OPERATOR METHOD IN THE PROBLEM OF THE H-POLARIZED WAVE DIFFRACTION BY TWO SEMI-INFINITE GRATINGS PLACED IN THE SAME PLANE

Purpose: Problem of the H-polarized plane wave diffraction by the structure, which consists of two semi-infinite strip gratings, is considered. The gratings are placed in the same plane. The gap between the gratings is arbitrary. The purpose of the paper is to develop the operator method to the structures, which scattered fields have both discrete and continuous spatial spectra. Design/methodology/approach: In the spectral domain, in the domain of the Fourier transform, the scattered field is expressed in terms of the unknown Fourier amplitude. The field reflected by the considered structure is represented as a sum of two fields of currents on the strips of semi-infinite gratings. The operator equations are obtained for the Fourier amplitudes. These equations use the operators of reflection of semi-infinite gratings, which are supposed to be known. The field scattered by a semi-infinite grating can be represented as a sum of plane and cylindrical waves. The reflection operator of a semi-infinite grating has singularities at the points, which correspond to the propagation constants of plane waves. Consequently, the unknown Fourier amplitudes of the fi eld scattered by the considered structure also have singularities. To eliminate these latter, the regularization procedure has been carried out. As a result of this procedure, the operator equations are reduced to the system of integral equations containing the integrals, which should be understood as the Cauchy principal value and Hadamar finite part integrals. The discretization has been carried out. As a result, the system of linear equations is obtained, which is solved with the use of the iterative procedure. Findings: The operator equations with respect to the Fourier amplitudes of the field scattered by the structure, which consists of two semi-infinite gratings, are obtained. The computational investigation of convergence has been made. The near and far scattered fields are investigated for different values of the grating parameters. Conclusions: The effective algorithm to study the fields scattered by the strip grating, which has both discrete and continuous spatial spectra, is proposed. The developed approach can be an effective instrument in solving a series of problems of antennas and microwave electronics. Key words: semi-infinite grating, operator method, singular integral, hypersingular integral, regularization procedure

An inhomogeneous problem for an elastic half-strip: An exact solution

We construct exact solutions of two inhomogeneous boundary value problems in the theory of elasticity for a half-strip with free long sides in the form of series in Papkovich–Fadle eigenfunctions: (a) the half-strip end is free and (b) the half-strip end is firmly clamped. Initially, we construct a solution of the inhomogeneous problem for an infinite strip. Subsequently, the corresponding solutions for a half-strip are added to this solution, whereby the boundary conditions at the end are satisfied. The Papkovich orthogonality relation is used to solve the inhomogeneous problem in a strip.

PARTICULARS OF A WAVE FIELD IN A SEMI-INFINITE WAVEGUIDE WITH MIXED BOUNDARY CONDITIONS AT ITS EDGE

The work is devoted to the analysis of the wave field, which is excited by the reflection of the first normal propagation Rayleigh-Lamb wave from the edge of an elastic semi-infinite strip, part of which is rigidly clamped, and part is free from stresses. The boundary value problem belongs to the class of mixed boundary value problems, the characteristic feature of which is the presence of a local feature of stresses at the point of change of the type of boundary conditions. To solve this boundary value problem, the paper proposes a method of superposition, which allows to take into account the feature of stresses due to the asymptotic properties of the unknown coefficients. Asymptotic dependences for coefficients are determined by the nature of the feature, which is known from the solution of the static problem. The criterion for the correctness of the obtained results was the control of the accuracy of the law of conservation of energy, the error of which did not exceed 2% of the energy of the incident wave for the entire considered frequency range. The paper evaluates the accuracy of the boundary conditions. It is shown that the boundary conditions are fulfilled with graphical accuracy along the entire end of the semi-infinite strip, except around a special point ($\epsilon$). In this case, along the clamped end of the semi-infinite strip in the vicinity of a special point of stress remain limited. The presence of the region $\epsilon$ and the limited stresses are due to the fact that the calculations took into account the $N$ members of the series that describe the wave field, and starting from the $N+1$ member of the series moved to asymptotic values of unknown coefficients, the number of which was also limited to $2N$. As the value $N$ increased, the accuracy of the boundary conditions increased, the region $\epsilon$ decreased, and the magnitude of the stresses near the singular point increased.