Ultrasonic wave propagation in cancellous and cortical bone: prediction of some experimental results by Biot's theory.

Pulse transmission ultrasound was used to determine the longitudinal wave speed along the direction of trabecular alignment in 32 water-saturated anisotropic tibial bovine cancellous bone samples and in one cortical bone sample also from the bovine tibia. These results are compared to published ultrasound wave-speed data obtained from bovine femoral specimens. Nonlinear regression was used to fit Biot's theory to the data. The correlation coefficient for regression analysis between the experimental ultrasound velocities and the velocities predicted by Biot's theory was r = 0.78.     

Measurements of ultrasound velocity and attenuation in numerical anisotropic porous media compared to Biot’s and multiple scattering models

This article quantitatively investigates ultrasound propagation in numerical anisotropic porous media with finite-difference simulations in 3D. The propagation media consist of clusters of ellipsoidal scatterers randomly distributed in water, mimicking the anisotropic structure of cancellous bone. Velocities and attenuation coefficients of the ensemble-averaged transmitted wave (also known as the coherent wave) are measured in various configurations. As in real cancellous bone, one or two longitudinal modes emerge, depending on the micro-structure. The results are confronted with two standard theoretical approaches: Biot’s theory, usually invoked in porous media, and the Independent Scattering Approximation (ISA), a classical first-order approach of multiple scattering theory. On the one hand, when only one longitudinal wave is observed, it is found that at porosities higher than 90% the ISA successfully predicts the attenuation coefficient (unlike Biot’s theory), as well as the existence of negative dispersion. On the other hand, the ISA is not well suited to study two-wave propagation, unlike Biot’s model, at least as far as wave speeds are concerned. No free fitting parameters were used for the application of Biot’s theory. Finally we investigate the phase-shift between waves in the fluid and the solid structure, and compare them to Biot’s predictions of in-phase and out-of-phase motions.     

The role of surface tension in elastic wave scattering in an inhomogeneous poroelastic medium

It is well known that many porous media such as rocks have heterogeneities at nearly all scales. We applied Biot's poroelastic theory to study the propagation of elastic waves in isotropic porous matrix with spherical inclusions. It is assumed that the heterogeneity dimension exceeds significantly the pore size. Modified boundary conditions on poroelastic interface are used to take into account the surface tension effects. The effective wavenumber is calculated using the Waterman and Truell multiple scattering theory, which relates the effective wave number to the amplitude of the wave field scattered by a single inclusion. The calculations were performed for a medium containing fluid-filled cavities or porous inclusions contrasting in saturating fluid elastic properties. The results obtained show that when we consider elastic wave propagation in poroelastic medium containing soft inclusions, it is necessary to take into account the capillary pressure. The influence of the surface tension depends on the diffraction parameter and it is a maximum in the low frequency range.     

The role of surface tension in elastic wave scattering in an inhomogeneous poroelastic medium

It is well known that many porous media such as rocks have heterogeneities at nearly all scales. We applied Biot's poroelastic theory to study the propagation of elastic waves in isotropic porous matrix with spherical inclusions. It is assumed that the heterogeneity dimension exceeds significantly the pore size. Modified boundary conditions on poroelastic interface are used to take into account the surface tension effects. The effective wavenumber is calculated using the Waterman and Truell multiple scattering theory, which relates the effective wave number to the amplitude of the wave field scattered by a single inclusion. The calculations were performed for a medium containing fluid-filled cavities or porous inclusions contrasting in saturating fluid elastic properties. The results obtained show that when we consider elastic wave propagation in poroelastic medium containing soft inclusions, it is necessary to take into account the capillary pressure. The influence of the surface tension depends on the diffraction parameter and it is a maximum in the low frequency range.     

