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.     

Propagation of Normal Waves in an Isolated Porous Fluid-Saturated Biot Layer

For a porous fluid-saturated Biot layer with boundaries free from stresses and pressure, the wave field is found and dispersion equations are derived. The roots of the dispersion equations and the dependence of the phase velocities of the normal waves on the wave number are investigated by analytic methods. It is shown that the phase velocities of most of the normal waves decrease with increasing wave number. Special investigations are conducted in the case of bend and plate waves and their phase velocities for high and low frequencies. It is also shown that on the boundary of a porous Biot half-space, the Rayleigh wave does not always originate, and conditions for the existence of such a wave are established. Bibliography: 7 titles.     

On the propagation of elastic waves in two-phase media

At the present time a number of papers has been already devoted to the dynamics of two-phase media. One may mention the papers by Frenkel' [1], Rakhmatulin [2], Biot [3,4], Zwikker and Kosten [5], and others. However, the basic problem of the setting up of the equations of motion in two-phase media still cannot be considered solved and requires additional study and experimental verification.

This paper is concerned with the study of the simplest case of motion, which is the propagation of elastic waves in a homogeneous isotropic medium consisting of a solid and a fluid phase. The problems of the reflection of plane waves and surface waves at the free boundary of the half-space are solved. It is shown that the stress-strain relations established by Frenkel' are equivalent to the analogous relations proposed by Biot and that the equations of motion of the latter are more general.     

The phase velocity of acoustic perturbations near a vibration excitation source

The plane problem of the forced vibrations of an elastic half-space due to the action of a time-periodic force, acting along the tangent to the surface of a medium is considered. It is shown that in the near zone the phase velocity of surface acoustic perturbations varies considerably with distance from the vibration source and considerably exceeds the Rayleigh wave velocity.     

Jumps of displacements and stresses as seismic sources in Biot's medium

Wave fields excited in a homogeneous isotropic Biot medium by point sources described in terms of discontinuities of displacements and stresses are determined. The results are represented in the form of relations involving Fourier-Bessel or Mellin integrals and in the form of Stokes-type formulas. The interrelations between these representations are established. Among all possible point sources exciting Biot's medium, the elementary sources, in terms of which any complicated linear source can be described, are selected. The special case where the wave fields in the two phases of the Biot medium are independent of one another is considered, and the corresponding sources in the Biot medium are compared with the known sources in elastic and fluid media.Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 239, 1997, pp. 164–196.This work was supported by the Russian Foundation for Basic Research under grant No. 96-05-65904.     

On the problem of determining the parameters of a layered elastic medium and an impulse source

Considering the linear system of elasticity equations describing the wave propagation in the half-space ? ₊ ³ = {x ∈ ?³ | x ₃ > 0} we address the problem of determining the density and elastic parameters which are piecewise constant functions of x ₃. The shape is unknown of a point-like impulse source that excites elastic oscillations in the half-space. We show that under certain assumptions on the source shape and the parameters of the elastic medium the displacements of the boundary points of the half-space for some finite time interval (0, T) uniquely determine the normalized density (with respect to the first layer) and the elastic Lamé parameters for x ₃ ∈ [0, H], where H = H(T). We give an algorithmic procedure for constructing the required parameters.     

Motion of a Force Source over an Elastic Surface

The motion of a variable point force P(t) over the free surface of an elastic half-space is investigated with a view toward studying wave processes in an elastic medium. Special attention is given to the determination of surface wavefronts, because the detection of the latter from seismogram data can be utilized to locate the position of a source in space and time. This type of planar problem has been investigated previously, for example, in [1], but the analytical solutions of the problem of a stationary source moving with a constant velocity have been obtained only in the limit t without regard for the transient process of wave propagation.     

The Love's problem for hyperbolic thermoelasticity

In our paper we investigated the initial-boundary value problem for elastic layer situated on half space of another elastic medium. In this medium the thermomechanical interactions were taken into consideration. The system of equations with initial-boundary conditions describes the phenomenon of wave propagation with finite speed. In our problem there are two surfaces ie. free surface and contact surface between layer and half space. On the free surface are setting boundary conditions for normal and tangent surface force. We consider two types of contact between layer and half-space: rigid contact and slip contact. The initial-boundary value problem was solved by using integral transformations and Cagniard-de Hoope methods. From the solution of this problem follows that in layer and half space exist some kind of thermoelastic waves. We investigated moreover the conditions which should be fullfiled for propagation of Rayleigh and Love's type waves on the contact surface between layers and half space. The results obtained in our investigation were used in technical applications especially engineering design and diagnostics of roads and airfields. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Martin Boundary of a Killed Random Walk on a Half-Space

A complete representation of the Martin boundary of killed random walks on a half-space ℤ ^d−1×ℕ^* is obtained. In particular, it is proved that the corresponding Martin boundary is homemorphic to the half-sphere . The method is based on a combination of ratio limits theorems and large deviation techniques.     

On the problem of determining the structure of a layered medium and the shape of an impulse source

For the equation of wave propagation in the half-space ? ₊ ² + = {(x, y) ∈ ?² | y > 0} we consider the problem of determining the speed of wave propagation that depends only on the variable y and the shape of a point impulse source on the boundary of the half-space. We show that, under some assumptions on the shape of the source and the structure of the medium, both unknown functions of one variable are uniquely determined by the displacements of boundary points of the medium. We estimate stability of a solution to the problem.     

