The reflection phenomena of quasi-vertical transverse waves in piezoelectric medium under initial stresses

The propagation of plane vertical transverse waves at an interface of a semi-infinite piezoelectric elastic medium under the influence of the initial stresses is discussed. The free surface of the piezoelectric elastic medium is considered to be adjacent to vacuum. We assumed that the piezoelectric material is anisotropic of the type of a transversely isotropic crystals (hexagonal crystal structure, class 6 mm). For an incident of vertical transverse plane wave, four types (two for the displacement and two for the electric potential) of reflected plane waves, called quasi-longitudinal (qP) and quasi-shear vertical (qSV) waves are shown to be exist. The relations governing the reflection coefficients of these reflected waves for various boundary conditions (mixed-free-fixed) are derived. It has been shown analytically that reflected coefficients of (qP) and (qSV) waves depend upon the angle of incidence, the parameters of electric potential, the material constants of the medium as well as the initial stresses presented in the medium. The numerical computations of reflection coefficients for different values of initial stresses have been carried out by computer for aluminum nitride (AlN) as an example and the results are given in the form of graphs. Finally, particular cases are considered in the absence of the initial stresses and the electric potential. Some of earlier studies have been compared to the special cases and shown good agreement with them.     

On Discontinuity Waves in Linear Piezoelectricity

A body composed of a linear piezoelectric medium is considered. It is shown that the condition of local propagation for a singular hypersurface S of any given order r, with r≥1, can be expressed in terms of a suitable acoustic tensor. This tensor does not depend on the order r and coincides with the one used for plane progressive waves in the homogeneous case. Thus, just as in Linear Elasticity, the laws of propagation of such discontinuity waves are the same as those for plane progressive waves. For any r≥1 singular hypersurfaces are characteristic for the linear piezoelectric partial differential equations, whereas for r=0 singular hypersurfaces may be non-characteristic for such equations. A condition is written which characterizes the strong waves of order 0 that are characteristic. For the latter waves the aforementioned acoustic tensor can be used to express the condition of local propagation. This revised version was published online in July 2006 with corrections to the Cover Date.     

Wave Reflection in Triclinic Crystalline Medium

In this paper, the reflection of a plane wave at a traction free boundary of a half -space composed of triclinic crystalline material is considered. It is shown that an incident plane wave generates three plane waves, namely quasi-P (qP), quasi-SV (qSV) and quasi-SH (qSH) waves governed by the propagation condition involving the acoustic tensor. A simple procedure is presented for the calculation of all the three phase velocities of these waves. It is demonstrated that the direction of particle motion is neither parallel nor perpendicular to the direction of propagation. A procedure is established for the calculation of the amplitude vector in terms of the phase velocity, the propagation vector, and the stiffness coefficients of the medium. Closed form solutions are obtained for the reflection coefficients of qP, qSV and qSH waves. Using the parameters of Vosges sandstone exhibiting triclinic symmetry, the graphical representations of the reflection coefficients due to an incident qP wave are given. It is observed that, in triclinic medium, the reflection coefficients are significantly different from those in an isotropic medium.     

Two dimensional wave problems in rotating elastic media

The rotation of an elastic medium makes it act anisotropically and dispersively. The eigenvectors for plane wave propagation are in general complex and thus the waves are elliptically polarized. In general the waves are neither pure shear nor pure compressional waves, and their speeds depend on the ratio of rotational frequency of the medium and the angular frequency of the wave.The class of problems discussed here involves waves propagating perpendicularly to the axis of rotation and in particular we discuss plane strain modes. The reflection and refraction of plane waves is considered.The plane waves are used to construct a general solution in cylindrical coordinates. The solution is given in terms of Bessel functions. The cylindrical solution is applied to scattering by circular cylinders. The problem of free oscillations is mentioned briefly.     

