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.     

The transmission of a plane acoustic wave through a non-uniform thermoelastic layer

The reflection and refraction of a plane acoustic wave by a thermoelastic plane layer, non-uniform in thickness, bounded by non-viscous heat-conducting liquids, generally different, is considered. The system of equations for small perturbations of the thermoelastic medium is reduced to a system of ordinary differential equations, the boundary-value problem for which is solved by two methods: the spline-collocation method and the power-series method. Analytic expressions are obtained which describe the wave fields outside the layer. The results of calculations of the intensity transmission coefficient of the acoustic wave are presented.     

Simplices passing through a hole

We study small holes through which regular 3-, 4-, and 5-dimensional simplices can pass.     

Conduction and convection phenomena through a slab with thermal heterogeneities

This paper addresses the computation of the three-dimensional transient heat transfer through a layered solid and/or fluid formation containing irregular inclusions. The use of appropriate Green’s functions for a flat layer formation in a boundary element method formulation avoids the discretization of the layer interface boundaries.     

Non-linear reflection of a plane wave in an elastic medium

Exact formulae are derived for the reflected and refracted waves which occur for the inclined incidence of a plane horizontally polarized transverse wave of arbitrary profile on a horizontal interface between two elastic half-spaces experiencing non-linear friction when they move with respect to one another. A smooth function of general form is adopted as the friction function, which depends on the difference between the horizontal velocities of the elements of the boundaries of the half-spaces considered. It is shown that if the friction function depends non-monotonically on the relative velocity of displacement of the sides of a slit, then even when the profile of the incident wave is smooth, the reflected and refracted waves may contain strong discontinuities.     

Covering a graph with cycles passing through given edges

We propose a conjecture: for each integer k ≥ 2, there exists N(k) such that if G is a graph of order n ≥ N(k) and d(x) + d(y) ≥ n + 2k - 2 for each pair of non-adjacent vertices x and y of G, then for any k independent edges e₁, …, e_k of G, there exist k vertex-disjoint cycles C₁, …, C_k in G such that e_i ∈ E(C_i) for all i ∈ {1, …, k} and V(C₁ ∪ ···∪ C_k) = V(G). If this conjecture is true, the condition on the degrees of G is sharp. We prove this conjecture for the case k = 2 in the paper. © 1997 John Wiley & Sons, Inc. J Graph Theory 26: 105–109, 1997     

The evolution of a viscous two-fluid flow down an inclined plane

An evolutionary problem on the viscous (isothermal) flow of two connected fluids down an inclined plane is investigated. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Scattering of a plane wave on a periodic curve with trapping domains

The problem of scattering on a periodic curve is considered. The asymptotic solution of the problem is constructed, and its principal terms are presented. The justification of the asymptotic solution found is provided. Bibliography: 7 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 230, 1995, pp. 75–85. Translated by I. V. Kamotskii     

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.     

Diffraction of a plane acoustic wave on an impedance cone. Surface waves

Diffraction of a plane wave by a circular impedance cone is studied. Surface waves propagating from the vertex of the cone are discussed. The excitation coefficient is computed with the help of the incomplete separation of variables in the problem. Bibliography: 14 titles.     

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.     

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.     

Long cycles passing through a specified path in a graph

For a graph G and an integer k ≥ 1, let ς_k(G) = d_G(v_i): {v₁, …, v_k} is an independent set of vertices in G}. Enomoto proved the following theorem. Let s ≥ 1 and let G be a (s + 2)-connected graph. Then G has a cycle of length ≥ min{|V(G)|, ς₂(G) − s} passing through any path of length s. We generalize this result as follows. Let k ≥ 3 and s ≥ 1 and let G be a (k + s − 1)-connected graph. Then G has a cycle of length ≥ min{|V(G)|, − s} passing through any path of length s. © 1998 John Wiley & Sons, Inc. J. Graph Theory 29: 177–184, 1998     

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.