Elastic Waves In Anisotropic Media Research Articles

Beating resonance patterns and extreme power flux skewing in anisotropic elastic plates.

Elastic waves in anisotropic media can exhibit a power flux that is not collinear with the wave vector. This has notable consequences for waves guided in a plate. Through laser-ultrasonic experiments, we evidence remarkable phenomena due to slow waves in a single-crystal silicon wafer. Waves exhibiting power flux orthogonal to their wave vector are identified. A pulsed line source that excites these waves reveals a wave packet radiated parallel to the line. Furthermore, there exist precisely eight plane waves with zero power flux. These so-called zero-group-velocity modes are oriented along the crystal's principal axes. Time acts as a filter in the wave-vector domain that selects these modes. Thus, a point source leads to beating resonance patterns with moving nodal curves on the surface of the infinite plate. We observe this pattern as it emerges naturally after a pulsed excitation.

Numerical simulations of pure quasi-P-waves in orthorhombic anisotropic media

The accurate simulation of anisotropic media is critical in seismic imaging and inversion. In recent years, some scholars have dedicated efforts to the study of precise elastic waves in anisotropic media; however, it is easy to separate P-wave and S-wave from elastic wave fields in isotropic media but difficult to separate them in anisotropic media. To address this issue, others have proposed pseudo-pure-wave equations based on the theory of wave-mode separation, but shear wave interference still exists. Therefore, we derived the first-order pure quasi-P-wave equation with no shear wave component in orthorhombic anisotropic media (ORT) which is common in the Earth’s crust and has very important research value. The presence of a pseudodifferential operator in the equation poses a challenge for solving. In order to solve the pure wave equation, we decomposed the original pseudodifferential operator into an elliptic differential operator and a scalar operator, both of which are easily solvable. In addition, we extended the equation from ORT media to tilted ORT (TORT) media. The example results indicate that our pure quasi-P-wave equation can yield a more stable and accurate P-wave field. The pure wave equation we propose can be applied in reverse time migration (RTM), the least squares RTM (LSRTM), and even the full waveform inversion (FWI).

Calculation of the amplitudes of elastic waves in anisotropic media in Cartesian or ray-centred coordinates

We derive various expressions for the amplitude of the ray-theory approximation of elastic waves in heterogeneous anisotropic media, and show their mutual relations. The amplitude of a wavefield with general initial conditions can be expressed in terms of two paraxial vectors of geometrical spreading in Cartesian coordinates, and in terms of the 2×2 matrix of geometrical spreading in ray-centred coordinates. The amplitude of the Green tensor can be expressed in six different ways: (a) in terms of the paraxial vectors corresponding to two ray parameters in Cartesian coordinates, (b) in terms of the 2×2 paraxial matrices corresponding to two ray parameters in ray-centred coordinates, (c) in terms of the 3×3 upper right submatrix of the 6×6 propagator matrix of geodesic deviation in Cartesian coordinates, (d) in terms of the 2×2 upper right submatrix of the 4×4 propagator matrix of geodesic deviation in ray-centred coordinates, (e) in terms of the 3×3 matrix of the mixed second-order spatial derivatives of the characteristic function with respect to the source and receiver Cartesian coordinates, and (f) in terms of the 2×2 matrix of the mixed second-order spatial derivatives of the characteristic function with respect to the source and receiver ray-centred coordinates. The step-by-step derivation of various equivalent expressions, both known or novel, elucidates the mutual relations between these expressions.

The Double Absorbing Boundary method for a class of anisotropic elastic media

The Double Absorbing Boundary (DAB) method was recently introduced as a new approach for solving wave problems in unbounded domains. It has common features to local high-order Absorbing Boundary Conditions (ABC) on one hand, and to Perfectly Matched Layers (PML) on the other, and has relative advantages with respect to both. In the DAB method, the unbounded domain is truncated to produce a finite computational domain, which is then enclosed by a thin layer, as in the PML. Local high-order ABC are then imposed on both the inner and outer boundaries of the layer. Auxiliary variables are defined on the two boundaries as well as inside the layer, and participate in the numerical scheme. Only low order derivatives appear in the formulation, which allows the use of standard discretization methods in space and time. In previous studies, the DAB was developed for acoustic waves which are solutions to the scalar wave equation, and for elastic waves in isotropic media. Here the approach is extended to time-dependent elastic waves in anisotropic media. The geometrical configuration assumed is that of a semi-infinite wave guide, truncated via the DAB layer. Standard Finite Elements (FE) are used for space discretization and the damped Newmark scheme is used for time discretization. The problems considered here are restricted to those with periodic lateral boundary conditions, and with anisotropic media which do not support inverse modes, since removing these limitations may lead to a numerical instability. The performance of the scheme is demonstrated via numerical examples, including uniform orthotropic and layered orthotropic media.