Rotation of elliptic optical beams in anisotropic media

We investigate the linear propagation of a paraxial optical beam in anisotropic media. We start from the eigenmode solution of the plane wave in the media, then subsequently derive the wave equation for the beam propagating along a general direction except the optic axes. The wave equation contains a second-order mixed derivative term originating from the anisotropy, and this term can result in the rotation of the beam-spot. The rotation effect is investigated by solving analytically the wave equation with an initial elliptical Gaussian beam for both uniaxial and biaxial media. For both media, it is found that there exists a specific direction, which is dependent on anisotropy of the media, on the cross-section perpendicular to propagation direction to determine the rotation of the beam-spot. When the major axes of the elliptical beam-spot of the input beam are parallel to the specific direction, the beam-spot will not rotate during propagation, otherwise, it will rotate with the direction and the velocity determined by input parameters of the beam.     

Numerical simulation of the dependence of quantitative ultrasonic parameters on trabecular bone microarchitecture and elastic constants

Finite-difference numerical simulation of ultrasound propagation in complex media such as cancellous bone represents a fertile alternative to analytical approaches because it can manage the complex 3D bone structure by coupling the numerical computation with 3D numerical models of bone microarchitecture obtained from high-resolution imaging modalities. The objective of this work was to assess in silico the sensitivity of ultrasound parameters to controlled changes of microarchitecture and variation of elastic constants. The simulation software uses a finite-difference approach based on the Virieux numerical scheme. An incident plane wave was propagated through a volume of bone of approximately 5 x 5 x 8 mm(3). The volumes were reconstructed from high-resolution micro-computed tomography data. An iterative numerical scenario of "virtual osteoporosis" was implemented using a dedicated image processing algorithm in order to modify the initial 3D microstructures. Numerical computations of wave propagation were performed at each step of the process. The sensitivity to bone material properties was also tested by changing the elastic constants of bone tissue. Our results suggest that ultrasonic variables (slope of the frequency-dependent attenuation coefficient and speed of sound) are mostly influenced by bone volume fraction. However, material properties and structure also appear to play a role. The impact of modifications of the stiffness coefficients remained lower than the variability caused by structural variations. This study emphasizes the potential of numerical computations tools coupled to realistic 3D structures to elucidate the physical mechanisms of interaction between ultrasound and bone structure and to assess the sensitivity of ultrasound variables to different bone properties.     

Investigation of an anisotropic tortuosity in a biot model of ultrasonic propagation in cancellous bone

The modeling of ultrasonic propagation in cancellous bone is relevant to the study of clinical bone assessment. Historical experiments revealed the importance of both the viscous effects of bone marrow and the anisotropy of the porous microstructure. Of those propagation models previously applied to cancellous bone, Biot's theory incorporates viscosity, but has only been applied in isotropic form, while Schoenberg's anisotropic model does not include viscosity. In this paper we present an approach that incorporates the merits of both models, by utilizing the tortuosity, a key parameter describing pore architecture. An angle-dependent tortuosity for a layered structure is used in Biot's theory to generate the "Stratified Biot Model" for cancellous bone, which is compared with published bone data. While the Stratified Biot model was inferior to Schoenberg's model for slow wave velocity prediction, the proposed model improved agreement fast wave velocity at high propagation angles, particularly when sorted for porosity. An attempt was made to improve the fast wave agreement at low angles by introducing an angle-dependent Young's Modulus, which, while improving the agreement of predicted fast wave velocity at low angles, degraded agreement at high angles. In this paper the utility of the tortuosity in characterizing the architecture of cancellous bone is highlighted.     

各向异性黏弹性介质伪谱法波场模拟

当地震信号通过复杂地球介质时，地层除了表现为各向异性，还表现为内在的黏弹性特征 因此，为准确描述地震波在地球介质中的传播特征，理想的地球介质模型应该能够模拟岩石 的各向异性特征和衰减特征.给出了各向异性黏弹性介质模型的波动方程，推导了伪谱法波场正演模拟的递推公式，并利用伪谱法实现了地震波波场数值模拟.表明了该介质模型中地 震波场特征与各向异性主轴方位和介质的黏滞性参数之间的关系. 关键词： 黏弹性 各向异性 伪谱法     

Application of an effective medium theory for modeling ultrasound wave propagation in healing long bones