The Classical and Generalized Love Problems in a Domain of Low-Frequency Interference Waves of Signal Type. II

Computations of low-frequency interference waves of SH type are performed for the generalized Love problem. The model of a medium consists of a thin elastic layer, a thick layer, and a half-space. The layers and half-space are in rigid contact with each other. It is assumed that the waves pass through the thick layer twice, there and back. A new method, suggested by G. I. Petrashen, for computing wave fields allows us to compute the fields of SH waves and to draw conclusions concerning their behavior. The computations are carried out for < 1 and > 1, where is the ratio of the velocities in the thin and thick layers. In either case, the thin layer significantly distorts a signal transmitted deep into the medium. Bibliography: 6 titles.     

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.     

Sources of the center of dilatation and center of compression types in the Biot model

For a homogeneous isotropic model of porous Biot media, wave fields of spherically symmetric point sources are determined. The conditions under which a point source of the center of compression type can be replaced by two sources, one of which is a pair of oppositely directed forces and the other is a center of radially directed tangential forces, are obtained. Bibliography: 9 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 230, 1995, pp. 196–213. Translated by L. A. Molotkov.     

弹性固体和充满两种互不相溶粘性流体的多孔固体之间界面的反射和折射及其衰减波

在充满两种互不相溶粘性流体的多孔固体中,研究弹性波的传播.用3个数性的势函数描述3个纵波的传播,用1个矢性的势函数单独描述横波的传播.根据这些势函数,在不同的组合相中,定义出质点的位移.可以看出,可能存在3个纵波和1个横波.在一个弹性固体半空间与一个充满两种互不相溶粘性流体的多孔固体半空间之间,研究其界面上入射纵波和横波所引起的反射和折射现象.由于孔隙流体中有粘性,折射到多孔介质中的波,朝垂直界面方向偏离.将入射波引起的反射波和折射波的波幅比,作为非奇异的线性代数方程组计算.进一步通过这些波幅比,计算出各个被离散波在入射波能量中所占的份额.通过一个特殊的数值模型,计算出波幅比和能量比系数随入射角的变化.超过SV波的临界入射角,反射波P将不再出现.越过界面的能量守恒原理得到了验证.绘出了图形并对不同孔隙饱和度以及频率的变化,讨论它们对能量分配的影响.     

Boundary layer method in the problem of far propagation of surface SV-waves

The SV polarized wave field is investigated in an elastic gradient layer of constant width. A point source is situated on the boundary of the layer. Rigid contact conditions are assumed to be valid on the boundary between the layer and an elastic half-space. It is shown that the interference field in the principal approximation far from the source does not depend on the relation between the phase velocity and the transversal and longitudinal velocities in the half-space. Bibliography: 11 titles.     

Transient analysis of a potential pulse in a piezoelectric (622) crystal plate

Transient analysis of the wave propagation due to an electric potential pulse concentrated at a point on the boundary of an infinite piezoelectric plate of the crystal class (622) resting on an infinite elastic medium is considered. The electric potential is studied for an exponential pulse and step pulse. Numerical results are obtained for theβ-quartz plate that rests on aluminium half-space.     

Effective models of stratified media containing porous biot layers

Periodic stratified media in which either two porous Biot layer, or an elastic and a porous layers, or a fluid and a porous layer alternate are considered. The effective models of these media are constructed and investigated. In the case of alternating porous layers, the effective model is a generalized transversely isotropic Biot medium. In this medium, the density of the fluid phase and the mean density acquire tensor character. It is shown that the effective model of a porous-fluid medium is, on the one hand, a generalized transversely isotropic Biot medium of special type and, on the other hand, a generalization of the effective model of a stratified elastic-fluid medium.Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 239, 1997, pp. 140–163.This work was supported by the Russian Foundation for Basic Research under grant Nos. 96-01-00666 and 96-05-66207.     

Application of Spectral Methods to Inverse Dynamic Problem of Seismicity of a Stratified Medium

Under study is the wave propagation process in the half-space y₃ = 0 with the Cartesian coordinates y₁, y₂, and y₃ which is filled with an elastic medium. The parameters of the medium are discontinuous and depend only on the coordinate y₃. The wave process is induced by an external perturbation source that generates a plane wave moving from the domain y₃ > h > 0. It is proved that the direct dynamic problem is uniquely solvable in the corresponding function space, and a special presentation is found for the solution. The problem of determination of the acoustic impedance of the medium from the wave field measurements on the surface is investigated by the spectral methods of the theory of differential equations.     

Boundary control by an elastic force at one endpoint of the string in the presence of a model nonlocal boundary condition of one of four types,and its optimization

In the present paper, in terms of a generalized solution of the wave equation, we perform an exhaustive study of the problem on the boundary control by an elastic force u _x (0, t) = µ(t) at one endpoint x = 0 of a string in the presence of a model nonlocal boundary condition of one of four types relating (with the sign “+” or “?”) the values of the displacement u(x, t) or its derivative u _x (x, t) at the boundary point x = l of the string to their values at some interior point \(\mathop x\limits^ \circ \) of the string (0 < \(\mathop x\limits^ \circ \) < l). We prove necessary and sufficient conditions for the existence of such boundary controls. Under these conditions, we optimize the controls by minimizing the boundary energy integral and then write out the optimal boundary controls in closed analytic form.     

Decaying Long-Time Asymptotics for the Focusing NLS Equation with Periodic Boundary Condition

We consider the initial boundary value problem for the focusingnonlinear Schrödinger equation in the quarter plane , inthe case of decaying initial data (for , as )and Dirichlet boundary data (for ) approaching a periodic (single-frequency) background as .We first provide admissibility conditions for the normal derivativeof the solution on the boundary, under the assumption that itbehaves asymptotically in a similar (single-frequency) manner.We then show that for the range , the long-time asymptotics of the solution inside the quarterplane exhibits decaying oscillations of Zakharov–Manakovtype.