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.     

夹层圆柱壳在移动内压作用下的临界速度研究

基于夹层壳理论和三维弹性动力学理论,研究了无限长夹层圆柱壳在移动内压作用下的临界速度．首先,基于夹层壳理论,考虑夹芯的压缩和剪切变形以及面板的剪切变形,研究了轴对称简谐波在无限长夹层圆柱壳中的传播问题;其次,基于三维弹性动力学理论,将位移变量用Legendre正交多项式系表示,同时引入位置相关函数,将求解导波问题化为简单的特征值问题．利用这两种方法得到了最低模态的频散曲线,最小相速便是内压移动的临界速度．最后,用算例和数值模拟来验证方法的有效性．结果表明,两种理论得到临界速度吻合得较好;当波数较小时,两种理论得到的频散曲线吻合得很好,当k→∞时,夹层壳理论和弹性动力学理论得到的极限相速分别趋于面板和夹芯的剪切波波速．波数较小时,两种理论分析夹层圆柱壳的导波问题是有效的．数值模拟预测的临界速度与理论分析的结果吻合得很好．     

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.     

Solitary wave solutions for a general Boussinesq type fluid model

The possible solitary wave solutions for a general Boussinesq (GBQ) type fluid model are studied analytically. After proving the non-Painlevé integrability of the model, the first type of exact explicit travelling solitary wave with a special velocity selection is found by the truncated Painlevé expansion. The general solitary waves with different travelling velocities can be studied by casting the problems to the Newtonian quasi-particles moving in some proper one dimensional potential fields. For some special velocity selections, the solitary waves possess different shapes, say, the left moving solitary waves may possess different shapes and/or amplitudes with those of the right moving solitons. For some other velocities, the solitary waves are completely prohibited. There are three types of GBQ systems (GBQSs) according to the different selections of the model parameters. For the first type of GBQS, both the faster moving and lower moving solitary waves allowed but the solitary waves with“middle” velocities are prohibit. For the second type of GBQS all the slower moving solitary waves are completely prohibit while for the third type of GBQS only the slower moving solitary waves are allowed. Only the solitary waves with the almost unit velocities meet the weak non-linearity conditions.     

缓变深度分层流体中的准周期波和准孤立波

本文讨论具缓变深度二流体系统中的非线性波,该系统由一不规则底部与一水平固壁间的两层常密度无粘流体所组成.文中用约化摄动法导出了所考虑模型的变系数Korteweg-de Vries方程,并用多重尺度法求出了该方程的近似解,发现底部固壁的不规则变化将产生所谓准周期波和准孤立波.它们的周期、波速和波形将发生缓慢变化,文中给出了准周期波的周期随深度的变化关系式以及准孤立波波幅、波速随深度的变化关系式,底部水平情形和单层流体情形可看成是本文的特例.     

Propagation of plane waves in microstretch fluid

The propagation of plane harmonic waves are studied in a microstretch fluid medium. It is found that five basic waves can propagate at distinct speeds in an infinite linear homogeneous isotropic microstretch fluid. Out of these five waves, one is a longitudinal micro-rotational wave, two are coupled longitudinal waves and remaining two are coupled transverse waves. The longitudinal micro-rotational wave travels independently and is not influenced by the microstretching of the medium, while the coupled longitudinal waves arise due to the presence of microstretching and coupled transverse waves arise due to the presence of micro-rotation in the medium. Speed of propagation of all the waves are found to be complex valued and dispersive at low frequency, but almost non-dispersive at high frequency. Due to complex valued speeds of propagation, all the waves are attenuating but differently. Coupled sets of longitudinal waves reduce to a longitudinal wave of micropolar fluid in the absence of microstretching. Reflection phenomena of a set of coupled longitudinal waves incident obliquely at the free surface of a microstretch fluid half-space has been investigated. Closed formulae for the reflection coefficients are presented and computed numerically for a particular medium. The real and imaginary parts of the complex speeds of all the waves and their corresponding attenuation coefficients have also been studied numerically and depicted graphically against frequency parameter.     

The harmonic oscillations of a viscoelastic rod of triangular cross-section

Two exact solutions of the plane strain problem of the harmonic oscillations of a viscoelastic rod, the cross-section of which is a right triangle, are proposed. Either the normal displacement and the shear stress or the shear displacement and the normal stress of the side surface of the rod are given. Six dimensionless parameters which affect the dynamic deformation process are derived. Two parameters characterize the contribution of the viscous properties with respect to the elastic properties, two others define the logarithmic decrement of the longitudinal and shear harmonic waves, and two other parameters affect the wavelength of the corresponding wave and the velocity of motion of the wave front of these waves. The velocities of both types of waves and their wavelengths turn out to be greater than the velocities and wavelengths of the corresponding elastic waves. It is shown that, for certain values of the viscosity and the oscillation frequency, pseudo-resonance frequencies are possible which are higher than the resonance frequencies for an elastic medium.     