Quantitative ultrasound has recently drawn significant interest in the monitoring of the bone healing process. Several research groups have studied ultrasound propagation in healing bones numerically, assuming callus to be a homogeneous and isotropic medium, thus neglecting the multiple scattering phenomena that occur due to the porous nature of callus. In this study, we model ultrasound wave propagation in healing long bones using an iterative effective medium approximation (IEMA), which has been shown to be significantly accurate for highly concentrated elastic mixtures. First, the effectiveness of IEMA in bone characterization is examined: (a) by comparing the theoretical phase velocities with experimental measurements in cancellous bone mimicking phantoms, and (b) by simulating wave propagation in complex healing bone geometries by using IEMA. The original material properties of cortical bone and callus were derived using serial scanning acoustic microscopy (SAM) images from previous animal studies. Guided wave analysis is performed for different healing stages and the results clearly indicate that IEMA predictions could provide supplementary information for bone assessment during the healing process. This methodology could potentially be applied in numerical studies dealing with wave propagation in composite media such as healing or osteoporotic bones in order to reduce the simulation time and simplify the study of complicated geometries with a significant porous nature.     

A Wave Automaton for Wave Propagation in Inhomogeneous Anisotropic Media

This paper presents an extension built on a hexagonal grid of the wave automaton, which was introduced in past few years for describing wave propagation in inhomogeneous media. This new method is capable of computing wave propagation in 2D anisotropic media without the need for introducing interpolating schemes. After a comparison of isotropic single scattering with analytical results using Mie theory, the method is used to compute the field scattered by one anisotropic particle for various orientations of its principal axes. Scattering by a collection of anisotropic particles is also presented.     

Stability of solitary waves in nonlinear composite media

The system of equations for planar waves in elastic composite media in the presence of anisotropy is considered. In anisotropic case two two-parametric families of solitary waves are found in an explicit form. In case of the absence of anisotropy these two families coalesce into the unique three parametric family. The solitary wave solutions are found to be orbitally stable in a certain range of their phase speeds (range of stability) both in an anisotropic as well as in an isotropic materials. It is also shown that the initial value problem for the governing equations is locally well posed which is needed to prove the stability result. The local well-posedness of the initial value problem along with stability of solitary waves implies global existence result provided the initial data lie in a neighbourhood of a stable solitary wave. This complements the previous results of blow-up for this type of equations.     

Anisotropic local-field effects of nanoparticles in plasmonic optics and magneto-optics

A theory of anisotropic optical local-field effects caused by resonantly polarizable small particles in multilayer polarizable media is developed. Considered is the model of a rectangular lattice of ellipsoidal nanoparticles with taking account of “image forces” at an interface in a layered medium. The lattice sums for anisotropic dipolar interactions are found using the Green’s function method in the quasi-point dipole approximation, and the effective polarizabilities of particles in a layer located near an interface are calculated self-consistently. The manifestation of an anisotropic local field of nanoparticles in optical radiation and propagation of evanescent waves responsible for optical near-field effects is investigated. Applications of the obtained results in the polar magneto-optical Kerr effect and reflectance anisotropy spectroscopy in propagating the polarized light along the normal to layers are considered. The resonant features in the spectra due to enhancement of the optical effects under excitation of surface (local) plasmons in nanoparticles of a noble metal are studied.     

Matrix method for obtaining Rayleigh wave equations for anisotropic media with hexagonal syngony

Equations for the existence of Rayleigh waves ona free boundary of anisotropic media of hexagonal syngony are explicitly derived by the matriciant method. The propagation of waves along the X and Y axes and the velocity of wave propagation are considered.     

Fast compressional wave attenuation and dispersion due to conversion scattering into slow shear waves in randomly heterogeneous porous media

Within the viscosity-extended Biot framework of wave propagation in porous media, the existence of a slow shear wave mode with non-vanishing velocity is predicted. It is a highly diffusive shear mode wherein the two constituent phases essentially undergo out-of-phase shear motions (slow shear wave). In order to elucidate the interaction of this wave mode with propagating wave fields in an inhomogeneous medium the process of conversion scattering from fast compressional waves into slow shear waves is analyzed using the method of statistical smoothing in randomly heterogeneous poroelastic media. The result is a complex wave number of a coherent plane compressional wave propagating in a dynamic-equivalent homogeneous medium. Analysis of the results shows that the conversion scattering process draws energy from the propagating wave and therefore leads to attenuation and phase velocity dispersion. Attenuation and dispersion characteristics are typical for a relaxation process, in this case shear stress relaxation. The mechanism of conversion scattering into the slow shear wave is associated with the development of viscous boundary layers in the transition from the viscosity-dominated to inertial regime in a macroscopically homogeneous poroelastic solid.     