Plane waves in a semi-infinite fluid saturated porous medium

The field equations governing the propagation of waves in an incompressible liquid-saturated porous medium are investigated and a general solution is presented. It has been revealed that coupled longitudinal and transverse waves propagate in the porous medium. The propagation of transverse waves in the fluid phase is completely due to the interaction between the solid and fluid phases. The dispersion relationship and attenuation features are discussed. Unlike other investigations, all explicit forms of the arguments are derived. The reflection of the plane harmonic waves at the plane, traction-free boundary, which shows the influence of the dissipation on the velocity, and the attenuation coefficients of the reflected waves is studied. It is of interest that pore pressure is produced in the process of reflection, even in the case of the incidence of transverse waves.     

Propagation of Transient Acoustic Waves in Layered Porous Media: Fractional Equations for the Scattering Operators

Acoustic waves scattering from a rigid air-saturated porous medium is studied in the time domain. The medium is one dimensional and its physical parameters are depth dependent, i.e., the medium is layered. The loss and dispersion properties of the medium are due to the fluid-structure interaction induced by wave propagation. They are modeled by generalized susceptibility functions which express the memory effects in the propagation process. The wave equation is then a fractional telegraphist’s equation. The two relevant quantities are the scattering operators—transmission and reflection operators—which give the scattered fields from the incident wave. They are obtained from Volterra equations which are fractional equations for the scattering operators.     

Reflection and refraction of plane quasi-longitudinal waves at an interface of two piezoelectric media under initial stresses

This paper is devoted to study a problem of reflection and refraction of quasi-longitudinal waves under initial stresses at an interface of two anisotropic piezoelectric media with different properties. One of the two media is aluminum nitride, which is considered the down piezoelectric medium and the above medium is chosen as PZT-5H ceramics. The two piezoelectric media welded are assumed to be anisotropic of a type of a transversely isotropic crystals (hexagonal crystal structure, class 6 mm). The equations of motion and constitutive relations for the piezoelectric media have been written. Suitable boundary conditions are used to obtain the reflection and refraction coefficients. For an incidence of quasi-longitudinal plane waves, four independent-type amplitude ratios of elastic displacement components for plane waves, called quasi-longitudinal (qP) and quasi-shear vertical (qSV) waves, are shown to exist. Also, it is observed that there exist four dependent amplitude ratios of electric potential, which are proportional to the previous four types. Finally, it is found that the coefficients of reflection and refraction are functions of angle of incidence, elastic constants, piezoelectric potential parameters and the initial stresses. Numerical computations and the results obtained are depicted graphically. In the end, a particular case has been reduced from the present study. This investigation is considered important because the initial stresses in such practical problems are inevitable and may result in frequency shift, a change in the velocity of surface waves and controlling the selectivity of a filter compensation of the devices.     

Propagation of Transient Acoustic Waves in Layered Porous Media: Fractional Equations for the Scattering Operators

Acoustic waves scattering from a rigid air-saturated porous medium is studied in the time domain. The medium is one dimensional and its physical parameters are depth dependent, i.e., the medium is layered. The loss and dispersion properties of the medium are due to the fluid-structure interaction induced by wave propagation. They are modeled by generalized susceptibility functions which express the memory effects in the propagation process. The wave equation is then a fractional telegraphists equation. The two relevant quantities are the scattering operators—transmission and reflection operators—which give the scattered fields from the incident wave. They are obtained from Volterra equations which are fractional equations for the scattering operators.     

流体饱和标准线性粘弹性多孔介质中的平面波

研究了流体饱和不可压标准线性粘弹性多孔介质中平面波的传播和反射问题．在固相骨架小变形的假定下,得到了粘弹性多孔介质中波动方程的一般解,讨论了弥散关系和波的衰减特性．结果表明：在流体饱和不可压粘弹性多孔介质中,仅存在一个耦合纵波和一个耦合横波,纵波和横波的波速、衰减率等取决于孔隙流体与固相骨架间的相互作用以及固相骨架本身的粘性．同时,研究了半空间自由边界上入射波(纵波、横波)的反射问题。得到了非均匀反射波的波速、反射系数、衰减率等的表达式及其相关的数值结果．     

Wave propagation in an orthotropic micropolar elastic solid

Two-dimensional plane wave propagation in an orthotropic micropolar elastic solid is studied. There exist three types of coupled waves in xy-plane, whose velocities depend upon the angle of propagation and material parameters. A problem on reflection of these plane waves from a stress-free boundary is considered. The reflection coefficients of various reflected waves are computed numerically for a particular model of the solid. The effects of anisotropy upon the velocities and reflection coefficients are depicted graphically for different angles of propagation.     

