Rayleigh wave scattering by a plane strip in a deep ocean

A theoretical formulation to study the problem of scattering of Rayleigh waves due to the presence of a rigid plane strip in a deep ocean is presented. A rigid plane strip (0 ≤z ≤ H, 0 ≤x ≤ l) is fixed in the surface of the ocean occupyingz ≥ 0. Fourier transformation and Wiener-Hopf technique are used to arrive at the solution. The scattered Rayleigh waves behave as cylindrical waves emerging out of the corner of the strip and its image in the free surface of the ocean. The scattered waves are obtained in terms of Bessel functions whose behaviour near and far from the strip is well-known. The numerical calculations for the scattered waves show that their amplitude increases rapidly for a small increase in the value of the wave number. Scattering of Rayleigh waves due to a thin plane vertical barrier and a thin barrier in the free surface of the ocean has been considered as the special cases.     

Surface wave scattering by a rectangular impedance cylinder located on a reactive plane

The electromagnetic scattering of the surface wave by a rectangular impedance cylinder located on an infinite reactive plane is considered for the case that the impedances of the horizontal and vertical sides of the cylinder can have different values. Firstly, the diffraction problem is reduced into a modified Wiener–Hopf equation of the third kind and then solved approximately. The solution contains branch‐cut integrals and two infinite sets of constants satisfying two infinite systems of linear algebraic equations. The approximate analytical or numerical evaluations of corresponding integrals and numerical solution of the linear algebraic equation systems are obtained for various values of parameters such as the surface reactance of the plane, the vertical and horizontal wall impedances, the width and the height of the cylinder. Copyright © 2004 John Wiley & Sons, Ltd.     

The scattering of a plane flexural wave by a sector of a thin elastic plate with a supported edge

A solution is constructed of the problem of the diffraction of a plane flexural wave at the vertex of a thin elastic plate cut out in the form of a sector, on the edge of which the condition of a supported edge (hinged support) is specified.     

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.     

Diffraction of a plane wave by a narrow-angle cone in the case of the dirichlet boundary conditions

By using Smyshlayev's formula for the amplitude of a wave scattered by the vertex of an arbitrary cone, an approximate formula for this amplitude is deduced for the case of a narrow-angle cone. Bibliography: 7 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 218, 1994, pp. 3–11. This work was supported by the Russian Foundation of Fundamental Research (Grant 93-011-16148). Translated by N. S. Zabavnikova.     

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.     

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.     

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.     

Diffraction of the plane wave by a transparent wedge: The Sommerfeld-Malyuzhinets approach

The time-harmonic diffraction of the plane wave by a transparent wedge of an arbitrary opening is examined by the Sommerfeld-Malyuzhinets approach. The velocities of wave propagation inside and outside the wedge are equal. An explicit solution is presented, which is studied asymptotically as kr → ∞ and kr → 0. Bibliography: 2 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 354, 2008, pp. 157–172.     

Many-body wave scattering problems in the case of small scatterers

Formulas are derived for solutions of many-body wave scattering problems by small particles in the case of acoustically soft, hard, and impedance particles embedded in an inhomogeneous medium. The case of transmission (interface) boundary conditions is also studied in detail. The limiting case is considered, when the size a of small particles tends to zero while their number tends to infinity at a suitable rate. Equations for the limiting effective (self-consistent) field in the medium are derived. The theory is based on a study of integral equations and asymptotics of their solutions as a→0. The case of wave scattering by many small particles embedded in an inhomogeneous medium is also studied.     

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.     

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.     

Light scattering by a homogeneous sphere near a plane boundary

The scattering of electromagnetic waves by a homogeneous sphere near a plane boundary is presented in this paper. The vector wave equations derived from Maxwell’s equations are solved by means of the two orthogonal solutions to the scalar wave equation. Hankel transformation and Erdélyi’s formula are used to satisfy the planar boundary conditions and the determination of the unknown coefficients in the scattered field and internal fields is achieved by matching the electromagnetic boundary conditions on the surface of the sphere. Existence and uniqueness of the solution of the series involving these unknown coefficients are shown.     

The diffraction of a plane compressional wave by a spherical cavity in an elastic medium

Zusammenfassung Ein sphärischer Hohlraum in einem unendlichen homogenen und isotropen elastischen Medium wird von einer ebenen Druckstosswelle überstrichen, deren Front sich gleichförmig fortbewegt. Das durch Ablenkung am Hohlraum erzeugte Verschiebungs- und spannungsfeld wird mittels einer Integraltransformation bestimmt. Insbesondere wird die starre Teilbewegung der Begrenzung des Hohlraums ermittelt. Das Problem wird für eine einstufige Druckwelle gelöst. Die Ergebnisse lassen sich als Einflussgrössen betrachten, mit denen sich die Duhamel-Integrale für allgemeinere Druckwellen berechnen lassen.

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This work was performed under Contract No. AF 29(601)-5132 (Project No. 1080, Task No. 108007) for Air Force Special Weapons Center, Kirtland Air Force Base, New Mexico, USA.     

The Diffraction of an obliquely incident surface wave by a submerged plane barrier

Zusammenfassung Das Problem der Beugung einer Oberflächenwelle, die schief auf ein völlig versenktes, ebenes Hindernis stösst, wobei das ebene Hindernis senkrecht zur ruhenden Oberfläche steht, wird auf die Lösung der Helmholtz-Gleichung reduziert. Die Lösung wird nach dem Wiener-Hopf-Verfahren durchgeführt und ergibt die Reflexionsbeiwerte.     

Light scattering by a homogeneous sphere near a plane boundary II

ABSTRACT

The problem of light scattering by a homogeneous sphere above a plane boundary is considered in this paper. Hankel transformation and Erdélyi's formula are used to satisfy the boundary conditions on the plane and the determination of the unknown coefficients in the scattered field is achieved by matching the electromagnetic boundary conditions on the surface of the sphere. Existence and uniqueness of the solution involving these unknown coefficients are shown and the extinction efficiency factor is presented.