Time-dependent waves of rayleigh type near the surface of an inhomogeneous elastic body

The well studied, high-frequency Rayleigh waves polarized in a plane normal to a cross section of the surface of an inhomogeneous elastic body with phase speed close to the speed of transverse waves are generalized to the case of the time-dependent equations of elasticity. For the wave field uniform asymptotics are obtained in the form of space-time ray expansions of two types: with a real eikonal (for transverse waves diffracted at the surface) and with a complex eikonal (for a longitudinal wave damped away from the surface).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 156, pp. 168–183, 1986.     

Effect of inhomogeneity of the medium on the directivity of the simplest sources of elastic waves

The initial data for the first correction term of the ray expansion of a longitudinal wave excited by a point center of expansion are found by the boundary-layer method. The same is done for a transverse wave from a center of rotation.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 140, pp. 77–87, 1984.     

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.     

Higher-Order Asymptotic Approximations for Surface Love Waves of SH Type in Tranversely Isotropic Elastic Media

A uniform asymptotics of the surface Love modes for a special case of anisotropy (tranverse isotropy) of an elastic media is obtained. In constructing the asymptotics of surface waves, the space-time (ST) ray method is employed. The wave field of each Love mode is represented as the sum of the ST caustic expansion involving the Airy functions with a real eikonal and two correction terms that are ST ray solutions, which in fact are inhomogeneous waves with complex eikonals. The eikonals and coefficients of the caustic and ray series are sought in the form of expansions in powers of two variables. The first variable is the distance from the surface, whereas the other characterizes the proximity of the caustic of a ray field to the boundary surface. Thanks to the specific structure of the elasticity tensor for a transversely isotropic medium, the boundary surface is necessarily a plane. A recursion process of computation of higher terms of the asymptotic expansion allows one to trace the conversion of the formulas obtained to the known ray solutions for isotropic elastic media. Relations between the elasticity parameters of a medium are obtained that ensure the existence of SH Love waves in a transversely isotropic medium and that are consistent with the conditions of the positiveness of the elastic energy of deformation. Bibliography: 6 titles.     

On the Propagation of Love Waves in Anisotropic Elastic Media

The Love waves concentrated near the surface of an anisotropic elastic body are studied. A uniform asymptotics of the wave field is constructed with the use of the nonstationary caustic expansion (Yu. A. Kravtsov's ansatz) in the form of a space-time ray series. Using three types of waves, which propagate along any direction in an elastic medium, as a vector basis, sufficient conditions for the existence of a nonzero asymptotic solution of the problem under study are obtained. The procedure for constructing asymptotic series is illustrated with the model of a transversely isotropic medium. Bibliography: 9 titles.     

Two-parameter asymptotic forms for space-time Gaussian beams in an elastic medium

Asymptotic equations are constructed which contain two small parameters, for the longitudinal and transverse space-time Gaussian beams.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 173, pp. 87–95, 1988.     

The effect of stiffness on the transverse and longitudinal motions of a musical string: Commemorating the 100th anniversary of the birth of Kh.A. Rakhmatulin

The effect of stiffness on the propagation of longitudinal and transverse waves and vibrations in prestretched strings is considered. The contribution of the longitudinal and transverse components to the dynamic load is of the same order. The longitudinal vibrations occur both at natural frequencies and at frequencies of the transverse vibrations. Resonance phenomena are possible. Low stiffness, which is characteristic for musical strings, leads to a small change in the frequencies of the whole spectrum of transverse and longitudinal vibrations, but to a considerable change in the shape of the string at strike and mounting points and on the transverse wave front.     

On the ray method for Biot media

This paper is devoted to an investigation of wave propagation in a Biot porous medium, which consists of elastic and fluid phases. The space-time ray expansion of solutions of dynamic equations for a Biot medium is constructed (in the anisotropic inhomogeneous case). In the inhomogeneous isotropic case, a Rytov law analog is derived similarly to elasticity theory. Bibliography: 3 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 354, 2008, pp. 112–131.     

Space-time ray method of calculating waves in a weakly inhomogeneous,layered medium

A space-time ray method is presented for constructing the fields of Love and Rayleigh waves for a half space with parameters that vary slowly in a horizontal direction, while the free surface and boundaries of the layers are weakly curved. A surface wave is interpreted here as a harmonic wave packet with amplitude and frequency modulation.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 99, pp. 5–18, 1980.     

Passage of a wave through a thin layer

The problem is considered of the passage of an acoustic wave given by its ray expansion through a thin layer. The method of boundary-layer asymptotic expansions is applied to obtain ray representations of the reflected and transmitted waves (the small parameters are the thickness of the layer and the wave length).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklov AN SSSR, Vol. 173, pp. 20–29, 1988.     

Focusing of WKB solution of the equation [Δ+K2n2(x)]u=0, K→∞

For a given incoming wave converging to a point, a focal solution and ray expansion for the outgoing wave are constructed. These nonuniform expansions are matched to all orders.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematichskogo Instituta im. V. A. Steklova AN SSSR, Vol. 51, pp. 123–128, 1975.The author is grateful to V. M. Babich for useful discussions during the course of the work.     