Analytical methods for modeling of ultrasonic nondestructive testing of anisotropic media

Many modern structural materials exhibit anisotropic elastic behavior leading to complicated wave propagation phenomena. To ensure the reliability of ultrasonic nondestructive testing techniques, these material properties as well as the influence of microstructural inhomogeneities and the effects of interfaces on ultrasonic wave propagation have to be taken into account. In this respect, mathematical modeling provides an efficient method of assisting analysis. Two computationally efficient analytical approaches--a Gaussian beam and a point source superposition technique--are presented, which are well-suited for performing ultrasonic wave propagation and scattering simulations for anisotropic media. Results for homogeneous as well as inhomogeneous anisotropic media like composites and weld material are presented.     

各向异性磁化等离子体涂敷三维导体目标FDTD分析

把电流密度卷积-时域有限差分法(JEC-FDTD)推广应用于三维各向异性色散介质——磁化等离子体中，该算法同时解决了电磁波在各向异性和频率色散介质中传播的难题，给出了各向异性磁化等离子体中JEC-FDTD迭代公式.计算了各向异性磁化等离子体涂敷二面角反射器和球锥体后其RCS的变化情况，分析了电子回旋频率对其RCS的影响.计算结果表明磁化等离子体吸收性能更好. 关键词： 时域有限差分方法 各向异性磁化等离子体     

Acoustic wavefront imaging

We introduce a new class of experiments which provide graphic insights into the propagation of acoustic waves in anisotropic media. Simply stated, we have devised a means of observing the expanding acoustic wavefront from a point disturbance in a solid. The data may be viewed as a movie or a series of snapshots. The observed wavefronts represent the group-velocity surfaces of acoustic waves, which reflect the basic elastic anisotropy of the solid. The technique has been applied to coherent acoustic waves with frequencies in the megahertz range (at ambient temperatures) and to incoherent heat pulses in the hundred-gigahertz range (at liquid-helium temperatures). In this article, we first provide a pedagogical introduction to wave propagation in elastically anisotropic media, reviewing some early methods for visualizing acoustic waves. Next, we describe the “acoustic wavefront imaging” method and give representative results in crystals and composite materials. Finally, we show how this method relates to recent advances in phonon imaging and internal diffraction of ultrasound.     

Groundwave propagation over horizontally layered anisotropic media

Expressions for the surface impedance and admittance, and the corresponding Fresnel reflection coefficients are derived for uniform plane TM and TE waves which are obliquely incident upon plane layered anisotropic media. The permittivity and permabi-lity tensors of each layer are assumed to be symmetric, and the propagation vector of the incident wave lies in one of the planes of anisotropy. The results can be used directly in well known formulas for groundwave propagation over isotropic media, and so there is no need to rederive these formulas for the various horizontal or vertical, electrical or magnetic dipole sources. The effect of the anisotropy upon the wave tilt is also discussed.     

Dynamics of interface in three-dimensional anisotropic bistable reaction-diffusion system

This paper presents a theoretical investigation of dynamics of interface (wave front) in three-dimensional (3D) reaction-diffusion (RD) system for bistable media with anisotropy constructed by means of anisotropic surface tension. An equation of motion for the wave front is derived to carry out stability analysis of transverse perturbations, which discloses mechanism of pattern formation such as labyrinthine in 3D bistable media. Particularly, the effects of anisotropy on wave propagation are studied. It was found that, sufficiently strong anisotropy can induce dynamical instabilities and lead to breakup of the wave front. With the fast-inhibitor limit, the bistable system can further be described by a variational dynamics so that the boundary integral method is adopted to study the dynamics of wave fronts.