Reflection and Transmission of a Plane Wave by an Array of Cavities in the Interface of Two Solids

The reflection and transmission of a plane wave by a distribution of cavities in the interface of two solids of different mechanical properties are investigated. For the calculation of the reflection and transmission coefficients by a distribution of cavities, six auxiliary wave states are used in conjunction with the reciprocal identity. Specific results are presented for scattering by a doubly periodic array of cavities in the interface of solids of different elastic moduli and mass densities. For a typical cell, the boundary integral equations for scattering by a cavity at the interface of two solids are derived on the basis of continuity of displacements and tractions across the interface and by taking advantage of the geometrical periodicity. Solutions to the system of singular integral equations have been obtained by the boundary element method. Numerical results are presented as functions of the frequency for two angles of incidence.

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Boundary integral equations for the scattering of electromagnetic waves by a homogeneous dielectric obstacle

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210500021296 ◽

1993 ◽

Vol 123 (1) ◽

pp. 185-208 ◽

Cited By ~ 41

Author(s):

P. A. Martin ◽

Petri Ola

Keyword(s):

Integral Equation ◽

Integral Equations ◽

Singular Integral ◽

Electromagnetic Waves ◽

Boundary Integral Equations ◽

Surrounding Medium ◽

Singular Integral Equations ◽

Boundary Integral ◽

Tangential Vector ◽

Single Integral

SynopsisTime-harmonic electromagnetic waves are scattered by a homogeneous dielectric obstacle. The corresponding electromagnetic transmission problem is reduced to a single integral equation over S for a single unknown tangential vector field, where S is the interface between the obstacle and the surrounding medium. In fact, several different integral equations are derived and analysed, including two previously-known equations due to E. Marx and J. R. Mautz, and two new singular integral equations. Mautz's equation is shown to be uniquely solvable at all frequencies. A new uniquely solvable singular integral equation is also found. The paper also includes a review of methods using pairs of coupled integral equations over S. It is these methods that are usually used in practice, although single integral equations seem to offer some computational advantages.

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MATHEMATICAL MODEL OF WAVE DIFFRACTION BY THE SYSTEM OF STRIPES WITH DIFFERENT VALUES OF SURFACE IMPEDANCE

Journal of Numerical and Applied Mathematics ◽

10.17721/2706-9699.2021.1.17 ◽

2021 ◽

pp. 131-137

Author(s):

V. D. Dushkin ◽

V. N. Melnik

Keyword(s):

Mathematical Model ◽

Integral Equations ◽

Surface Impedance ◽

Boundary Integral Equations ◽

Logarithmic Singularity ◽

Singular Integral Equations ◽

Initial Boundary ◽

Initial Boundary Value Problem ◽

Parametric Representation ◽

Boundary Integral

A mathematical model of diffraction of E-polarized and H-polarized waves on a finite system of not perfectly conducting tapes is obtained. The value of the surface impedance on the two sides of the stripes is different. The initial boundary value problem for the Helmholtz equation with boundary conditions of the third kind was reduced to a system of boundary integral equations. This system of boundary integral equations consists of singular integral equations of the first kind and integral equations of the second kind with a logarithmic singularity. The method of parametric representation of integral operator was used to perform transformations. The values of the physical characteristics of the process are expressed through the solutions of the obtained systems of integral equations. Numerical solution of these equations is performed using a computational scheme based on the discrete singularities method.

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APPLICATION OF THE METHOD OF BOUNDARY INTEGRAL EQUATIONS FOR NON-STATIONARY PROBLEM OF THERMAL CONDUCTIVITY IN AXIALLY SYMMETRIC DOMAIN

EUREKA Physics and Engineering ◽

10.21303/2461-4262.2017.00285 ◽

2017 ◽

Vol 1 ◽

pp. 61-68

Author(s):

Grigoriy Zrazhevsky ◽

Vera Zrazhevska

Keyword(s):

Thermal Conductivity ◽

Integral Equations ◽

Boundary Integral Equations ◽

Symmetric Domain ◽

Singular Integral Equations ◽

Initial Boundary ◽

Boundary Integral ◽

Fourier Series Expansion ◽

Axially Symmetric ◽

Initial Boundary Problem

The article considers the non-stationary initial-boundary problem of thermal conductivity in axially symmetric domain in Minkowski space, formulated as equivalent boundary integral equation. Using the representation of the solution in the form of a Fourier series expansion, the problem is reformulated as an infinite system of two-dimensional singular integral equations regarding expansion coefficients. The paper presents and investigates the explicit form for fundamental solutions used in the integral representation of the solution in the domain and on the border. The obtained results can be used in the construction of efficient numerical boundary element method for estimation of structures behavior under the influence of intense thermal loads in real-time.

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