Nonstationary love waves of theSH type in an anisotropic elastic medium. A kinematic approach. II

In this paper, the study of the high-frequency Love waves (simular to the well-known transverse waves of theSH type) near the surface of an anisotropic elastic body is continued. The formulation of the boundary-value problem, independent of a specific form of the elasticity tensor, provides the possibility of developing a kinematic approach, which is essential for constructing the asymptotics of these high-frequency waves. To this end, an algorithm is proposed that allows one to relate the transversality of the polarization of surface waves to the directions of their propagation on the surface and to obtain the conditions necessary for the origination of such waves. The algorithm suggested makes it possible to indicate those types of symmetry of media (special cases of anisotropy) for which the directions obtained correspond to the field of rays of Love waves. In these cases, the space-time ray method provides a mathematical tool for constructing uniform asymptotics of the surface waves in question. Bibliography: 8 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol 218, 1994, pp. 206–219. This work was supported by the Russian Foundation of Fundamental Research (Grant 93-011-16148). Translated by Z. A. Yanson.     

On nonstationary love waves near the surface of an anisotropic elastic body

Space-time (ST) ray expansions in the presence of a caustic are used for the construction of an asymptotics of surface waves similar to the SH Love waves. Certain conditions which ensure the existence of surface waves with SH polarization are deduced. Some special cases of anisotropic media are indiceted where these conditions are satisfied. Bibliography: 6 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 203, 1992, pp. 166–172. Translated by Z. A. Yason.     

Gaussian beams of love waves in an anisotropic medium and their summation

Gaussian beams of surface waves are constructed by means of the Kravtsov ansatz. The Gaussian beams are summed which enables us to describe analytically the wave field of a surface Love wave in the vicinity of an arbitrary singularity. Bibliography: 13 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 186, pp. 6–13, 1990. Translated by V. D. Azhotkin.     

Propagation of quasielastic Rayleigh waves in an inhomogeneous,anisotropic, elastic medium

The propagation of modulated oscillations along the smooth surface free from normal stresses of an inhomogeneous, anisotropic, elastic body with parameters which vary little over a characteristic wavelength is considered.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 89, pp. 234–245, 1979.The author thanks the participants of the seminar on the propagation and diffraction of waves of the Leningrad Branch of the Mathematics Institute (LOMI), in particular, V. S. Buldyrev and V. M. Babich, for fruitful discussions and their attention to the work.     

Reconstruction of cracks in an inhomogeneous anisotropic elastic medium

In this paper we give in two and three dimensions a reconstruction formula for determining cracks buried in an inhomogeneous anisotropic elastic body by making elastic displacement and traction measurements at the boundary. The information is encoded in the local Neumann-to-Dirichlet map. With the help of the Runge property, the local Neumann-to-Dirichlet map is connected to the so-called indicator function. This function can be expressed as an energy integral involving some special solutions, called reflected solutions. The heart of our method lies in analyzing the blow-up behavior at the crack of the indicator function, which is by no means an easy task for the inhomogeneous anisotropic elasticity system. To overcome the difficulties, we construct suitable approximations of the reflected solutions that capture their singularities. The indicator function is then analyzed by the Plancherel formula.     

An energy approach to the proof of the existence of Rayleigh waves in an anisotropic elastic half-space

An approach based on investigating the energy functional is applied for the first time to the classical problem of Rayleigh waves in an anisotropic half-space with a free boundary. The main object of the investigation is an ordinary differential operator in a variable characterizing the depth. An investigation of the spectrum by variational methods enables a new proof to be given of the existence of a Rayleigh wave in a linear elastic half-space with arbitrary anisotropy, which does not rest on the Stroh formalism.     

Stable elastic waves in anisotropic materials

Zusammenfassung Diese Arbeit befasst sich mit der Fortpflanzung von stabilen Wellen in einem anisotropen Medium infolge einer sich bewegenden Ladung an der Grenze des Materials. Wir behandeln besonders mobile Ladungen, deren Geschwindigkeit höher ist als diejenige der Wellen im Material. Das Problem wird als ebenes Spannungsproblem behandelt.