Longitudinal waves in partially saturated porous media: the effect of gas bubbles

The problem of the propagation of longitudinal waves in a liquid-saturated porous medium when there are gas bubbles present is considered. The decay factor and the phase velocity of Frenkel–Biot waves of the first and second kind are found as a function of the frequency in the linear approximation. It is shown that, in the neighbourhood of the resonance frequency of the bubbles, longitudinal Frenkel–Biot waves change their form. A wave of the first kind is transformed from a fast wave at low frequencies into a slow wave at high frequencies. The dispersion curve of a wave of the second kind consists of two branches – a “low-frequency” branch, the oscillations of which possess the classical properties, and a “high-frequency” branch, which is a weakly decaying high-velocity mode. The frequency dependences of the ratio of the mass velocities of a gas-liquid mixture and of a porous matrix, and also of the perturbations of the stress in the matrix and the pressure in the mixture, are constructed. It is shown that the “high-frequency” branch of a wave of the second kind is characterized by the in phase motion of the gas-liquid mixture and of the porous matrix, while their mass velocities are close, which explains the weak decay of this mode of oscillations. An analytical expression is obtained for the “boundary frequency”, which determines the offset of the “high-frequency” branch of the dispersion curve of the wave of the second kind.     

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.     

The propagation of elastic waves in a viscous-fluid-saturated porous medium

Elastic shock waves in a viscous-fluid-saturated porous medium are investigated. The porosity is only taken into account with respect to pores communicating with one another, and isolated pores are considered as elements of the elastic part of the porous skeleton. It is shown, using the theory of discontinuity, that in such a medium there are two types of vortex-free waves and one equivoluminal wave. Differential equations and their solution for determining the change in the wave-front intensity are obtained. The effect of the fluid viscosity and porosity on the propagation of spherical waves is demonstrated using an example.     

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].     

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.     

An Effective Model of a Fractured Medium

A homogeneous isotropic elastic medium intersected by three systems of fractures on which the jumps of stresses are proportional to displacements is considered. An effective model of this medium is described by equations differing from the respective equations of the elastic medium by additional terms. On the basis of the equations of the effective model, the wave field excited by a point source is established. An investigation of the integral representation of the wave field shows that the velocities of the longitudinal and transversal waves and of the Rayleigh wave are functions of the frequency and the wave numbers. Formulas for the phase and group velocities of these waves are derived. Bibliography: 3 titles.     

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.     

Linear Elastic Wave Propagation in Unsaturated,Unconsolidated Soils

By means of a macroscopic linear model for a poroelastic medium with three deformable components the acoustic behavior of four unconsolidated soil types, filled by water and air, is studied. Necessary material parameters are mainly gathered from the German standard DIN 4220. It provides some useful parameters, as for example, van Genuchten parameters, for thirty-one different soil types including sands, silts and clays. There appear four body waves: three longitudinal waves and one shear wave. The slowest compressional wave does only exist if at least two pore fluids appear and is driven by the capillary pressure between the pore fluids. The wave analysis yields the dependence of velocities and attenuations of these waves on the saturation and on the frequency. This is compared to the so far most frequently studied case of partially saturated sandstones. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Low-frequency acoustic characteristics of biological tissues

The laws of propagation of elastic waves of different types in biological tissues in the acoustic frequency range have been theoretically and experimentally investigated. The contributions of the imaginary and real components of the complex modulus of elasticity to the elastic wave velocity are analyzed. It is shown that in soft tissues, low-frequency elastic disturbances are propagated chiefly by shear (transverse) waves. The geometric dispersion of the elastic wave velocity has been investigated in experiments on gel model systems; the results of the measurements are in agreement with the theoretical dispersion curve.     

Non-linear reflection of a plane wave in an elastic medium

Exact formulae are derived for the reflected and refracted waves which occur for the inclined incidence of a plane horizontally polarized transverse wave of arbitrary profile on a horizontal interface between two elastic half-spaces experiencing non-linear friction when they move with respect to one another. A smooth function of general form is adopted as the friction function, which depends on the difference between the horizontal velocities of the elements of the boundaries of the half-spaces considered. It is shown that if the friction function depends non-monotonically on the relative velocity of displacement of the sides of a slit, then even when the profile of the incident wave is smooth, the reflected and refracted waves may contain strong discontinuities.     

General relations for waves in one-dimensional elastic systems

Common features inherent in waves propagating in one-dimensional elastic systems are pointed out. Local laws of energy and wave momentum transfer when the Lagrangian of an elastic system depends on the generalized coordinates and their derivatives up to the second order inclusive are presented. It is shown that in a reference system moving with the phase velocity, the ratio of the energy flux density to the wave momentum flux density is equal to the phase velocity. It is established that for systems, the behaviour of which is described by linear equations or by nonlinear equations in the unknown function, the ratio of the mean values of the energy flux density to the wave momentum density is equal to the product of the phase and group velocities of the waves.     

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.