Scattering of elastic waves through a heterogeneous medium

In this paper we study the elastic wave diffraction in R ³ through a heterogeneous medium, with periodic structure, which occupies a bounded domain. We show that, as the period tends to zero, the solution tends, in some sense, to the solution corresponding to the diffraction by an obstacle made of the classical “homogenized medium”. An analogous result is also proved for the scattering frequencies and the associated scattering functions.     

Propagation of nonlinear capillaary waves in an electrically conducting liquid

The article deals with the propagation of periodic capillary waves with finite amplitude on the surface of an electrically conducting liquid subjected to the effect of a magnetic field. It is shown that the evolution of wave packets is described by perturbed nonlinear Schrödinger equations. Its asymptotic solution is obtained, and it is established that the influence of MHD effects manifests itself in reduced frequency and amplitude of the propagating waves.Kiev. Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 21, pp. 97–99, 1990.     

Nonlinear breaking of waves in an electrically conducting and radiating gas

The effects of radiative transfer are treated by the use of a differential approximation which is valid over the entire optical depth range from the transparent limit to the optically thick limit. The singular surface theory is used to determine the modes of wave propagation and to evaluate the behaviour at the wave head. It is shown that there are two modes of wave propagation namely (i) the radiation induced waves which are always damped, and (ii) the modified magnetogasdynamic waves which break at the wave front if the initial discontinuity is sufficiently strong. The effects of thermal radiation, the magnetic field intensity, the finite electrical conductivity and the initial wave front curvature on the non-linear breaking of modified magnetogasdynamic waves are discussed.

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Zusammenfassung Der Einfluß der Strahlung wird mit Hilfe von Differential-Näherungen behandelt, die im ganzen Bereich von optischen Tiefen Gültigkeit haben, von der Transparenz bis zur optisch dicken Grenze. Die Theorie singulärer Flächen wird benützt um die Wellenausbreitung und das Verhalten an der Wellenfront zu behandeln. Es wird gezeigt, daß es zwei Formen von Wellenausbreitung gibt, nämlich (i) Wellen erzeugt durch Strahlung, die immer gedämpft sind, und (ii) die modifizierte magnetogasdynamische Welle, die an der Front immer bricht, wenn die ursprüngliche Diskontinuität stark genug ist. Es werden die Einflüsse der Wärmestrahlung, der magnetischen Feldstärke, der endlichen elektrischen Leitfähigkeit und der ursprünglichen Wellenkrümmung auf die nicht-lineare Frontbildung untersucht.

     

Scattering of plane harmonic waves on a cylindrical cavity with an elliptical cross section in an orthotropic medium

The diffraction of a plane longitudinal harmonic wave on a cavity with a smooth curvilinear cross section in a rectilinearly orthotropic medium is solved by using small perturbations in the elastic moduli and introducing generalized wave potentials. Results are presented from a numerical analysis of the dynamic stresses in the near diffraction field and at the boundary of an elliptical cavity including variations in the relative incident wavelength, eccentricity of the cavity, and degree of anisotropy of the medium. Donetsk State University. Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 29, pp. 102–110, 1999.     

The motion of a rigid spherical inclusion in an elastic medium under the action of plane waves

Two problems on the motion of a rigid spherical inclusion in an elastic medium under the action of a nonstationary, longitudinal, plane, compressional wave and a harmonic shear wave are considered. Using the method of vector eigenfunctions and the integral Laplace transform with respect to time their precise solutions are constructed. Numerical results are given which illustrate the dependence of characteristics of motion of the inclusion on parameters of the falling wave.Translated from Dinamicheskie Sistemy, No. 9, pp. 37–47, 1990.     

Shock waves in an elastic medium with cubic anisotropy

Low-intensity shock waves, propagating along the principal diagonal of a cube in an incompressible elastic medium possessing cubic symmetry, are considered. The form of the shock adiabatic in the phase plane of shears is obtained. Sections corresponding to non-decreasing entropy at the discontinuity and the conditions of evolutionarity of the discontinuity on it are indicated. The structure of the shock waves is investigated.     

Scattering of emerging normal elastic waves for a crystalline plate in a liquid medium

Frequency dependences are given for the imaginary components of the three lowest energy scattering branches of the dispersion spectra of symmetric and antisymmetric normal emerging waves propagating along the elastically equivalent directions in a plate of single-crystal Rochelle salt immersed in water. Donetsk State University. Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 30, pp. 156–161, 1999.     

Successive scatterings of stress waves by a train of parallel elastic cylinders in an elastic medium

Zusammenfassung Die Streuung von Spannungswellen durch eine Reihe paralleler elastischer Kreiszylinder, eingebettet in einem elastischen Kontinuum, wurde untersucht. Alle Zylinder sind unendlich lang und haben einheitlichen Querschnitt und identische Materialeigenschaften, verschieden von denjenigen des Kontinuums. Die Achsen der Zylinder liegen in einer Parallelebene zum Propagationsvector einer ebenen Spannungswelle. Durch Anwendung der Kirchhoffschen Methode des retardierten Potentials [1–4] wurden in dem beschriebenen System Wellenfrontspannungen und Wellenfrontverschiebungen gefunden, welche von der Streuung der ersten und zweiten Generation verursacht wurden.Die Ergebnisse zeigen die Intensität der Spannungen; Verschiebungen klingen sehr schnell ab und verschwinden beinahe mit dem fünften Zylinder. Die Resultate wurden mit dem Abklingen von Wellen in einem geschichteten System verglichen, das aus zwei Materialien hergestellt ist. Die Analyse der Wellenzerstreuung kann auch auf Systeme ausgedehnt werden, welche mehrere Materialphasen von allgemeiner Form enthalten.

