Reflected-refracted and head sh waves formed on a thin layer that is in welded contact with two elastic half-spaces

The reflection and refraction of SH waves by an elastic layer sandwiched between two elastic half-spaces are studied by using the numerical simulation on the basis of contour integration in the complex plane of the horizontal component of the slowness vector. The propagation of body, channel, head, and screened body waves are simulated in time and spectral domains. The wave fields associated with the propagation in the layer have strong attenuation, provided that the wave length is smaller than the value of the thickness of the layer. The stationary wave field of such waves is of resonance nature. Moreover, the maximum of the modulus of the spectral function is shifted to higher frequences as the epicentric distance increases. Thus, the attenuation of such waves depends on spectral characteristics of a source-receiver system. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 225, 1997, pp. 91–120. Translated by T. N. Surkova.     

Propagation of Scholte-Gogoladze surface waves along a curved fluid-solid interface

A high-frequency ray theory is presented for a type of small-amplitude waves (termed Scholte-Gogoladze waves), localized in a thin layer around an interface between elastic and fluid domains. The interface is assumed to be smooth, with typical radii of curvature much larger than the excitation wavelength. The technique employed in the paper is based on a boundary-layer version of the classical WKB-expansion (see V. M. Babich and N. Ya. Kirpichnikova, The Boundary-Layer Method in Diffraction Problems (Springer, Berlin 1979)). Bibliography: 20 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 324, 2005, pp. 229–246.     

On a fast method for computing the field of waves reflected by a thin elastic layer

A method for computing the field of waves reflected by a thin elastic layer is proposed. The method takes into account all kinds of exchange type reflections and is based on the summation of kinematic and dynamic analogues.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova, Vol. 179, pp. 152–162, 1989.     

Excitation of a tube wave in a borehole by a slow wave propagating in a fluid layer

Conversion of a slow wave propagating in a fluid layer inside an elastic medium into a tube wave propagating in a borehole that intersects this layer is considered. It is shown that the total field of the latter wave consists of three summands of different physical nature. Seismograms of the slow and tube waves are presented. Bibliography: 5 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 230, 1995, pp. 115–124. Translated by P. V. Krauklis.     

Attenuation of normal waves in a cavity with a liquid in the presence of an elastic shell

The attenuation of normal waves is investigated in a cylindrical cavity of an unbounded viscoelastic medium, in which there is a thin elastic shell filled with a viscous compressible liquid. It is assumed that the absorption of waves in a liquid or solid is small. The effect of the shell thickness on the value of the absorption coefficient of the normal waves is studied.Translated from Matematicheskie Metody i Fiziko-Mekhanicheskie Polya, No. 25, pp. 49–52, 1987.     

The Classical and Generalized Love Problems in a Domain of Low-Frequency Interference Waves of Signal Type. II

Computations of low-frequency interference waves of SH type are performed for the generalized Love problem. The model of a medium consists of a thin elastic layer, a thick layer, and a half-space. The layers and half-space are in rigid contact with each other. It is assumed that the waves pass through the thick layer twice, there and back. A new method, suggested by G. I. Petrashen, for computing wave fields allows us to compute the fields of SH waves and to draw conclusions concerning their behavior. The computations are carried out for < 1 and > 1, where is the ratio of the velocities in the thin and thick layers. In either case, the thin layer significantly distorts a signal transmitted deep into the medium. Bibliography: 6 titles.     

Diffraction of surface SV-waves on the line of jump in elastic parameters

A set of problems on the diffraction of elastic surface waves of vertical polarization (SV-waves) on the line of jump of elastic parameters is under consideration. Expressions for reflected and transmitted wave fields are obtained by means of the parabolic equation method. The corresponding transformation coefficients are found for the reflection and transmission. Bibliography: 7 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 342, 2007, pp. 77–105.     

Nonstationary interference wave fields of SV type in a free elastic layer and the problem of ultrasonic simulation of plane seismic fields by using plate models

The problem of quantitatively studying wave fields in a free elastic layer excited by a nonstationary point source is discussed. Much attention is given to the low-frequency part of the Fourier spectrum. The roots of the dispersion equation of the layer are studied comprehensively, and on this basis efficient methods of quantitatively estimating interference wave modes are developed. Particular attention is given to the ultralow-frequency band of the spectrum. This allows us to make progress toward the mathematical justification of methods of seismic modeling of plane wave fields on plate models. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 225, 1997, pp. 121–239. Translated by N. S. Zabavnikova.     

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.     

An influence of vertical structure of density and flows on free internal waves

In the framework of general linear theory we study the influence of a thin stratification of fields of density and velocity of plane-parallel shear flow on dispersion curves and vertical profiles of modes of free internal gravitational waves. It is shown that an accounting of thin structural singularities can essentially change kinematic characteristics of such waves.Translated from Dinamicheskie Sistemy, No. 8, pp. 99–107, 1989.     

Comparison of displacement and stress fields upon diffraction of elastic waves at different defects: Cracks and thin rigid intrusions

This paper considers the problem of diffraction of elastic waves at rectilinear defects (cracks or thin rigid intrusions) in an unbounded elastic medium. A discontinuous solution of the Helmholtz equation is used to reduce the problem to a singular integral equation in unknown jumps. There is a detailed analysis and comparison of the stressed state close to the defects and the diffraction field at large distances from the defect for defects of different types. Translated fromDinamicheskie Sistemy. Vol. 12, pp. 14–23, 1993.     

