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     

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.     

Possibility of Love wave propagation in a porous layer under the effect of linearly varying directional rigidities

The present paper investigates the Love wave propagation in an anisotropic porous layer under the effect of rigid boundary. Effect of initial stresses on the propagation of Love waves in a fluid saturated, anisotropic, porous layer having linear variation in directional rigidities lying in contact over a pre-stressed, inhomogeneous elastic half-space has also been considered. The dispersion equation of phase velocity has been derived and the influence of medium characteristic such as porosity, rigid boundary, initial stress, anisotropy and inhomogeneity over it has been discussed. The velocities of Love waves have been calculated numerically as a function of KH (where K is the wave number and H is the thickness of the layer) and are presented in a number of graphs.     

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.     

Love waves in a fluid-saturated porous layer under a rigid boundary and lying over an elastic half-space under gravity

In this paper, mathematical modeling of the propagation of Love waves in a fluid-saturated porous layer under a rigid boundary and lying over an elastic half-space under gravity has been considered. The equations of motion have been formulated separately for different media under suitable boundary conditions at the interface of porous layer, elastic half-space under gravity and rigid layer. Following Biot, the frequency equation has been derived which contain Whittaker’s function and its derivative that have been expanded asymptotically up to second term (for approximate result) for large argument due to small values of Biot’s gravity parameter (varying from 0 to 1). The effect of porosity and gravity of the layers in the propagation of Love waves has been studied. The effect of hydrostatic initial stress generated due to gravity in the half-space has also been shown in the phase velocity of Love waves. The phase velocity of Love waves for first two modes has been presented graphically. Frequency equations have also been derived for some particular cases, which are in perfect agreement with standard results. Subsequently the lower and upper bounds of Love wave speed have also been discussed.     

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.     

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.     

不规则界面对扭转表面波在初应力各向异性多孔弹性层中传播的影响

研究了扭转表面波在一个半无限非均匀半空间中的传播,半空间上覆盖着具有初始应力的各向异性多孔弹性层,弹性层的刚度和密度线性地变化,造成了界面的不规则性．半空间中界面的不规则性,用一个矩形形式表示．可以发现,扭转表面波在这样假定的介质中传播,得到了没有不规则性时的扭转表面波的速度方程．还可以发现,对于均匀半空间覆盖的层状介质,扭转表面波的速度与Love波的速度相一致．     

Surface acoustic waves in the testing of layered media. The waves’ sensitivity to variations in the properties of the individual layers

Research on the use of surface acoustic waves for the nondestructive testing of layered media is reviewed. A model to describe horizontally polarized surface acoustic waves in layered anisotropic (monoclinic) media is constructed. A modified transfer-matrix method is developed to obtain a solution. Non-canonical type waves with horizontal transverse polarization are investigated. Dispersion curves are constructed for a multilayer composite in contact with an anisotropic half-space. It is shown that the variation of the physical characteristics and the geometry of any of the internal layers leads to a variation in the dispersion curves. This opens up the possibility of using dispersion analysis for the nondestructive testing of the properties of the individual layers.     

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.     

Soliton-like lamb waves

The velocity and polarization of acoustic Lamb waves, propagating in the directions of elastic symmetry of single-layer and double-layer anisotropic media at vanishingly low frequencies (soliton-like waves), are investigated. The method of fundamental matrices is used to construct solutions. The conditions for soliton-like Lamb waves to exist are analysed.     

Shock Waves in Anisotropic Cylinders

We study small-amplitude longitudinal and torsional shock waves in circular cylinders consisting of an anisotropic medium such that the velocities of the longitudinal and torsional waves are close to each other. Previously, simple waves were considered in the same situation and conditions were found for these waves to overturn and for the corresponding shock waves to form. Here we present the study of shock waves: the shock adiabat and the evolutionary conditions. The results obtained can also be related to shock waves in unbounded media with quadratic nonlinearity.     