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This work was supported by the United States Atomic Energy Commission.     

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.     

On the growth of transverse waves in anisotropic elastic materials

Zusammenfassung Es wird gezeigt, dass die Amplituden ebener, transversaler Beschleunigungswellen, bei Fortpflanzung in einem anisotropischen und hyperelastischen Körper, entweder sich unendlich erweitern, oder bis auf Null abklingen können, je nach den Eigenschaften des Elastizitäts-Tensors zweiter Ordnung; auch wird weiterhin gezeigt, dass das Verhalten gewisser Komponenten der Unstetigkeits-Begleiterscheinungen dritter Ordnung tatsächlich verknüpft ist mit dem Verhalten der primären transversalen Wellen.     

Love waves of SH type in an inhomogeneous transversely isotropic elastic medium

The asymptotics of high-frequency Love waves, which are analogous to transverse surface SH waves, is considered for a special type of anisotropy (transverse isotropy) of elastic media. The wave field is represented as a sum of the space-time (ST) caustic expansion and two additional ST ray series for faster (relative to the transverse surface wave) body waves, decaying exponentially with depth. Near the surface, the coefficients of the ST caustic and ray series, as well as the eikonals of waves, are determined in the form of expansions in a small parameter, which characterizes the proximity of the caustic of the ray field to the surface. With regard for the specific structure of the elasticity tensor of a transversely isotropic medium, the surface is treated as a plane. Interrelations between the parameters of elasticity, which are consistent with the conditions of the positivity of the elastic deformation energy and provide for the origination of the surface waves considered, are obtained.Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 239, 1997, pp. 243–262.This work was supported by the Russian Foundation for Basic Research under grant No. 96-01-00666.     

On quasiphotons of Rayleigh waves (the case of an anisotropic elastic body)

A procedure of constructing analytic formulas for quasiphotons is developed. Quasiphotons are special asymptotic solutions of linear equations, which describe wave processes. These asymptotic solutions correspond to concentrated wave packets propagating along rays on the surface of an elastic body. Bibliography: 20 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 332, 2006, pp. 7–18.     

The propagation of the energy of elastic waves in anisotropic media

The particular features of the propagation of the seismic energy of elastic waves in anisotropic media with four constants of elasticity, depending on the directions of motion of the waves and the ratios of the constants of elasticity for all practical media of the anisotropy class considered, are investigated. A direct connection is established between the formation of acute-angled edges on the fronts of quasi-transverse waves from point sources and the distinctive features of the propagation of the energy of the waves under certain conditions for the constants of elasticity.     

Scattering of cylindrical waves by an elastic inclusion in an electrically conducting compressible medium

We study the problems of diffraction of cylindrical waves at an elastic homogeneous weakly conducting cylinder in a liquid weakly conducting medium and at a magnetohydrodynamic liquid cylinder in a magnetoelastic medium.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 35, 1992, pp. 127–131.     

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.     

On the dispersion of nonstationary surface waves in anisotropic elastic media

The whispering gallery modes propagating along the surface of an anisotropic elastic body are investigated with the use of space-time caustic expansions and space-time ray series. Each surface mode modulated in amplitude and frequency, is interpreted as a wave packet, with its amplitude’s maximum moving at a group velocity. On the boundary surface, asymptotic expressions for the group velocity (as a function of time and coordinates) are derived, which are in agreement with analogous formulas for Rayleigh waves of SV type in the isotropic case. Bibliography: 6 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 332, 2006, pp. 299–312.     

The stability of quasi-transverse shock waves in anisotropic elastic media

The stability of weak quasi-transverrse shock waves in a weakly anistropic elastic medium with respect to arbitrarily oriented perturbations is investigated in the linear approximation. It is shown that fast quasi-transverse shock waves are stable.