Wave propagation in dual-phase-lag anisotropic thermoelasticity

The present paper is concerned with the propagation of plane waves in a transversely isotropic dual-phase-lag generalized thermoelastic solid half-space. The governing equations are solved in x–z plane to show the existence of three plane waves. Reflection of these plane waves from thermally insulated as well as isothermal stress-free surfaces is studied to obtain a system of three non-homogeneous equations in reflection coefficients of reflected waves. For numerical computations of speeds and reflection coefficients, a particular material is modeled as transversely isotropic dual-phase-lag generalized thermoelastic solid half-space. The speeds of plane waves are computed numerically for a certain range of the angle of propagation and are shown graphically against the angle of propagation for the cases of dual-phase-lag (DPL) thermoelasticity, coupled thermoelasticity and Lord–Shulman generalized thermoelasticity. Reflection coefficients of various reflected plane waves are computed numerically for thermally insulated as well as isothermal cases and are shown graphically against the angle of incidence for the cases of DPL thermoelasticity, coupled thermoelasticity and Lord–Shulman generalized thermoelasticity.     

Reflection and refraction of attenuated waves at boundary of elastic solid and porous solid saturated with two immiscible viscous fluids

The propagation of elastic waves is studied in a porous solid saturated with two immiscible viscous fluids.The propagation of three longitudinal waves is represented through three scalar potential functions.The lone transverse wave is presented by a vector potential function.The displacements of particles in different phases of the aggregate are defined in terms of these potential functions.It is shown that there exist three longitudinal waves and one transverse wave.The phenomena of reflection and refraction due to longitudinal and transverse waves at a plane interface between an elastic solid half-space and a porous solid half-space saturated with two immiscible viscous fluids are investigated.For the presence of viscosity in pore-fluids,the waves refracted to the porous medium attenuate in the direction normal to the interface.The ratios of the amplitudes of the reflected and refracted waves to that of the incident wave are calculated as a nonsingular system of linear algebraic equations.These amplitude ratios are used to further calculate the shares of different scattered waves in the energy of the incident wave.The modulus of the amplitude and the energy ratios with the angle of incidence are computed for a particular numerical model.The conservation of the energy across the interface is verified.The effects of variations in non-wet saturation of pores and frequencies on the energy partition are depicted graphically and discussed.     

Propagation of discontinuities against a static background

The solution of the ideal gasdynamic equations describing propagation of a shock wave initiated, for example, by the motion of a piston against an inhomogeneous static background is considered. The solution is constructed in the form of Taylor series in a special time variable which is equal to zero on the shock wave. In the case of weak shock waves divergence of the series serves as the constraint for such an approach. Then the solution is constructed by linearizing the equations about the solution with a weak discontinuity. In the case of a given background the last solution can be always found exactly by solving successively a set of transport equations, all these equations are reduced to linear ordinary differential equations. The presentation begins from the one-dimensional solutions with plane waves and ends by discussion of spatial problems.     

Mean wave propagation in a slab of one-dimensional discrete random medium

A study has been made of the propagation of time harmonic waves through a one-dimensional medium of discrete scatterers randomly positioned over a finite interval L. The random medium is modeled by a Poisson impulse process with density λ. The invariant imbedding procedure is employed to obtain a set of initial value stochastic differential equations for the field inside the medium and the reflection coefficient of the layer. By using the Markov properties of the Poisson impulse process. exact integro-differential equations of the Kolmogorov-Feller type are derived for the probability density function of the reflection coefficient and the field. When the concentration of the scatterers is low, a two variable perturbation method in small λ is used to obtain an approximate solution for the mean field. It is shown that this solution, which varies exponentially with respect to λL, agrees exactly with the mean field obtained by Feldy's approximate method.     

