Scalar problem of plane wave diffraction by a system of nonintersecting screens and inhomogeneous bodies

The scalar problem of plane wave diffraction by a system of bodies and infinitely thin screens is considered in a quasi-classical formulation. The solution is sought in the classical sense but is defined not in the entire space ?³ but rather everywhere except for the screen edges. The original boundary value problem for the Helmholtz equation is reduced to a system of weakly singular integral equations in the regions occupied by the bodies and on the screen surfaces. The equivalence of the integral and differential formulations is proven, and the solvability of the system in the Sobolev spaces is established. The integral equations are approximately solved by the Bubnov-Galerkin method. The convergence of the method is proved, its software implementation is described, and numerical results are presented.     

Propagation of a disturbance in a plane neural network

Translated fromIssledovaniya po Prikladnoi Matematike, No. 18, 1992, pp. 111–124.     

Hydromagnetic flow over a plane wall with uniform suction

Zusammenfassung Diese Abhandlung behandelt die Strömung einer inkompressiblen, zähen und elektrisch leitenden Flüssigkeit über eine ebene Wand mit homogener Absaugung und senkrechtem Magnetfeld. Eine exakte Lösung der hydromagnetischen Gleichungen kann erhalten werden, wenn die magnetische Druckzahl kleiner als eins ist.Aus den Rechnungen findet man, dass sowohl die Reibungsschubspannung als auch die Wärmeübertragung an der Wand unabhängig sind von der hydromagnetischen Wirkung, aber dass die Grenzschichtdicke zunimmt mit der Zunahme des hydromagnetischen Einflusses.Es sei bemerkt, dass keine stationäre Lösung erhalten wird für den Fall des Ausblasens.     

Effects of plane strain on inhomogeneous turbulence

The influence of the mean plane strain on the turbulence transportation is investigated by large eddy simulation (LES) in the shearless turbulence mixing layer. It is found that the mean strains enhance the turbulent fluctuations in the mixing region. Compression in the inhomogeneous direction can greatly increase the transport of turbulent kinetic energy by triple correlation terms, while stretching in the inhomogeneous direction decreases the turbulence transportation. The gradient diffusion models for turbulent transportation are evaluated and it is found that the intermittency consideration can improve the prediction ability of the gradient-type models for the triple correlation terms. Project supported by the Sino-French Laboratory in Beijing and the National Natural Science Foundation of China (Grant No. 19572041).     

Diffraction of a plane wave by a step discontinuity in an impedance plane

In the present paper, the high-frequency diffraction of a plane wave by a right-angle step discontinuity in an impedance plane is analyzed with the help of the uniform geometric theory of diffraction, beginning with the Maluzhenets solution. The principal term of an asymptotic solution of the problem, which is uniform with respect to the angle of incidence of the plane wave and the angle of observation, is derived. The excitation of primary and multiply diffracted fields radiated from the upper edge, as well as surface waves, is considered (the lower edge does not radiate cylindrical or surface waves owing to a right-angle step). For simplicity, the details of computation are given here for a right-angle step discontinuity. A similar procedure is applied to other examples with more complicated geometry. Bibliography: 7 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 250, 1998, pp. 97–108. Translated by V. V. Zalipaev.     

Diffraction of a plane wave by a narrow cone

The problem of diffraction of a plane scalar wave by a narrow cone is considered. The shape of the cone is arbitrary. The boundary condition is the Dirichlet or Neumann one. The wave scattered by the cone vertex arises as a result of the diffraction process. The subject of this paper is to calculate the wave amplitude. If the cone is narrow, it is possible to obtain simpler approximate formulas in comparison with Smyshlayev's one. The exactness of the approximate formulas is checked numerically. The etalon is a solution in explicit form in the axially symmetric case. The calculation shows that our formula is more exact in the case of the Dirichlet boundary condition than Felsen's formula. The approximate formula is a generalization of Felsen's one for circular cone to an arbitrary narrow cone in the case of the Neumann boundary condition. Bibliography: 6 titles. Dedicated to N. N. Uraltseva on her jubilee Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 221, 1995, pp. 67–74. Translated by D. B. Dement'ev.     

