Propagation of Seismic Waves in Block Elastic-Fluid Media. III

The propagation of seismic waves in block two- and three-dimensional media is investigated. These media are composed of identical cells in which there are several fluid blocks and one elastic block. For these media, effective models, which are anisotropic fluids, are established. Formulas for the velocities of propagation in these fluids are derived and investigated. A special investigation is carried out in the cases where the elastic block occupies almost the entire cell or where the relative volume of the elastic block is very small. Bibliography: 9 titles. Dedicated to P. V. Krauklis on the occasion of his seventieth birthday __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 308, 2004, pp. 147–160.     

Investigation of wave propagation velocities in fluid mixtures

In order to determine the wave propagation velocities in fluid mixtures, the mixtures are approximated by block structures. These structures consist of identical cells containing eight blocks. The blocks may be filled with different fluids. In block structures, the passage to the limit is carried out under the conditions that the sizes of blocks tend to zero but the relative sizes of blocks remain constant. In the general case, the average wave field satisfies the equations of anisotropic fluids. Two special cases of mixtures of two fluids are considered. In the first case, both fluids are intermixed completely. In the second case, there are periodic inclusions of one fluid into the other. In both cases, the fluid mixtures are homogeneous and isotropic, and formulas for the velocities are obtained. These formulas determine the dependence of the velocities on the percent composition and the parameters of two mixed fluids. The velocity of propagation in the fluid mixture does not exceed the greatest velocity but may be less than the least velocity in mixed fluids. Bibliography: 6 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 332, 2006, pp. 149–162.     

Estimating inequalities for velocities of wave propagation and for effective densities in fluid mixtures

Wave propagation in block fluid media is investigated on the basis of effective models, which are anisotropic fluids. For velocities of wave propagation and for effective densities, estimating inequalities are established. The propagation velocity in a fluid mixture cannot be greater than the greatest velocity in mixed fluids, but can be less than the least velocity in mixed fluids. Bibliography: 7 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 324, 2005, pp. 180–189.     

Attenuation of waves propagating in fluid mixtures

Wave propagation in fluid mixtures is investigated on the basis of effective models of block and layered media. These models are anisotropic fluids described by wave equations. In the equations, additional terms describing wave attenuation are introduced. The attenuation is related to a friction force proporitional to the difference of tangent displacements on the boundaries. Owing to attenuation, the total energy of the wave field decreases steadily and the amplitudes of waves are diminished expotentially with time, which is determined by attenuation coefficients. The attenuation coe.cients are found in the cases where two fluids are mixed completely and where the particles of one fluid are inclusions into the other. The approach suggested enables one to consider more complicated fluid mixtures as well. Bibliography: 7 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 324, 2005, pp. 148–179.     

Propagation of Seismic Waves in Block Elastic-Fluid Media. II

A two-dimensional medium consisting of alternating elastic and fluid blocks along the x and z axes is considered. For this block medium, an effective model described by a system of equations is constructed by the method of matrix averaging. An investigation of the equations of this model enables one to separate two body waves from the wave field, to construct their fronts, and to obtain expressions for their velocities along the axes. The effective model is considered in the cases where the block medium is converted to a layered elastic-fluid medium, where all the blocks are of the same size, and where an elastic or a fluid medium occupies the entire volume. Bibliography: 7 titles.__________Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 297, 2003, pp. 254–271.     

Propagation of Seismic Waves in Block Elastic-Fluid Media. I

An approach of averaging block elastic-fluid media is proposed, and an effective model for a block medium in which every cell consists of three elastic blocks and one fluid block is constructed. An investigation of the model equations shows that in this model two longitudinal waves and one wave with a concave front set propagate. The limiting cases where the fluid block is narrowed down to a point or where the fluid block occupies the whole cell are considered in the paper. Bibliography: 10 titles.__________Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 297, 2003, pp. 230–253.     

On Effective Models of Porous Stratified Media with a Sliding Contact Between Layers

In order to study wave propagation in porous layered media with a sliding contact between the elastic phases on the interfaces, effective models of these media are investigated. For these models, the front sets of four waves excited by point sources are established and formulas for the wave velocities along the axes are derived. The methods of constructing the front sets applied in this paper allow one to point out special features of these front sets such as loops and juts. The particular case where all of the layers are identical and a sliding contact occurs between layers is also considered. Bibliography: 8 titles.     

饱水孔隙介质的质量耦合波动问题

本文按照混合物理论严格地推导出了饱和孔隙介质的一般波传播理论.该理论的重要性在于包含了质量耦合作用,并为研究该问题提供了理性基础和实用方程.本文对所得方程中的系数的物理意义和热力学限制进行了讨论.通过比较认为本文的理论和Biot古典理论基本上一致.本文还对完全透水、完全不透水和具有刚性固体骨架的介质的无边界条件下的波传播问题进行了研究,得到了一些有意义的结论.     