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This work was performed under the sponsorship of Southwest Research Institute Internal Research Program under SwRI Project 02-9009.     

Scattering of elastic waves by nonaxisymmetric three-dimensional dipping layer

Using a boundary method, we investigated the scattering of elastic plane harmonic SH, SV, P, and Rayleigh waves by three-dimensional nonaxisymmetric dipping layers embedded in an elastic half-space. The valley was subjected to incident Rayleigh wave and oblique incident SH, SV, and P waves. The method utilized spherical wave functions to express the unknown scattered field. These functions satisfy the equation of motion and radiation conditions at infinity but they do not satisfy the stress-free boundary conditions at the surface of the half-space. The boundary and continuity conditions are imposed locally in the least-square-sense at several points on the layer interface and on the surface of the half-space. A comparative study was done to examine the validity and limitations of the two-dimensional approximations (antiplane and plane strain models) of three-dimensional models. It is demonstrated that the two-dimensional approximations may be inadequate to represent actual displacement field for three-dimensional irregularities.     

Interaction of infinite thin elastic cylindrical shell and pulsating spherical inclusion in potential subsonic flow of ideal compressible liquid

In the present paper a method is proposed to investigate behaviour of the axis symmetric system consisting of an infinite thin elastic cylindrical shell, filled by potential flow of ideal compressible liquid and containing pulsating spherical inclusion. This coupled problem is considered in the framework of classical mechanics models: linear potential flow theory and thin elastic shells theory based on the Kirchhoff - Love hypotheses. The approach suggested consists in possibility of rewriting general solution of the corresponding mathematical physics equations from one to the other coordinate system. It enables to satisfy boundary conditions on both the spherical and the cylindrical surfaces and to get an exact analytical solution (as a Fourier series) of the coupled interaction problem. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Weak fluctuation modes of an electrically and thermally conducting viscous compressible flow in a homogeneous magnetic field

Zusammenfassung Eine Perturbationstheorie erster Ordnung zeigt, daß die Gleichungen, die die Bewegung eines elektrisch und thermisch leitenden zähen kompressiblen Strömungsfeldes bestimmen, welches endliche elektrische Leitfähigkeit hat, vier verschiedene Typen von Störungsfeldern enthalten kann; diese werden von vier unabhängigen partiellen Differentialgleichungen bestimmt. Die drei ersten Typen sind: Wirbel-Stromdichte- und magnetisch-akustische Wellen-Typen. Sie sind hyperbolischer Art und pflanzen sich nicht isotrop als Wellen fort. Der vierte Typ ist ein Entropie-Typ, der sich nicht ausbreitet und der von einer parabolischen Gleichung bestimmt wird. Jeder dieser verschiedenen Typen kann nicht weiter zerlegt werden wie im Falle inkompressibler Strömung. Zusätzlich wird gezeigt, daß die magnetisch-akustischen Wellen im Falle endlicher elektrischer Leitfähigkeit neue und interessante Eigenschaften besitzen.     

Study of oscillations in a viscoelastic cylindrical shell in an elastic medium

An equation was obtained and solved for oscillations in a viscoelastic cylindrical shell located in the ground. A numerical example is examined.Moscow Electronic Engineering Institute. Translated from Mekhanika Polimerov, No. 1, pp. 178–181, January–February, 1974.     

Investigation of low-frequency normal waves in a Biot layer surrounded by an elastic medium

The porous Biot layer surrounded by two elastic half-plates is considered. For this medium, two dispersion equations are established. These dispersion equations correspond to symmetric and antisymmetric parts of the medium. The investigation of these equations is conducted in the region of low frequencies. The imaginary roots of these equations in this region determine low-frequency normal waves. Bibliography: 6 titles.     

The stressed state of a thin coating on an elastic inclusion of optimal shape in an elastic medium

We examine the properties of the stressed state of the thin intermediate zone separating a nonhomogeneous inclusion from the matrix. Explicit expressions are given for the components of the stress tensor in the thin coating depending on the properties of the phases and the load parameters.Simferopol' University. Translated from Dinamicheskie Sistemy, No. 10, pp. 26–30, 1992.     

Scattering of waves by an expanding surface

In the present paper, the perturbance of a harmonic plane wave by an expanding surface originating at a fixed point is considered. This surface generates a wave with spherical front. An analytic expression of the wave field in the vicinity of the wave front is obtained both in the scalar and in the electromagnetic case. Bibliography: 7 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 250, 1998, pp. 35–48. Translated N. S. Zabavnikova.