The embedding formula for an electromagnetic diffraction problem

Embedding formulas are powerful tools that enable one to reduce the dimension of the space of variables for a diffraction problem. Let a scatterer be finite, planar, and perfectly conducting. The idea of the method is to replace the initial problem of diffraction of a plane wave by construction of an edge Green’s function, i.e., to solve a problem with a source located near the edge of a scatterer. An embedding formula is an integral relation connecting the solution of the initial plane wave incidence problem with the edge Green’s function. Earlier, embedding formulas have been derived for acoustic and elasticity problems. Here we derive an embedding formula for an electromagnetic problem. Bibliography: 11 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 324, 2005, pp. 247–261.     

Tabulation of the functions of V. A. Fock

Algorithms of two programs for computing the functions introduced by V. A. Fock are described. Estimates are given of the accuracy of the algorithms applied, and results of numerical computations of Fock functions on the basis of these programs are presented. The application of the programs constructed to compute diffraction fields according to two distinct asymptotic formulas is demonstrated.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Institute im. V. A. Steklova AN SSSR, Vol. 89, pp. 97–111, 1979.     

Shortwave scattering by a diffraction echelette-grating

The two-dimensional problem of the scattering of a plane wave by a periodic perfectly conducting grating, an echelette with a right angle, is considered in the case of a high-frequency approximation (the wavelength is assumed to be small as compared with the period of the grating). The situation where the incident plane wave glides along one of the faces of a wedge is discussed. A ray-optical solution of the problem (a shortwave asymptotic result) is derived by the method of summing multiple diffracted fields, which is well known in the geometric theory of diffraction. The main result of this paper consists of obtaining simple formulas for the efficiency e_n of a diffraction order with maximal value of e_n, derived in the shortwave approximation. Numerical results are presented, and important optical properties obtained by asymptotic analysis are described. Bibliography: 10 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 250, 1998, pp. 109–135. Translated by V. V. Zalipaev.     

The energy conservation law in acoustics of a moving nonstationary inhomogeneous medium

Expressions for the energy density of a sound wave and the sound energy flow vector in a moving nonstationary inhomogeneous medium are obtained. The energy conservation law is stated.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova, Vol. 179, pp. 52–57, 1989.     

Nonstationary normal waves in a thin,curved waveguide of variable cross section

The propagation of nonstationary normal waves in a thin waveguide with variable properties along its course is considered. Along space-time rays the eikonal and transport equations are integrated, and asymptotic formulas for these waves are obtained. It is observed that for some types of initial data the effect of overturning of a wave packet of a normal wave is possible.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 89, pp. 210–218, 1979.     

Optical theorem for the scattering of waves in an elastic plate

The scattering of waves in an elastic plate is investigated. A generalization of the optical theorem is found for the case of a noncompact scatterer. The formulas obtained can be used for indirect control of the correctness of computing diffraction fields.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 104, pp. 20–23, 1981.The author is grateful to D. P. Kouzov for his constant attention to the work.     

Nonstationary normal waves in a thin,curved waveguide in a multiray zone (uniform asymptotics)

In a thin waveguide with properties that vary along its course, the propagation of nonstationary normal waves in the presence of caustics for space-time rays is considered. The connection of the critical section of the waveguide with such caustics is determined. Uniform asymptotic formulas are obtained for the wave field in a multiray zone, and the passage into geometric rays outside a neighborhood of the caustics is traced.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 99, pp. 127–137, 1980.In conclusion, we note that the matrix machinery used in the present work is not connected with the special nature of the present problem and can always be applied when the initial Ansatz contains a collection of special functions which go over into one another under differentiation.     

