M}*r*

* . It can be shown that the following asymptotic relationships*

_{M}###

*P*

*M*

###

*MP* *r* *e*

*r* , and

##

_{M}*P*
*P*

*M* *r* *e*

*r*

, ˆ

ˆ ˆ

ˆ

###

are valid. Here,*e*

###

*is*

_{M}a unit vector oriented in the direction to point *M*; i.e.

* _{M}* /

*M*

*r*

*e*

###

sinsin,cossin,cos###

. Taking into account the above relationships and asymptotic behavior of the Hankel function, we can obtain the asymptotic representations for)
,
(*M* *P*

*gE* ,

*P*
*E*
*n*
*P*
*M*
*g*
( , )

in the form

###

###

_{}

_{} 4 exp 2 ,

_{1}1

*i*

*P*

*M*

*gE*

###

###

###

###

##

##

##

##

##

*i*

*rP*

*eM*

*i*

*rP*

*eM*

##

, ˆ exp ,

exp

###

1 ###

1 ˆ### ;

###

###

_{}

_{} 4 exp 2 , 1 1

*i*

*n*

*P*

*M*

*g*

*P*

*E*

###

###

###

###

###

###

###

*i*

*nP*

*eM*

*i*

*rP*

*eM* , exp ,

_{1}1

##

*n*

_{P}*eM*

##

###

*i*

###

*r*

_{P}*eM*

###

###

##

*i*
,
ˆ
exp

, _{1} _{ˆ}

ˆ

1

###

We call quantity *FE*

###

, that equals###

, *E*

*F*

###

###

###

###

###

###

###

*S*

*M*
*P* *e*

*r*

*i* ,

exp 1

1 1

##

*i*

##

*rP*

*eM*

##

##

##

, ˆ

exp

###

1 ˆ###

###

###

###

###

###

###

*n*

_{P}*e*

_{M}*i*

*r*

_{P}*e*

_{M}*W*

*W*

*k*

, exp

, 1

1 1

2

1 _{}

###

###

###

###

###

###

###

###

*P*
*M*

*P*
*M*

*P* _{n}

*P*
*u*
*e*

*r*
*i*
*e*

*n*ˆ, exp 1 ˆˆ, ( )

_{}

###

###

###

###

###

###

###

*nP*

*eM*

*i*

*rP*

*eM*

*W*

, exp

, 1

1 2

0 _{}

###

###

###

###

###

###

###

*P*

*P*
*M*
*P*
*M*

*P* *ds*

*P*
*v*
*e*
*r*
*i*
*e*

*n* _{}

ˆ , ( )

exp

, 1 ˆ

ˆ

the pattern of scattering field *u*_{1}

###

*M*. Knowing the solution of the system of singular integral equations (1)-(2), we calculate the currents induced on the irregular surface and far-field pattern for the scattered field in the far zone [4].

**3.** **Numerical results **

The induced currents and reflected electromagnetic field
patterns are calculated for *E-* polarization**,**_{2} 504*i***, **contour

*S*

specified in the *XOY*

plane by the equation *f*

###

*x*1cos

*x*,

###

### 2

### 0

###

*x*

###

,*a*

###

### 2

###

.Figure 2 shows the distribution of the calculated function of the induced current

*I*

### (

*x*

### )

###

###

*u*

###

*P*

### /

###

*n*

on the boundary *S*

along the *x* axis for *ak*_{1}3 , *a*

_{}

1.5, where ###

is the wavelength of the incident field. The results are obtained for various angles###

_{0}and

###

_{0}of the electromagnetic wave incidence:

### (

curve*1*

### )

###

_{0}

###

### 0

### ,

00

###

### 85

###

### ; (

curve*2*

### )

_{0}30

### ,

00 60

### ;

### (

curve*3*

### )

0 45### ,

0 0 45

### ; (

curve*4*

### )

0 60### ,

0 0 30

### .

Figures 3-4 show for *ak*_{1} 3 , *a*

_{}

1.5 reflected field
pattern

*F*

*E*

###

###

### ,

###

in polar coordinates for the range

### 85

### 85

###

###

###

###

and fixed values of###

. Curves*1-5*

correspond to 85,60,45,30,5. Figures 3-4 represent the results of calculations for the reflected field when a plane wave incident at the angles 0 0,0 85 (Figure 3) and

_{,} _{45}

45 _{0}

0

(Figure 4).

Figure 2

Figure 3

Figure 4

Figure 5

Figure 7

**4.** **Conclusion **

1. The exact model of plane electromagneticwave incident on inhomogeneous conductive surface is investigated. 2. The problem is reduced to the system of singular integral

equations. The analytic solution is provided for a singularity in the kernels of integral equations.

3. The numerical algorithm for calculation of scattering characteristics of incident electromagnetic field was

**5.** ** References **

[1] A. S. Ilinskiy, T. N. Galishnikova, “Singular Integral Equations in the Wave Scattering Problems”. Proc. Of URSI Commission B International Symposium on Electromagnetic Theory (EMTS 2016), 14-18 August, 2016, Espoo, Finland. Pp. 635-637 (electronic publication).

[2] A. S. Il’inskii and T. N. Galishnikova, “The Integral Equation Method in Problems of Diffraction by a Finite Impedance Section of a Medium Interface”, Moscow University Computational Mathematics and Cybernetics (Allerton Press), vol. 32, no. 4, pp.187-193, 2008.

[3] A. S. Il’inskii and T. N. Galishnikova, “Investigation of Diffraction of an Electromagnetic Wave Arbitrarily Incident on a Locally Inhomogeneous Medium Interface”, Journal of Communications Technology and Electronics, vol. 58, no. 1, pp. 40-47, 2013.

[4] A. S. Il’inskii and T. N. Galishnikova, “Integral Equation Method in Problems of Electromagnetic-Wave Reflection from Inhomogeneous Interfaces between Media”, Journal of Communications Technology and Electronics, vol. 61, no. 9, pp. 981-994, 2016.