Propagation of perturbations in fluids excited by moving sources

A large series of A.A. Dorodnicyn’s works deals with rigorous mathematical formulations and development of efficient research techniques for mathematical models used in inhomogeneous fluid dynamics. Numerous problems he studied in these directions are closely related to stratified fluid dynamics, which were addressed in a series of works having been published in this journal by this paper’s authors and their coauthors since 1980. This paper describes the results of a series of works analyzing the propagation of small perturbations in various stratified and/or uniformly rotating inviscid fluids. It is assumed that each of the fluids either occupies an unbounded lower half-space with a free surface or is a semi-infinite two-component fluid layer. The perturbations are excited by a moving source specified as a periodic plane wave traveling along the interface of the fluids. Problems for five mathematical fluid models are formulated, their explicit analytical solutions are constructed, and their existence and uniqueness are discussed. The asymptotics of the solution as t → +∞ are studied, and the long-time wave patterns developing in five fluid models are compared.     

Numerical analysis of wave propagation in fluid-filled deformable tubes

The theory of Biot describing wave propagation in fluid saturated porous media is a good effective approximation of a wave induced in a fluid-filled deformable tube. Nonetheless, it has been found that Biot's theory has shortcomings in predicting the fast P-wave velocities and the amount of intrinsic attenuation. These problems arises when complex mechanical interactions of the solid phase and the fluid phase in the micro-scale are not taken into account. In contrast, the approach proposed by Bernabe does take into account micro-scopic interaction between phases and therefore poses an interesting alternative to Biot's theory. A Wave propagating in a deformable tube saturated with a viscous fluid is a simplified model of a porous material, and therefore the study of this geometry is of great interest. By using this geometry, the results of analytical and numerical results have an easier interpretation and therefore can be compared straightforward. Using a Finite Difference viscoelastic wave propagation code, the transient response was simulated. The wave source was modified with different characteristic frequencies in order to gain information of the dispersion relation. It was found that the P-wave velocities of the simulations at sub-critical frequencies closely match those of Bernabe's solution, but at over-critical frequencies they come closer to Biot's solution. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Non-reflecting boundary conditions for 2 Danisotropic elastodynamics

Our numerical method of generating the boundary operator is based on conception of both analytical and discrete transparent boundary conditions. The elastic wave propagation in 2D orthotropic media is considered to show transparency of open circular boundary with proposed NRBC for all tested cases, even where PML approach fails. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

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.     

流变学流体的蠕动传输:食道中食物块的运动模型

研究食道中蠕动传输的流体力学.对任意的波形和任意的管道长度,建立起流变学流体蠕动传输的数学模型.用粘性流体的Ostwald-de Waele幂定律,描述非Newton流体的流动特性.解析公式化模型,详细且精确地给出食物块在食道中蠕动传输相关的一些重要性质.分析中应用了润滑理论,本研究特别适合于Reynolds数不大的情况.将食道看作环形的管道,通过食道壁周期性的收缩来传输食物块.就单个波和周期性收缩一组波的传播,研究与传输过程有关变量的变化,如压力、流速、食物颗粒轨迹以及流量等.局部压力的变化,对流变指数n有着高度的敏感性.研究结果清晰地表明,食物块在食道中蠕动传输时,Newton流体或流变学流体构成的连续流体,以组合波传播比大间隔单波传播,传输效率要高得多.     

流体饱和多孔介质中波传播问题的有限元分析

采用基于混合物理论的多孔介质模型,给出流体饱和两相多孔介质波动问题的有限元分析方法·　采用罚方法导出的有限元动力方程,时间积分可采用显式和隐式积分两种方案·　用编制的有限元程序分析了一维柱体在跃阶载荷和脉冲载荷作用下的波传播问题,得到该两种瞬态载荷作用下固体和流体相位移、速度以及固体相有效应力和孔隙压力随时间的变化关系,并对波的传播现象进行了分析·　所得结果与理论相吻合·     

Modelado numérico para estudiar interfases fluido-sólidas ante excitaciones dinámicas

This work shows the wave propagation in fluid-solid interfaces due to dynamic excitations, such interface waves are known as Scholte's waves. We studied a wide range of elastic solid materials used in engineering. The interface connects an acoustic medium (fluid) and another solid. It has been shown that by means of an analysis of diffracted waves in a fluid, it is possible to deduce the mechanical characteristics of the solid medium, specifically, its propagation velocities. For this purpose, the diffracted field of pressures and displacements, due to an initial pressure in the fluid, are expressed using boundary integral representations, which satisfy the equation of motion. The initial pressure in the fluid is represented by a Hankel's function of second kind and zero order. The solution to this problem of wave propagation is obtained by means of the Indirect Boundary Element Method, which is equivalent to the well-known Somigliana's representation theorem. The validation of the results was performed by means of the Discrete Wave Number Method. Firstly, spectra of pressures to illustrate the behavior of the fluid for each solid material considered are included, then, the Fast Fourier Transform algorithm to display the results in the time domain is applied, where the emergence of Scholte's waves and the amount of energy that they carry are highlighted.     