Characteristics with singularities and the boundary values of the asymptotic solution of the Cauchy problem for a degenerate wave equation

We consider the Cauchy problem with spatially localized initial data for the twodimensional wave equation degenerating on the boundary of the domain. This problem arises, in particular, in the theory of tsunami wave run-up on a shallow beach. Earlier, S. Yu. Dobrokhotov, V. E. Nazaikinskii, and B. Tirozzi developed a method for constructing asymptotic solutions of this problem. The method is based on a modified Maslov canonical operator and on characteristics (trajectories) unbounded in the momentum variables; such characteristics are nonstandard from the viewpoint of the theory of partial differential equations. In a neighborhood of the velocity degeneration line, which is a caustic of a special form, the canonical operator is defined via the Hankel transform, which arises when applying Fock’s quantization procedure to the canonical transformation regularizing the above-mentioned nonstandard characteristics in a neighborhood of the velocity degeneration line (the boundary of the domain). It is shown in the present paper that the restriction of the asymptotic solutions to the boundary is determined by the standard canonical operator, which simplifies the asymptotic formulas for the solution on the boundary dramatically; for initial perturbations of special form, the solutions can be expressed via simple algebraic functions.     

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.     

Focusing problem and the asymptotics of the spectral function of the Laplace-Beltrami operator. II

In the paper a formal high-frequency solution of the problem of a point source of oscillations near a reflecting boundary is constructed. The boundary is geodesically concave which makes it possible to use methods developed in diffraction theory by V. A. Fock and J. B. Keller. On the basis of the formal solution of the problem of a point source of oscillations, it is possible to construct the asymptotics of the spectral functions of the Laplace-Beltrami operator.Translated fromZapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 89, pp. 14–53, 1979.     

Diffraction on a semi-infinite screen of a plane nonstationary wave the amplitude of which increases linearly along the front set

An exact analytic solution of the two-variables nonstationary problem of diffraction on an ideal half-infinite screen is obtained by the Smirnov-Sobolev method. The source of the field is an incident plane acoustic wave with a δ-function profile. The wave amplitude is a linear function increasing along the front set. Bibliography: 6 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 342, 2007, pp. 138–152.     

Diffraction echo signal generated by the incidence of a plane wave on a prolate spheroid

The problem of the diffraction of a plane wave by a prolate spheroid placed in a fluid is considered in the high-frequency approximation. The spheroid is hereby either a solid body or a shell. The contribution to the echo signal, i.e., the back-reflected field, of diffraction waves of creep and break-off type is investigated. Computational formulas for the geometric divergences of surface rays and also of creep and break-off waves are obtained.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituts im. V. A. Steklova AN SSSR, Vol. 140, pp. 18–35, 1984.     

弹性波在平面多连通域中的绕射与动应力集中

本文用复变函数论方法研究了弹性波在平面多连通域中的绕射问题,给出了这一问题解的完备逼近序列及边备条件的一般表示。问题归结为无穷代数方程组的求解,使用电子计算机可直接求得解答。特别是,对弱耦合问题,本文提出了渐近求解方法并且使用这个方法详细地讨论了P波对圆孔群的绕射问题。基于绕射波场的解,文中给出了任意形状空腔动应力集中系数的一般算式。     

Dispersion properties of nonstationary SV Rayleigh waves near the surface of an inhomogeneous elastic body

The physical characteristics of the surface of nonstationary SV Rayleigh waves are studied and formulas are obtained for the phase and group velocities (as functions of time and the observation point on the surface) of the Rayleigh waves. These formulas are in accord with the familiar formulas for the phase and group velocities in the stationary case.Translated from Zapiski Nauchnykh Seminarov Leningradzkogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova, AN SSSR, Vol. 173, pp. 172–179, 1988.     

Shortwave grazing scattering of a plane wave by a smooth periodic boundary. I. Diffraction of the penumbral field by a smooth convex contour

The diffraction problem of the penumbral field by a smooth convex contour is solved. New equations are obtained in the Fok region and in the vicinity of the limiting ray. Relations are provided on their basis for multiple diffraction of the penumbral field by humps of a periodic boundary.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 165, pp. 59–90, 1987.     

