Scattering of a plane wave on a wavy surface

Numerical methods are proposed for solving problems of diffraction of a plane electromagnetic wave on a nonhomogeneous interface of two media. Strictly periodic cylindrical surfaces are considered, as well as surfaces that are nonplanar on a finite section. Algorithms are developed for computing the characteristics of the reflected field in the far zone. Numerical results are reported for various electrodynamic models, and their limits of applicability are investigated.Translated from Matematicheskie Modeli Estestvoznaniya, Published by Moscow University, Moscow, 1995, pp. 86–111.     

Scattering of a plane acoustic wave on a periodic array of piston radiators embedded into a rigid screen

The stationary scattering of a plane acoustic wave on a periodic or a doubly periodic set of piston radiators elastically connected with a rigid screen is considered. The screen separates two half-spaces filled with media possessing different acoustic properties. The model considered takes into account energy losses in the elastic ties between the piston radiators and rigid screen. The problem solution in the case where only one half-space is filled with an acoustic medium is found as a special case. The results of numerically analyzing the reflection coefficients of a plane wave as functions either of the frequency of the incident wave or of the angle of incidence of the plane acoustic wave for different values of the parameters involved are presented. Bibliography: 4 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 230, 1995, pp. 125–137. Translated by G. L. Nikitin.     

Scattering of the plane wave by a transparent wedge

In the paper it is proved that the problem of scattering of the plane wave by a transparent wedge has a unique solution, provided that the radiation condition should be meant in the following form: if one subtracts from the solution the incident wave and all reflected and refracted waves, then the remainder satisfies the radiation condition in integral form. The problem is scalar, the velocities of the wave inside and outside the wedge are not equal, the wave process is described by the classical Helmholtz equations, and the conjugation boundary condition is satisfied on the sides of the wedge. Bibliography: 8 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 354, 2008, pp. 5–18.     

DN‐Scattering of a plane wave by wedges

We continue our study of a nonstationary scattering by wedges. In this paper we consider nonstationary scattering of plane waves by a ‘hard–soft’ wedge. We prove the uniqueness and existence of a solution to the corresponding DN‐Cauchy problem in appropriate functional spaces. We also give the explicit form of the solution and prove the Limiting Amplitude Principle. Copyright © 2011 John Wiley & Sons, Ltd.     

Wood's anomalies in the scattering problem of a plane wave on a smooth periodic boundary

A series of multiple diffracted fields in a high-frequency asymptotic expansion of the solution to the problem of grazing scattering of a plane wave by a smooth periodic boundary is summed in the case of appearance of Wood's anomalies. Bibliography: 8 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 186, pp. 87–100, 1990. Translated by V. V. Zalipaev.     

Scattering of a plane wave by shallow buried cylindrical lining in a poroelastic half-space

The shallow buried tunnel is frequently encountered in underground engineering. The dynamic response of a tunnel under incident wave is of great importance for guiding the safety design in tunnel engineering. In this paper, a model for predicting the dynamic response of a shallow buried tunnel in saturated soil is proposed based on nonlocal Biot theory. The analytical solution is obtained using the wave function expansion method. To consider practical engineering problem, a set of material parameters for saturated soil and tunnel lining are selected for the numerical analysis. The influence of nonlocal parameter, which is introduced in nonlocal Biot theory to consider the pore size effect and pore dynamic effect, on dynamic stress concentrate factor in the lining is investigated in detail. The dynamic responses affected by the other factors, such as incident wave angle, frequency of incident wave and buried depth of the tunnel, have also been implemented. The dynamic stress concentrate factor distributed in the lining is also shown and the position and orientation appearing maximum concentrate factor can be easily determined from the contour plot, which can provide a visual guideline for safety design of a tunnel.     

Numerical analysis of the diffraction of a plane wave on a periodic grating of loaded strip elements

Translated from Aktual'nye Voprosy Prikladnoi Matematiki, pp. 138–144, 1989.     

The attractor problem for nonlinear wave equations in plane domains

We consider a model example of a quasilinear wave equation in the unit square with zero boundary conditions and use the method of quasinormal forms to prove that there are quite a few dichotomic cycles and tori bifurcating from zero equilibrium. A conjecture concerning the attractor structure is presented. Translated fromMatematicheskie Zametki, Vol. 68, No. 2, pp. 217–229, August, 2000.     

Canonical frame of a curve on a conformal plane

We show how differential geometry of smooth curves on the conformal plane can be studied by Élie Cartan’s method of exterior forms and moving frames. We find the canonical form of the derivation equations of a curve (which is not a circle) in the case of a semi-isotropic frame. We give a new proof of the theorem that states that curves of constant (in particular, zero) conformal curvature are loxodromes. We integrate the system of structure equations of the isotropy subgroup of a point.     

Short-wavelength grazing scattering of a plane wave by a smooth periodic boundary. II. Diffraction by an infinite periodic boundary

The short-wavelength asymptotic behavior of the field near a reflecting boundary (the Fock zone and the neighborhood of the limit ray) is constructed for the problem of the diffraction of a plane wave by a smooth periodic boundary.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Seklova AN SSSR, Vol. 173, pp. 60–86, 1990.     

An identification problem for a time periodic nonlinear wave equation in non cylindrical domains

The existence of a time-periodic solution of a free boundary nonlinear wave equation in non cylindrical domains is established. The problem arises in the study of the identification of the coefficient of the wave equation and of the boundary of the region from the observed values of the solution in a fixed subregion.     

一种保持周长的平面曲线流

The purpose of this paper is to investigate a new type of evolution problem for closedconvex plane curves which will preserves the perimeter of the curve but expands the enclosedarea and the final limiting curve is a circle in the Hausdorff metric in the plane.     

Scattering on a line. I. Existence of wave operators and asymptotic completeness in the periodic case

Existence of wave operators for scattering on a line in R³ is proved. It is shown that the completeness of have operators is violated in the case of a line with periodic boundary conditions. However, the asymptotic completeness and unitary property of wave operators are preserved. Bibliography:11 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 186, pp. 136–141, 1990. Translated by Ya. V. Kurylev.     

Moduli of plane curve singularities with a single characteristic exponent

This paper studies the moduli space corresponding to irreducible germs of plane analytic curve with a single characteristic exponent. We stratify the moduli space corresponding to such germs using an analytical invariant introduced by Zariski. Then, we compute the minimum Tjurina number on each stratum as well as the dimension of the strata.