Generalization of the method of functionally invariant solutions for dynamic problems of the plane theory of elasticity of anisotropic media

A new approach for constructing functionally invariant solutions for dynamic problems of the plane theory of elasticity of anisotropic media is proposed. Solutions of the equations of motion in displacements and potentials, which express plane waves and waves from a point source, and also complex solutions of a general type are obtained and investigated. The problem of the reflection of plane waves from the boundary of a half-space is solved for comparison with earlier results [1]. The solutions obtained agree with the physical meaning of the problems and with the solutions for isotropic media.     

Instability of solitary waves in non-linear composite media

A problem on the dynamic instability of soliton solutions (solitary waves) of Hamilton's equations, describing plane waves in non-linear elastic composite media with or without anisotropy, is considered. In the anisotropic case, there are two two-parameter families of solitary waves: fast and slow and, when there is no anisotropy, there is one three-parameter family. A classification of the instability of solitary waves of the fast family in the anisotropic case and of representatives of families of solitary waves, the velocities of which lie outside of the range of stability when there is anisotropy and when there is no anisotropy, is given on the basis of a numerical solution of a Cauchy problem for the model equations. In this paper, the fundamental equations describing plane waves in non-linear, anisotropic, elastic composites are derived, the Hamilton form of the basic equations is presented, the symmetries in the anisotropic and isotropic cases are considered, the conserved quantities and the soliton solutions, that is, the solitary waves are presented, the nature of the instability of representatives of all three families is investigated, brief formulation of the results is presented and problems of the instability of the fast family in the anisotropic case and of representatives of the families, the velocities of which lie outside of the range of stability in the presence and absence of anisotropy (explosive instability), are discussed.     

Love waves with a transverse structure

For an arbitrary layered isotropic structure, new exact solutions of the elastodynamic problem for the propagation of surface waves are presented. These solutions describe waves with rectilinear wave fronts propagating at the phase velocities of common SH-polarized Love waves. They linearly depend on a lateral transverse variable and, in addition to being standardly SH-polarized, have a longitudinally polarized anomalous component. The construction uses the assumption of the existence of standard Love waves. It is based on a potential representation of the wavefield and is quite elementary.     

Existence,blow-up and exponential decay for a nonlinear Love equation associated with Dirichlet conditions

In this paper we consider a nonlinear Love equation associated with Dirichlet conditions. First, under suitable conditions, the existence of a unique local weak solution is proved. Next, a blow up result for solutions with negative initial energy is also established. Finally, a sufficient condition guaranteeing the global existence and exponential decay of weak solutions is given. The proofs are based on the linearization method, the Galerkin method associated with a priori estimates, weak convergence, compactness techniques and the construction of a suitable Lyapunov functional. To our knowledge, there has been no decay or blow up result for equations of Love waves or Love type waves before.     

Shape identification of anisotropic diffraction gratings for TM-polarized electromagnetic waves

This paper concerns the shape identification problem of anisotropic periodic structures which are known as diffraction gratings. We study the so-called Factorization method as a tool for reconstructing the anisotropic periodic media from measured spectral data involving scattered electromagnetic waves in TM modes. This sampling method provides a simple criterion to compute a picture of shape of diffraction gratings in a rapid way. We propose a rigorous analysis for the method as well as numerical experiments to examine its performance.     

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.     

Explicit asymptotic modelling of transient Love waves propagated along a thin coating

An explicit asymptotic model for transient Love waves is derived from the exact equations of anti-plane elasticity. The perturbation procedure relies upon the slow decay of low-frequency Love waves to approximate the displacement field in the substrate by a power series in the depth coordinate. When appropriate decay conditions are imposed on the series, one obtains a model equation governing the displacement at the interface between the coating and the substrate. Unusually, the model equation contains a term with a pseudo-differential operator. This result is confirmed and interpreted by analysing the exact solution obtained by integral transforms. The performance of the derived model is illustrated by numerical examples