Determination of a wave field inside a hollow cone with a notch along the generator

The aim of this investigation is to determine the wave field inside a part of a conic domain filled with an acoustic medium subjected to the action of a nonstationary pressure. The method of solution is based on the discretization of the problem with respect to time by replacing the second derivative by a difference scheme and using new integral transformations with respect to other variables. A recurrent solution of the problem is obtained, and the calculation of a wave field for different geometric parameters of the domain is performed.     

Existence and uniqueness theorems in electromagnetic diffraction on systems of lossless dielectrics and perfectly conducting screens

New vector problem of electromagnetic wave diffraction by a system of non-intersecting three-dimensional inhomogeneous dielectric bodies and infinitely thin screens is considered in a quasiclassical formulation as well as the classical problem of diffraction by a lossless inhomogeneous body. In both cases, the original boundary value problem for Maxwell’s equations is reduced to integro-differential equations in the regions occupied by the bodies (and on the screen surfaces). The integro-differential operator is treated as a pseudodifferential operator in Sobolev spaces and is shown to be zero-index Fredholm operator. Uniqueness of solutions is proved under the realistic hypothesis of discontinuity of the dielectric permittivity the boundary of a volume scatterer. This result allowed to establish invertibility of the integro-differential operator in sufficiently broad spaces. For the problem of diffraction on dielectrics and surface conductors, theorem on smoothness of a solution is proved under assumption of data smoothness. The latter implies equivalence between the differential and integral formulations of the scattering problem. The matrix integro-differential operator is proved to be a Fredholm invertible operator. Thus, the existence of a unique solution to both problems is established.     

On spherical-wave scattering by a spherical scatterer and related near-field inverse problems

A spherical acoustic wave is scattered by a bounded obstacle.A generalization of the ‘optical theorem’ (whichrelates the scattering cross-section to the far-field patternin the forward direction for an incident plane wave) is proved.For a spherical scatterer, low-frequency results are obtainedby approximating the known exact solution (separation of variables).In particular, a closed-form approximation for the scatteredwavefield at the source of the incident spherical wave is obtained.This leads to the explicit solution of some simple near-fieldinverse problems, where both the source and coincident receiverare located at several points in the vicinity of a small sphere.     

Nonstationary mass transfer of a reacting spherical particle in laminar stream of a viscous fluid : PMM vol. 38, n 6, 1974, pp. 1041–1047

The process of the formation of a stationary mass transfer mode for a moving reacting particle is examined. An analytic expression valid for a nonstationary distribution of the concentration of matter in a steady stream of viscous fluid, flowing past a spherical particle, was obtained for the case when at a certain instant a chemical reaction of the first order begins at the surface of the sphere. The problem is solved for small finite Reynolds and Péclet numbers. The solution of the corresponding stationary problem has been obtained in [1]. Paper [2] examined a nonstationary heat transfer of a fluid spherical drop in an inviscid flow with spasmodic change of initial temperature at high Péclet numbers. Paper [3] contains an analysis of the problem of a nonstationary heat transfer of a rigid spherical particle for small Reynolds and Péclet numbers at spasmodic change of temperature of the particle surface. The results obtained in [3] can be used to describe the mass transfer for a moving reacting particle only in the case of a diffusion mode of the chemical reaction.     

Method of solving equations of motion of viscoelastic media

A new method is proposed for solving dynamic problems for viscoelastic media based on the introduction of potential functions and transformation of equations of motion. The equations obtained for potential functions are used for constructing the general solution in the case of the effect of moving loads on viscoelastic media with plane-parallel interfaces. The problem of the propagation of Rayleigh surface waves is solved independently of the form of the kernels of the linear operators; a formula is obtained for determining the velocity of the Rayleigh surface wave with an arbitrary form of the viscoelastic operators. A method of experimental determination of the kernels determining the linear viscoelastic operators is proposed.V. V. Kuibyshev Moscow Civil Engineering Institute. Translated from Mekhanika Polimerov, Vol. 9, No. 3, pp. 429–435, May–June, 1973.