Wave propagation through an interface between viscoelastic media in the presence of defects

The propagation of plane harmonic waves through an interface between viscoelastic media is considered using the equations of field theory of defects, the kinematic identities for an elastic continuum with defects, and the dynamic equations of gauge theory. The reflection and refraction coefficients of elastic displacement waves and the waves of a defect field characterized by a dislocation density tensor and a defect flux tensor are determined. Dependences of the obtained quantities on the parameters of the interfacing media are analyzed.     

Propagation,reflection, and transmission of SH-waves in slightly compressible,finitely deformed elastic media

The propagation, reflection, and transmission of SH waves in slightly compressible, finitely deformed elastic media are considered in this paper. The dispersion relation for SH-wave propagation in slightly compressible, finitely deformed layer overlying a slightly compressible, finitely deformed half-space is derived. The present paper also deals with the reflection and refraction (transmission) phenomena due to the SH wave incident at the plane interface between two distinct slightly compressible, finitely deformed elastic media. The closed form expressions for the amplitude ratios of reflection and refraction coefficients of the reflected and refracted SH waves are obtained from suitable boundary conditions. For the numerical discussions, we consider the Neo-Hookean form of a strain energy function. The phase speed curves, the variations of reflection, and transmission coefficients with the angle of incidence, and the plots of the slowness sections are presented by means of graphs.     

Degenerate weakly non-linear elastic plane waves

Weakly non-linear plane waves are considered in hyperelastic crystals. Evolution equations are derived at a quadratically non-linear level for the amplitudes of quasi-longitudinal and quasi-transverse waves propagating in arbitrary anisotropic media. The form of the equations obtained depends upon the direction of propagation relative to the crystal axes. A single equation is found for all propagation directions for quasi-longitudinal waves, but a pair of coupled equations occurs for quasi-transverse waves propagating along directions of degeneracy, or acoustic axes. The coupled equations involve four material parameters but they simplify if the wave propagates along an axis of material symmetry. Thus, only two parameters arise for propagation along an axis of twofold symmetry, and one for a threefold axis. The transverse wave equations decouple if the axis is fourfold or higher. In the absence of a symmetry axis it is possible that the evolution equations of the quasi-transverse waves decouple if the third-order elastic moduli satisfy a certain identity. The theoretical results are illustrated with explicit examples.     

曲面曲率对Rayleigh波传播特性的影响

对任意形状的均匀各向同性线弹性曲面物体,用 WKB～（1）方法求解了沿曲面传播的Rayleigh表面波的运动微分方程,同时考虑了波传播方向及其垂直方向曲面曲率对波的穿透性的影, 所获波动方程的势函数解答表明,在一般情况下垂直波传播方向的曲面曲率对波的穿透深度的影响是不容忽视的.进而以同种介质平面表面情况下的Rayleigh面波的传播特性为基准,给出了曲面曲率引起波数或波速变化的解析表达式.通过理论分析和数值算例,描述了曲面上Rayleigh面波传播行为的一些基本特征.     

Wave propagation in thermally conducting linear fibre-reinforced composite materials

The propagation of plane waves in a fibre-reinforced, anisotropic, generalized thermoelastic media is discussed. The governing equations in x–y plane are solved to obtain a cubic equation in phase velocity. Three coupled waves, namely quasi-P, quasi-SV and quasi-thermal waves are shown to exist. The propagation of Rayleigh waves in stress free thermally insulated and transversely isotropic fibre-reinforced thermoelastic solid half-space is also investigated. The frequency equation is obtained for these waves. The velocities of the plane waves are shown graphically with the angle of propagation. The numerical results are also compared to those without thermal disturbances and anisotropy parameters.     

Two-dimensional stress wave analysis in incompressible elastic solids

Two-dimensional stress waves in a general incompressible elastic solid are investigated. First, basic equations for simple waves and shock waves are presented for a general strain energy function. Then the characteristic wave speeds and the associated characteristic vectors are deduced. It is shown that there usually exist two simple waves and two shock waves. Finally, two examples are given for the case of plane strain deformation and antiplane strain deformation, respectively. It is proved that, in the case of plane strain deformation the oblique reflection problem of a plane shock is not solvable in general.