Inversion for weak triclinic anisotropy from acoustic axes

Acoustic axes are directions in anisotropic elastic media, in which phase velocities of two or three plane waves (P, S1 or S2 waves) coincide. Acoustic axes are important, because they can cause singularities in the field of polarization vectors and anomalies in the shape of the slowness surface. The maximum number of acoustic axes in triclinic anisotropy is 16, and their directions depend on anisotropy parameters in a complicate way. Under weak anisotropy approximation this dependence simplifies and the directions of acoustic axes can be used for the inversion for anisotropy parameters. The maximum acoustic axes under weak anisotropy is 16, the minimum number of acoustic axes is zero. In the inversion, we can retrieve 13 combinations of anisotropy parameters provided we use directions of 7 acoustic axes at least. Under weak anisotropy approximation, the directions of acoustic axes are insensitive to strength of anisotropy; hence we cannot invert for absolute values of weak anisotropy parameters, but only for their relative values. Numerical tests have shown that the inversion is applicable only to very weak anisotropy with strength of less than 5%, provided that the acoustic axes used in the inversion are determined with an accuracy of 0.1∘ or better. In this case the inversion yields an average error for elastic parameters of less than 10%. In order to invert for the total set of 21 anisotropy parameters it is necessary to combine the measurements of the directions of the acoustic axes with measurements of other attributes of elastic waves in anisotropic media.

Derivation of integral equations, describing the scattering of elastic waves in anisotropic media

Integral equations, describing the scattering of elastic waves on homogeneities of finite size, located in an infinite anisotropic medium are given. It is shown that they are equivalent to differential equations

A high-order super-grid-scale absorbing layer and its application to linear hyperbolic systems

We continue the development of the super-grid-scale model initiated in [T. Colonius, H. Ran, A super-grid-scale model for simulating compressible flow on unbounded domains, J. Comput. Phys. 182 (1) (2002) 191–212] and consider its application to linear hyperbolic systems. The super-grid-scale model consists of two parts: reduction of an unbounded to a bounded domain by a smooth coordinate transformation and a damping of those scales. For linear problems the super-grid scales are analogous to spurious numerical waves. We damp these waves by high-order undivided differences. We compute reflection coefficients for different orders of the damping and find that significant improvements are obtained when high-order damping is used. In numerical experiments with Maxwell’s equations, we show that when the damping is of high order, the error from the boundary condition converges at the order of the interior scheme. We also demonstrate that the new method achieves perfectly matched layer-like accuracy. When applied to linear hyperbolic systems the stability of the super-grid-scale method follows from its construction. This makes our method particularly suitable for problems for which perfectly matched layers are unstable. We present results for two such problems: elastic waves in anisotropic media and isotropic elastic waves in wave guides with traction-free surfaces.

2.5D modelling of elastic waves in transversely isotropic media using the spectral element method

The spectral-element method provides an accurate alternative to the finite-element method for modelling elastic waves in anisotropic media. With the aim of reducing the high computational overheads of 3D modelling, we have implemented 2.5D spectral-element modelling of elastic waves, initially for vertically transversely isotropic media, and then extended to a tilted transversely isotropic medium with a dipping symmetry-axis. We have investigated different categories of absorbing boundaries to minimise artificial boundary reflections, including viscous boundary conditions and two distinct 2.5D formulations of perfectly matched layers (PMLs). Both PML methods use complex coordinate stretching, but the first method is only applicable in the frequency-domain. The other method uses the decomposed gradient operator and can be applied in both the time-domain and frequency-domain. Traditional 2D modelling exhibits cylindrical wave propagation characteristics from a line source. Our 2.5D modelling results, presented as images of seismic wavefields, illustrate the more realistic spherical propagation from a point source.