Evolution of flexure-gravitational stimulated vibrations of a floating nonuniformly compressed elastic plate

We consider the evolution of vibrations of a thin nonuniformly compressed elastic plate that floats on a surface of a homogeneous ideal incompressible fluid of finite depth, by using the method of integral transformations. Waves are caused by pressures of the type of traveling waves on the surface of the plate, which is harmonic in one of the directions. An analysis of conditions of perturbations of waves that form oscillations of the plate and the wave motion of the fluid is given. A study is given of the dependence of the structure and the number of waves from stressing forces and characteristics of perturbed pressures.Translated from Dinamicheskie Sistemy, No. 7, pp. 58–62, 1988.     

Comparison of methods for computing interference elastic waves in thin-layered media. 2

The paper is an immediate continuation of the paper where the solution of the problem on the propagation of low-frequency waves in thin-layered media by the dispersion equation method was considered in detail. In the present article, the solution of a similar problem is given for an elastic layer and a half-space, which are in rigid contact, by the method of superposition of complex plane waves. Bibliography: 17 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 342, 2007, pp. 217–232.     

Wave fields in a thin layer of an (ideal) liquid covering an elastic half-space (the simplest model of a shelf)

The head interference wave associated with the propagation of the P-wave in an elastic half-space is studied by using as an example the propagation of pressure waves in a liquid layer covering an elastic half-space. The attenuation of such a wave with respect to the distance between a source and a receiver is smaller than that in the classical theory. The wave field is considered both in time and frequency domains. The stationary wave field of the head interference wave is of resonance nature. From the mathematical point of view, the resonance peaks occur when the roots of the dispersion equation pass through a branch point. The minimal attenuation of the stationary wave field is observed in a neighborhood of such resonance peaks. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 225, 1996, pp. 40–61. Translated by N. S. Zabavnikova.     

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     

Nonstationary love waves of the SH-type in an anisotropic elastic medium. A kinematic approach

In this paper, the propagation of Love waves in anisotropic elastic media is studied. These waves are a similar to the transverse surface SH waves in the isotropic case. Necessary conditions for the existence of Love waves of this polarization type near the surface Σ of an anisotropic elastic body are deduced. The algorithm developed here makes it possible to find the direction (s) of transverse surface wave propagation (at every point on the surface Σ). The algorithm employed is illustrated by some special anisotropic cases. The space-time method is used to construct the asymptotics of Love waves for those types of anisotropic media the eikonal equation of which is valid on the surface of an elastic body. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 210, 1994, pp. 262–276 Translated by Z. A. Yanson     

Numerical Study of Properties of Quasilocal Plane Waves of the Modal Type in the Case of a Thin Low-Velocity Layer that is in Contact with an Elastic Half-Space

Numerical study of properties of quasilocal plane waves of the modal type propagating deep into a medium is carried out with the example of the model of a low-velocity elastic layer in the case of rigid contact with the underlying half-space. It is established that the genesis of these waves is closely related to singular complex roots of the dispersion equation of the problem. Eighteen variants of the model differing by relative parameters of the problem, which have a physical meaning, are considered. For every variant, seismograms of modal and body waves are computed and a comparison of them in intensity is carried out. Bibliography: 4 titles. Dedicated to P. V. Krauklis on the occasion of his seventieth birthday __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 308, 2004, pp. 182–196.     

Applied aspects of studying the roots of dispersion equations on nonprincipal sheets of a complex plane

In quantitatively estimating interference wave fields by methods of function theory, the initial integration path is subjected to deformation in the complex (ζ) plane. As a result of this deformation, sometimes the contour λ may go into other sheets of the Riemann (ζ) surface. This forces us to study the integrands and, in particular, the dispersion equation of the problem on the other sheets of the (ζ) plane. Similar questions are discussed for the case where a layer SH is in contact with two elastic half-spaces. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 225, 1996, pp. 62–90. Translated by T. N. Surkova.     

Reflection and transmission of the surface <Emphasis Type="Italic">P</Emphasis>-waves on the line of jump in elastic parameters

Problems of diffraction of elastic surface waves of horizontal polarization (P-waves) on a line of jump in elastic parameters are considered. The corresponding coefficients of reflection, refraction, and transmission are obtained by means of the parabolic equation method. A comparison of the Rayleigh waves, SV-waves, and P-waves is carried out. Numerical values of the transformation matrix are found in the case where the whispering gallery wave transforms to a sum of whispering gallery waves and a homogeneous wave transforms to the first five modes of whispering gallery waves. Bibliography: 6 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 342, 2007, pp. 106–137.     

Approximate Description of the Dynamics of Thin Isotropic Elastic Coatings and Interlayers by Using Asymptotics of High Order of Accuracy

Asymptotically accurate low-frequency models for isotropic elastic coatings and interlayers are developed. The main constraint is the requirement on contact conditions for the layer and the base that at least one of the boundary conditions must include the displacement component in an explicit form. The displacement and stress fields in the three-dimensional elastic system are sought in the form of asymptotic expansion into power series of a small parameter — the ratio between the half-thickness of the layer and the minimum length of the wave in the longitudinal direction. The action of the layer is approximated by impedance boundary conditions, which are transferred to the contact surface with the basic, more thick body. These conditions are obtained with an asymptotic error up to and including the sixth order of magnitude. A numerical testing, which is carried out with the example of partial waves, shows a satisfactory accuracy, comparable with that of high-order theories of plates. The results obtained can be utilized in fast algorithms for calculating spectra of natural waves in half-spaces, thick laminated plates, and shallow shells with coatings and interlayers. The physical limit of applicability of the theory developed is the frequency of the first quasi-resonance in the corresponding deformable system. The number of alternating interlayers is unlimited. __________ Translated from Mekhanika Kompozitnykh Materialov, Vol. 41, No. 6, pp. 783–794, November–December, 2005.