On the propagation of elastic waves in two-phase media

At the present time a number of papers has been already devoted to the dynamics of two-phase media. One may mention the papers by Frenkel' [1], Rakhmatulin [2], Biot [3,4], Zwikker and Kosten [5], and others. However, the basic problem of the setting up of the equations of motion in two-phase media still cannot be considered solved and requires additional study and experimental verification.

This paper is concerned with the study of the simplest case of motion, which is the propagation of elastic waves in a homogeneous isotropic medium consisting of a solid and a fluid phase. The problems of the reflection of plane waves and surface waves at the free boundary of the half-space are solved. It is shown that the stress-strain relations established by Frenkel' are equivalent to the analogous relations proposed by Biot and that the equations of motion of the latter are more general.     

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.     

弹性动力学方程组的Delta波

该文研究了半线性弹性动力学方程组Delta波的传播。在[7]中Delta波存在的意义下,证明了如果非线性项一致有界且Lipschitz连续,则半线性弹性动力学方程组的Delta波存在。     

An algorithm for determining wave fields in multilayer elastic media

An algorithm for computing wave fields propagating in elastic media with plane-parallel interfaces is proposed. This algorithm is based on summation of kinematically equivalent waves. All kinds of multiple waves of the exchange type arriving to an observer at the fixed interval of time (t₁, t₂) are taken into account. Dynamic analogues are determined within each kinematic group. Bibliography: 12 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 186, pp. 154–171, 1990. Translated by N. S. Zabavnikova.     

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.     

New results for guided waves in heterogeneous elastic media

We prove the existence of guided waves propagating with a velocity strictly larger than the S (shear) wave velocity at infinity in the case of unbounded elastic media invariant under translation in one space direction and asymptotically homogeneous at infinity. These waves correspond to the existence of eigenvalues embedded in the essential spectrum of the self-adjoint elastic propagator.     

On elastic waves in a wedge

The existence of waves propagating along the edge of an elastic wedge has been established by many authors by physically rigorous arguments on the base of numerical computations. A mathematically rigorous proof for a wedge with aperture angle less than π/2 was presented by I. Kamotskii. We supplement the I. Kamotskii result and prove the existence of fundamental modes for some range of aperture angles greater than π/2. Bibliography: 7 titles.     

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.     

The property of inner symmetry in optimum multilayer structures under the action of elastic waves

The general properties of laminar, linearly elastic, locally isotropic bodies which are energetically optimum for a specified inclined incidence of a plane wave are investigated. It is shown that, in the case of special supplementary conditions of the type that the frequencies of the maximum attainable value of the energy characteristic should be present in the required spectrum or that the angle of incidence is sufficiently small, the problem reduces to an alternate configuration of no more than two of the given set of layers which are identical both in their physical properties and in their width. These properties were established earlier for the propagation of electromagnetic and purely longitudinal waves.     

On the scattering amplitudes for elastic waves

Reciprocity and scattering theorems for the normalized spherical scattering amplitude for elastic waves are obtained for the case of a rigid scatterer, a cavity and a penetrable scattering region. Depending on the polarization of the two incident waves reciprocity relations of the radial-radial, radial-angular, and angular-angular type are established. Radial and angular scattering theorems, expressing the corresponding scattering amplitudes via integrals of the amplitudes over all directions of observation, as well as their special forms for scatterers with inversion symmetry are also provided. As a consequence of the stated scattering theorems the scattering cross-section for either a longitudinal, or a transverse incident wave is expressed through the forward value of the radial, or the angular amplitude, correspondingly. All the known relative theorems for acoustic scattering are trivially recovered from their elastic counterparts.Part of this work was done during the time that the first author was visiting the Department of Mathematics of The University of Tennessee at Knoxville. The authors want to thank the Greek Ministry of Research and Technology for partially supporting the present work.     

On the calculation of wave fields propagating in multilayer media

A program for evaluating elastic waves in multilayer media is composed on the basis of the algorithm previously stated. The waves under consideration undergo all possible acts of reflection and refraction on the interfaces of the media. Exchanges on these interfaces are also taken into account. Two examples illustrating the calculation of theoretical seismograms of elastic waves are presented. Bibliography: 5 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 203, 1992, pp. 156–165. Translated by N. S. Zabavnikova     

Reflection and refraction of attenuated waves in semi-infinite elastic solid media

In earthquakes it is known that waves of one type starting from the origin inside the earth produce waves of other types after impact at the free surface of the earth. A theoretical discussion is given here dealing with reflection and refraction of waves which have their amplitudes increasing with depth similar in nature to the initial diverging earthquake waves which after reflection produce the apparent surface waves in which the displacement decreases with depth. In one case it has been found that at a free surface a plane wave of SV type, travelling parallel to the surface, is capable of generating surface waves after internal reflection which under a certain state of attenuation are identical with the Rayleigh Waves.     

On the structure of quasitransverse elastic shock waves

The structure of quasitransverse shock waves in a slightly anisotropic medium in the presence of dissipation due to viscosity is investigated. The existence of a shock structure “responsible” for ambiguity of the solution of a selfsimilar problem about waves excited in a half-space is demonstrated. The question of the existence of a structure for the remaining quasitransverse shock waves is discussed.     

On modeling the frequency transformation effect in elastic waves

We present a continual model of an elastic medium with a field of distributed defects and use it to study the frequency transformation effect in elastic waves. By an example of simulation of the one-dimensional waves arising under a harmonic action on a layer with defects, we demonstrate the possibility of describing frequency transformation in the framework of the model.     

A note on the dispersion of Love waves in layered monoclinic elastic media

The dispersion equation for Love waves in a monoclinic elastic layer of uniform thickness overlying a monoclinic elastic half-space is derived by applying the traction-free boundary condition at the surface and continuity conditions at the interface. The dispersion curves showing the effect of anisotropy on the calculated phase velocity are presented. The special cases of orthotropic and transversely isotropic media are also considered. It is shown that the well-known dispersion equation for Love waves in an isotropic layer overlying an isotropic half-space follows as a particular case.