2.5D modelling of elastic waves in anisotropic media using the spectral-element method – a preliminary investigation

The spectral-element method provides an accurate alternative to the finite element method for modelling elastic waves in anisotropic media. However its high computational overheads make it impractical for inversion. With the aim of reducing the overheads, we have implemented 2.5D spectral-element modelling of elastic waves, initially for vertically transversely isotropic (VTI) media. We have incorporated perfectly matched layers to minimise artificial boundary reflections, using the split gradient operator for the time domain and complex coordinate stretching for the frequency domain. Traditional 2D modelling does not reflect the true coupling between wave modes. Our preliminary results, presented as images of seismic waveforms, illustrate the expected coupling. We also present an intuitive explanation of our unique implementation of the spectral-element method.

Non-linear seismic wave propagation in anisotropic media using the flux-corrected transport technique

Both anisotropy and non-linearity of rocks are important properties related to the elastic wave propagation in the Earth. However, in the literature, there has been a continuous increase in the interest of anisotropy, but very few studies of non-linearity are reported. In this paper, we aim to tackle this imbalance by examining propagation characteristics of non-linear elastic waves in anisotropic media. As analytic solutions to anisotropic wave equation in non-linear media (or non-linear wave equation in anisotropic media) are not readily available, we use a numerical-based approach. However, we realize that numerical modelling of non-linear seismic waves suffers from problems such as steep gradients, shocks and unphysical oscillations. Accordingly, some special treatments have been presented in the literature to reduce these problems. In this paper, we present a second-order central finite difference scheme based on the modified flux-corrected transport (FCT) technique, and we also present the stability criterion for the FCT. Our modelling results show that the product of strength of non-linearity and the initial amplitudes of the seismic source is the main factor influencing propagation characteristics accumulated with traveltimes. Moreover, the effects of non-linear elastic wave propagation can be distorted by the initial frequency of the seismic source and become enhanced by the presence of anisotropy. Similarly, the effects induced by anisotropy are enhanced in non-linear media. Both in isotropic and vertical symmetry axis media, the non-linear effects, such as the waveform aberration (or waveform distortion), the resonant peak shift and the generation of harmonics, can be clearly seen, and the interaction between anisotropy and non-linearity is also obvious. In conclusion, both the anisotropy and the non-linearity should be considered to investigate characteristics of elastic wave propagation in solid media, especially when the source is strong (e.g. in source regions).

A new absorbing layer for elastic waves

A new perfectly matched layer (PML) for the simulation of elastic waves in anisotropic media on an unbounded domain is constructed. Theoretical and numerical results, showing that the stability properties of the present layer are better than previously suggested layers, are presented. In addition, the layer can be formulated with fewer auxiliary variables than the split-field PML.

Some basic properties of bulk elastic waves in anisotropic media

A short survey of some basic results in the theory of bulk elastic waves in anisotropic media is presented. A series of general properties of phase speed branches is deduced. Main features of the theory of acoustic axes are briefly discussed. Bulk wave modes in a semi-infinite medium with a free surface are considered: exceptional waves, the reflection problem and limiting inhomogeneous wave superposition are described.

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.

Theory of perfectly matched layer for elastic waves and their applications in cylindrical and spherical coordinates

Recently, a general theory of perfectly matched layer (PML) is developed for elastic waves in anisotropic media and is applied to waves in cylindrical and spherical coordinates through an improved scheme of complex coordinates [Liu, J. Acoust. Soc. Am. 105, 2075–2084 (1999)]. As is known for electromagnetic waves, Berenger’s original PML scheme does not apply to cylindrical and spherical coordinates. The straightforward extension of the complex coordinates for elastic waves to cylindrical and spherical coordinates requires extra unknowns for time-domain solutions, wasting computer memory and computation time. The main idea of the improved scheme in this work is the use of integrated complex variables. It is shown that for three-dimensional cylindrical and spherical coordinates, this improved PML scheme requires no more unknowns than in Cartesian coordinates. The number of unknowns can be further reduced through the use of symmetry in the partial differential equations. The PML scheme allows an arbitrary inhomogeneity in the medium, and is suitable for numerical solutions of wave equations by finite-difference, finite-element, and pseudospectral methods for elastic waves in inhomogeneous media with cylindrical and spherical structures. Finite-difference time-domain (FDTD) results are shown to demonstrate the efficacy of the PML absorbing boundary condition.