Diffraction of a plane wave by a circular disk

In this paper we solve the problem of diffraction of a normally incident plane wave by a circular disk. We treat both the hard and soft disk. In each case we obtain the solution as a series which converges when the product of the wave number and the radius of the disk is large. Our construction leads directly to asymptotic approximations to the solution for large wave number.     

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.     

Some problems of non uniform geometric probability relative to plane lattices

Dans ce travail on déterminera la probabilité pour qu’un cercle aléatoire coupe un réseau de points ou de droites ou de rectangles dans le plan euclidien si les coordonnées du centre du cercle suivent la loi de probabilité de densité la fonction

---

Work supported by M. P. I. (40%–60%)     

Boundary integral equations for a body with inhomogeneous inclusions

On the basis of the partially singular differential equations of the stationary problem of heat conduction and the quasi-static problem of thermoelasticity, written taking account of conditions of nonideal thermomechanical contact, we derive boundary integral equations for a body with inhomogeneous inclusions. We propose a method of solving these equations taking account of the order of the principal term of the asymptotics of the solution in neighborhoods of the corners of the contact surfaces. Translated fromMatematichni Metodi ta Fiziko-mekhanichni Polya, Vol. 39, No. 1, 1996, pp. 37–41.     

Local uniform stability for the semilinear wave equation in inhomogeneous media with locally distributed Kelvin–Voigt damping

We consider the semilinear wave equation posed in an inhomogeneous medium Ω with smooth boundary subject to a local viscoelastic damping distributed around a neighborhood ω of the boundary according to the Geometric Control Condition. We show that the energy of the wave equation goes uniformly and exponentially to zero for all initial data of finite energy taken in bounded sets of finite energy phase‐space. As far as we know, this is the first stabilization result for a semilinear wave equation with localized Kelvin–Voigt damping.     

Propagation failure of fronts in discrete inhomogeneous media with a sawtooth nonlinearity

Exact, steady-state, single-front solutions are constructed for a spatially discrete bistable equation with a piecewise linear reaction term, known as a sawtooth nonlinearity. These solutions are obtained by solving second-order difference equations with variable coefficients, which are linear under certain assumptions on the expected solutions. An algorithmic procedure for constructing solutions in general, for both homogeneous and inhomogeneous diffusion, is obtained using a combination of Jacobi-Operator theory and the Sherman–Morrison formula. The existence of solutions for the difference equation, implies propagation failure of fronts for the corresponding differential-difference equation. The interval of propagation failure, which is the range of values of the detuning parameter that render stationary fronts, is studied in detail for the case of a single defect in the medium of propagation. Explicit formulae reveal precise relationships between parameter values that cause traveling fronts to fail to propagate when the interface reaches the inhomogeneities in the medium. These explicit formulae are also compared to numerical computations using the proposed algorithmic approach, which provides a check of its computational usefulness and illustrates its capabilities for problems with more complicated choices of parameter values.     

Diffraction of a plane electromagnetic wave on a wedge-shaped inclusion

A technique is developed for reducing the problem of diffraction by a wedge-shaped inclusion to functional equations in the complex plane. An original formulation of the radiation conditions is proposed that enables one to formulate the diffraction problem in this case. An existence and uniqueness theorem is proven. For the case of a sufficiently small opening span of the angle a formula is derived which is a good approximation to the solution to the problem.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Stekova AN SSSR, Vol. 195, pp. 29–39, 1991.     

Diffraction of a plane wave on a slippery elastic wedge

Diffraction of a plane elastic wave on a slippery wedge is considered; by a slippery wedge we mean a wedge in which the tangent tension and the normal component of the displacement vector are equal to zero on its surface. It is known that one can construct an explicit solution of this problem. The Sommerfeld representation of this solution is found in construct the paper. Bibliography: 6 titles.