Reconstruction of 2-tensor fields, given in a unit circle, by their ray transforms based on LSM with B-splines

A numerical method for solving a tensor tomography problem of reconstructing a symmetric 2-tensor field, given in a unit circle, is proposed. The potential and (or) solenoidal part of the desired field with fixed properties on the boundary are found from transverse and (or) longitudinal ray transforms calculated along the straight lines crossing the support of the field. The solution is sought by means of the least-squares method with the use, as an approximating sequence, of local bases constructed on the basis of B-splines.     

Wave propagation in single-walled carbon nanotube under longitudinal magnetic field using nonlocal Euler–Bernoulli beam theory

In the present work, the effect of longitudinal magnetic field on wave dispersion characteristics of equivalent continuum structure (ECS) of single-walled carbon nanotubes (SWCNT) embedded in elastic medium is studied. The ECS is modelled as an Euler–Bernoulli beam. The chemical bonds between a SWCNT and the elastic medium are assumed to be formed. The elastic matrix is described by Pasternak foundation model, which accounts for both normal pressure and the transverse shear deformation. The governing equations of motion for the ECS of SWCNT under a longitudinal magnetic field are derived by considering the Lorentz magnetic force obtained from Maxwell’s relations within the frame work of nonlocal elasticity theory. The wave propagation analysis is performed using spectral analysis. The results obtained show that the velocity of flexural waves in SWCNTs increases with the increase of longitudinal magnetic field exerted on it in the frequency range; 0–20 THz. The present analysis also shows that the flexural wave dispersion in the ECS of SWCNT obtained by local and nonlocal elasticity theories differ. It is found that the nonlocality reduces the wave velocity irrespective of the presence of the magnetic field and does not influences it in the higher frequency region. Further it is found that the presence of elastic matrix introduces the frequency band gap in flexural wave mode. The band gap in the flexural wave is found to independent of strength of the longitudinal magnetic field.     

Methods of calculating further approximations in the asymptotic space-time Gaussian beam method

In the example of the equations of the inhomogeneous isotrophic theory of elasticity and of considerations of the longitudinal component of the displacement vector, a scheme is proposed for determining the successive approximations in the asymptotic expansion of the space-time Gaussian beam method.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 173, pp. 104–112, 1988.     

Differaction of a pulsed longitudinal shear wave in an orthrotropic massif with a cavita

We propose a method for numerical-analytic solution of a boundary-value problem for diffraction of a longitudinal shear wave in the form of a triangular pulse on a cylindrical tunnel cavity of circular crosssection in an unbounded rectilinearly orthotropic massif. The basis of the proposed approach consits of methods of spectral decomposition of periodically continued pulses with the introduction of small correcting perturbations of the elastic constants of the massif. The characteristics of the stress-strain state in the basic stationary problems are expressed in the terms of the generalized wave potentials which become the classical potentials of longitudinal and transverse waves for an isotropic medium. We study the influence of te space-time structure of the pulse on the concentration of the contour dynamical stresses under different phases of mutual potision of the leading edge of the pulse and the contour of the section of the cavity in a massif of isotropic and anisotropic basalt. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 23, 1992, pp. 67–74.     

Propagating waves in elastic material with voids subjected to electro-magnetic interactions

Constitutive relations and field equations are developed for an elastic solid with voids subjected to electro-magnetic field. The linearized form of the relations and equations are presented separately when medium is subjected to a large magnetic field and when it is subjected to a large electric field. The possibility of propagation of time harmonic plane waves in an infinite elastic solid with voids has been explored. It is found that when the medium is subjected to large magnetic field, there exist two coupled longitudinal waves propagating with distinct speeds and a transverse wave mode. However, when the medium is subjected to a large electric field, there may propagate five basic waves comprising of four coupled longitudinal waves propagating with distinct speeds and a lone transverse wave. The effects of magnetic and electric fields are observed on the propagation characteristics of the existing waves. Under the limiting cases of frequency and for different electric conductive materials, the speeds of various waves are investigated. The phase speeds of different waves and their corresponding attenuations have been computed against the frequency parameter and depicted graphically for a specific material.     

High-frequency point sources in an inhomogeneous elastic medium

Higher approximations of ray asymptotics are investigated by the boundary-layer method. For sources that are naturally called a center of pressure and a center of rotation, direction diagrams are found for transverse and longitudinal waves, respectively, which are absent in a homogeneous medium.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 99, pp. 28–42, 1980.     

The space-time caustic of an elastic short-wave field

A smooth caustic of space-time rays is considered; the tangency of the rays and the caustic has first order everywhere. In a neighborhood of the caustic an analytical expression for the wave field is obtained.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 117, pp. 147–161, 1981.The author thanks her scientific supervisor Professor V. M. Babich for valuable help with the work and useful suggestions in discussion of the problem.     

Eigenwaves in a relativistic heat-conducting viscous fluid according to extended irreversible thermodynamics

We propose an approach to extended irreversible thermodynamics leading to a set of transport equations for the dissipative variables characterizing a relativistic heat-conducting viscous fluid. After this, we study the propagation eigenmodes showing the possible existence of three non-trivial modes: two longitudinal and one transverse. The fast longitudinal wave is a pressure wave associated to a slow longitudinal thermal dissipation wave and a transverse viscous dissipation wave. The general study of the longitudinal case is not possible. Natural special cases are analyzed in detail.