Electromagnetic shock waves and their structure in anisotropic magnetic materials

The analogy between non-linear electromagnetic waves in magnetizable media and non-linear elastic waves in anisotropic media is justified and used. The analogy occurs when dispersion and dissipation are ignored. By using existing results [1] one can therefore immediately formulate all that relates to investigating relations in Riemann waves and electromagnetic shock waves. To describe the structure of electromagnetic shock waves in magnetic materials the Landau-Lifshits equation is employed, which differs considerably from the relations used to describe the structure of shock waves in elastic media. A consequence of this is that the set of permissible shock waves (i.e. possessing a structure) acquires a complex structure and differs considerably from the analogous set for elastic shock waves. It is shown below, in particular, that the set of permissible electromagnetic shock waves is not the same as the set of initially evolution discontinuities. The requirement that a structure should exist distinguishes, on certain parts of the shock adiabat, a set which is a dashed line with a very short dash length. In addition, there is a large number of individual points on the shock adiabat of the electromagnetic shock waves. Each point corresponds to a discontinuity with a separate velocity of motion, recalling the slow-combustion front in gas dynamics.

An Exact Method for Analyzing Elastic Waves in Anisotropic Media Excited by Harmonic Line Loads.

An exact matrix formulation for analyzing wave fields generated by harmonic line loads in anisotropic media is presented. In this formulation, a Fourier transform with respect to spatial coordinates is employed. An efficient technique to evaluate the inverse integral is implemented. Using this technique, the inverse integration in one direction is carried out analytically and the displacement is subsequently expressed explicitly as a one-dimensional integration. A quadrature scheme and complex path is then used to evaluate the integration to obtain the displacement in the spatial domain for both pure elastic materials and materials with dissipation. Numerical examples are presented to demonstrate the validity of the present method, and wave fields for the displacement are investigated for isotropic and anisotropic media.

Elastic waves in infinite and semi-infinite anisotropic media

The general theory of elastic waves in anisotropic media is briefly reviewed. The present status of existence considerations for surface waves is discussed. The main concepts involved in determining the resonant properties of reflection (transmission) near the leaky wave branch are introduced both for plane waves and for bounded beams. The exceptional waves and their relation to the degeneracy of phase velocities of bulk waves are discussed. Finally, the classification of acoustic axes, comprising all possible types of contact geometry of degenerate velocity sheets and the corresponding polarization singularities, is reviewed.

On Green’s functions for elastic waves in anisotropic media

The Green’s function, as a temporal Fourier transform of the point-source solution of the wave equation in an infinite medium, is obtained for the homogeneous, transversely isotropic solids, such as fiber composites or directionally solidificated steels of the columnar-grained polycrystalline structure. The surface integrals, representing the exact point-source solutions for the quasilongitudinal, quasitransverse, and purely transverse horizontally polarized shear waves, are approximated in terms of elementary functions, given that the elastic parameters characterizing the deviation of the solid from the isotropic medium are relatively small (weak anisotropy approximation). A brief discussion of the main features of the solution and a comparison to well-known eikonal (ray) approximations are presented.

Criteria for elastic waves in anisotropic media — a consolidation

The geometry relating to the tangent plane at a stationary point on a surface has been used to re-examine various criteria for the existence of parabolic points (inflexions in 2D-sections) on the outermost sheet of the slowness surface for elastic waves in anisotropic media. Previous results obtained by the authors, exact [4,5], and approximate [6], are related in detail. Further approximations, based on geometrical properties, are derived; one of these proves equivalent to a sufficient condition first applied by McCurdy [8]. Numerical investigation shows that, over a wide range of anisotropy, the simply applied approximate criterion of [6] is sufficient and within the accuracy of observation of the elastic stiffnesses.

Asymptotic equipartition of kinetic and strain energy for elastic waves in anisotropic media

Asymptotic equipartition of kinetic and strain energy for elastic waves in anisotropic media