Propagation of Seismic Waves in Block Fluid Media

The propagation of seismic waves in block two- and three-dimensional fluid media is investigated. For these media, effective models, which are anisotropic fluids, are established. Formulas for the velocities of wave propagation in these fluid media are derived and analyzed. Special investigation is conducted in the cases where blocks with different fluids alternate along the coordinate axes or where blocks filled with a fluid are surrounded by blocks with another fluid. In both cases, the dependence of the wave velocities in the entire medium on the differences of the densities and the wave velocities in fluid blocks is studied. 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. 124–146.     

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. 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.     

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.     

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.     

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.     

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.     

On acceleration waves in anisotropic thermoelastic media taking account of finiteness of the heat propagation velocity : PMM vol. 38, n 6, 1974, pp. 1098–1104

The propagation of acceleration waves in an anisotropic thermoelastic medium is studied. It is shown that taking account of the finiteness of the heat distribution velocity results in the appearance of four kinds of accelaration waves, whose velocities and damping coefficients depend in an essential way on the direction of wave surface propagation. A comparison between the velocities and damping coefficients of plane acceleration waves in a zinc crystal, obtained with and without the finiteness of the heat propagation velocity taken into account, is presented.The papers [1, 2] are devoted to the influence of the coupling of the strain and temperature fields on the nature of wave propagation in a homogeneous isotropic body in the case of an infinite heat distribution velocity. A number of features due to coupling of the fields is obtained therein, and it is shown in particular that weak and strong discontinuities damp out, and the order of damping is determined by an exponential factor.Taking account of finiteness of the heat distribution velocity results in the appearance of two kinds of longitudinal waves whose propagation velocities depend in an essential manner on the velocity of the heat perturbation [3, 4].     

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.     

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.     

On Wave Interactions II. Explosive Resonant Triads in Fluid Mechanics

This paper considers the occurrence of explosive resonant triads in fluid mechanics. These are weakly nonlinear waves whose amplitudes become unbounded in finite time. Previous work is expanded to include interfacial flow systems with continuously varying basic velocities and densities. The first paper in this series [10] discussed the surprisingly strong singular nature of explosive triads. Many of the problems to be studied here will be found to have additional singularities, and the techniques for analyzing these difficulties will be developed. This will involve the concept of a critical layer in a fluid, a level at which a wave phase speed equals the unperturbed fluid velocity in the direction of propagation. Examples of such waves in this context are presented.     

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.     

SMITH NORMAL FORMAL OF DISTANCE MATRIX OF BLOCK GRAPHS

A connected graph, whose blocks are all cliques (of possibly varying sizes), is called a { block graph.} Let D(G) be its distance matrix. In this note, we prove that the Smith normal form of D(G) is independent of the interconnection way of blocks and give an explicit expression for the Smith normal form in the case that all cliques have the same size, which generalize the results on determinants.     

Discontinuities in hyperbolic optimal problems

Optimal problems are discussed for systems governed by hyperbolic equations in cases where the characteristic (phase) velocities depend upon the control functions. It is assumed that the phase velocity depends upon both spatial coordinates and time.The problems in question lead to certain difficulties connected with the formulation of the Weierstrass necessary condition for a minimum. Actually, strong discontinuities may arise in their solutions when we perform avariation in a strip. These discontinuities provide additional possibilities which are useful in certain minimization problems.In the first part of the paper, the case of a single quasilinear equation of the first order is discussed; an optimal problem for one-dimensional wave equation is described in the second part. In both cases, it turns out that additional information is needed, and usually this information is provided by physical arguments. This results of the paper can be generalized to the case where the number of spatial coordinates exceeds one.The author is indebted to Dr. K. G. Guderley for his very helpful comments.     

Weakly Nonlinear Waves in Nonideal Fluids

We study the propagation of weakly nonlinear waves in nonideal fluids, which exhibit mixed nonlinearity. A method of multiple scales is used to obtain a transport equation from the Navier–Stokes equations, supplemented by the equation of state for a van der Waals fluid. Effects of van der Waals parameters on the wave evolution, governed by the transport equation, are investigated.     

Wave propagation in the soil due to a falling chimney

This paper describes a simulation of wave propagation in the soil as a result of an impact. In this specific case, the dynamic load is the velocity of a falling chimney. The impacting parts of the chimney transfer their dynamic forces to the adjacent continuum. This causes vibrations in the surrounding area. In this paper, measurements of velocities in the soil caused by a falling chimney are compared with numerical solutions using a two-dimensional FEM-SBFEM model. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

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)     

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.     

Oblique Interactions between Internal Solitary Waves

In this paper, we study the oblique interaction of weakly, nonlinear, long internal gravity waves in both shallow and deep fluids. The interaction is classified as weak when where Δ₁=|c_m/c_n?cosδ|, Δ₂=|c_n/c_m?cosδ|,c_m,n, are the linear, long wave speeds for waves with mode numbers m, n, δ is the angle between the respective propagation directions, and α measures the wave amplitude. In this case, each wave is governed by its own Kortweg-de Vries (KdV) equation for a shallow fluid, or intermediate long-wave (ILW) equation for a deep fluid, and the main effect of the interaction is an 0(α) phase shift. A strong interaction (I) occurs when Δ_1,2 are 0(α), and this case is governed by two coupled Kadomtsev-Petviashvili (KP) equations for a shallow fluid, or two coupled two-dimensional ILW equations for deep fluids. A strong interaction (II) occurs when Δ₁ is 0(α), and (or vice versa), and in this case, each wave is governed by its own KdV equation for a shallow fluid, or ILW equation for a deep fluid. The main effect of the interaction is that the phase shift associated with Δ₁ leads to a local distortion of the wave speed of the mode n. When the interacting waves belong to the same mode (i.e., m = n) the general results simplify and we show that for a weak interaction the phase shift for obliquely interacting waves is always negative (positive) for (1/2+cosδ)>0(<0), while the interaction term always has the same polarity as the interacting waves.