The Evolution of Nonlinear Wave Trains in Stratified Shear Flows

The propagation of an internal wave train in a stratified shear flow is investigated for a Boussinesq fluid in a horizontal channel. Linear effects are primarily reflected in the dispersion relation for the various modes. The phenomenon of Eckart resonance occurs for more realistic stratification profiles. The evolution of nonlinear internal wave packets is studied through a systematic perturbation analysis. A nonlinear Schrodinger equation for the envelope of the internal wave train is derived. Depending on the relative sign of the dispersive and nonlinear terms, a wave train may disperse or form an envelope soliton. The analysis demonstrates the existence of two types of critical layers: one the ordinary critical point where ū=c, while the other occurs where ū=c_g. In order to calculate the coefficients of the nonlinear Schrodinger equation a numerical code has been developed which computes the second-harmonic and induced mean motions. The existence of these envelope solitons and their dependence on environmental conditions are discussed.     

On the Rayleigh wave on a curvilinear boundary between elastic and fluid media

Along the boundary between elastic and fluid media, the surface Rayleigh wave propagates. The velocity of this wave v _R0 in the case of a plane boundary is less than the velocity of the Rayleigh wave v _R on a free plane boundary of an elastic medium and less than the velocity v _P0 in a fluid medium. To investigate the velocity v _R0 in the case of curvilinear boundaries, the propagation of Rayleigh waves under consideration along cylindrical and spherical surfaces is studied. The velocity of the Rayleigh wave depends on the curvature of the wave trajectory and the curvature in the direction perpendicular to the trajectory. Furthermore this velocity depends on the presence or absence of a fluid medium. Bibliography: 5 titles.     

Solutions to the Navier-Stokes Equations Describing Local Flows of Stratified and Compressible Fluids

A family of exact solutions of the Navier-Stokes equations is used to describe local flows of incompressible stratified and compressible fluids. For some of the flows, the coefficient of viscosity can depend on the temperature. An example of an incompressible stratified flow for which the analysis is applicable is the sheared swirling flow that is produced between two parallel plates that translate with different velocities and rotate with different angular velocities about different, but parallel, axes. The fluid may be stratified in the direction normal to the plates. These generalized von Karman flows are relevant to the study of strong local atmospheric disturbances, such as might be produced by the passage of a tornado. Also, when the coefficient of viscosity depends on the temperature, they can be used to analyze the flow of molten metals between surfaces that are in relative motion. An example of a compressible flow for which the analysis is applicable is that produced by a plane shock wave as it traverses a layer where the fluid is sheared in a direction normal to the shock.     

Sur la magnétohydrodynamique anisotrope relativiste

Summary This paper is concerned with the relativistic phenomenological theory of anisotropic magnetohydrodynamics. An anisotropic fluid scheme is defined and studied. The main system of anisotropic magnetohydroldynamics is deduced. This system may describe a collisionless anisotropic plasma embedded in a strong magnetic field. The main system is shown to yield to three types of waves as in isotropic (perfect) magnetohydrodynamics: the entropic waves, the magnetosonic waves and the Alfven waves. For the rays associated respectively to the magnetosonic and Alfven waves the fundamental property concerning the propagation of infinitesimal discontinuities of variables is established. The conditions under which the velocities of propagation of magnetosonic and Alfven waves are real are derived: these conditions imply as in the classical theory the absence of fire hose and mirror instabilities in the fluid. The study of wave cones allows, on the one hand to point out some particularities of the propagation of waves in anisotropic magnetohydrodynamics, and on the other hand to clear up the hyperbolicity character of differential operators associated to various waves.

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Entrata in Redazione il 23 aprile 1975.     

Propagation of shear waves in homogeneous and inhomogeneous fibre-reinforced media on a cylindrical Earth model

The present paper is concerned with the propagation of shear wave in a cylindrical structure comprised of two concentric media by two distinct cases. In first case, media are homogeneous and in second case, media are heterogeneous. The heterogeneities are caused due to the radial variation in both the media distinctly. The dispersion relation for shear wave propagation in homogeneous and heterogeneous, fibre-reinforced media have been derived analytically in the closed form using Debye Asymptotic Expansion and verified with the existing literature and classical results. Numerical computation and graphical demonstration have been carried out to show the remarkable effect of various parameter viz. fibre-reinforcement, heterogeneities and radii ratio of the media on the intensity of frequency of shear wave propagation against the dimensionless wave number. A comparative study between the reinforcement and isotropic material with or without heterogeneity has been discussed extensively and unravel some important peculiarities.