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.     

Reflection and refraction of plane waves under initial stresses at an interface

The paper deals with the phenomena of reflection and refraction of plane elastic waves at a plane interface between two semi-infinite elastic solid media in contact, when both the media are initially stressed. It has been shown analytically that both reflected and refracted P and SV waves depend on initial stresses present in the media. The numerical values of reflection and refraction coefficients for different initial stresses and the angle of incidence have been calculated by computer and the results are given in the form of graphs. Many results are found in the paper which are not seen in initially stress-free media.     

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.     

Reflection and transmission of quasi-plane waves at the interface of piezoelectric semiconductors with initial stresses

We examine the reflection and transmission phenomena of quasi-longitudinal plane(QP) waves in an AlN-ZnO laminated composite structure. The structure is designed under the influence of the initial stresses in which one carrier piezoelectric semiconductor(PSC) half-space is in welded contact with another PSC half-space.The secular equations in the transversely isotropic PSC material are derived from the general dynamic equation, taking the initial stresses into consideration. It is shown that the incident quasi-longitudinal wave(QP-mode) at the interface generates four types of reflected and transmitted waves, namely, QP wave, quasi-transverse(QSV) wave,electric-acoustic(EA) wave, and carrier plane(CP) wave. The algebraic equations are obtained by imposing the boundary conditions on the common interface of the laminated structure. Reflection and transmission coefficients of waves are obtained by implementing Cramer's rule. Profound impacts of the initial stresses and exterior electric biasing field on the reflection and transmission coefficients of waves are investigated and presented graphically.     

Propagation of plane waves in thermoelastic cubic crystal material with two relaxation times

A problem concerned with the reflection and refraction of thermoelastic plane waves at an imperfect interface between two generalized thermally conducting cubic crystal solid half-spaces of different elastic and thermal properties with two relaxation times has been investigated. The generalized thermoelastic theory with two relaxation times developed by Green and Lindsay has been used to study the problem. The expressions for the reflection and refraction coefficients which are the ratios of the amplitudes of reflected and refracted waves to the amplitude of incident waves are obtained for an imperfect boundary and deduced for normal stiffness, transverse stiffness, thermal contact conductance, slip and welded boundaries. Amplitude ratios of different reflected and refracted waves for different boundaries with angle of emergence have been compared graphically for different incident waves. It is observed that the amplitude ratios of reflected and refracted waves are affected by the stiffness and thermal properties of the media.     

Interfacial imperfection on reflection and transmission of plane waves in anisotropic micropolar media

This study is concerned with the reflection and transmission of plane waves at an imperfectly bonded interface between two orthotropic micropolar elastic half-spaces with different elastic and micropolar properties. There exist three types of coupled waves in xy-plane. The reflection and transmission coefficients of quasi-longitudinal (QLD) wave, quasi-coupled transverse microrotational (QCTM) wave and quasi-coupled transverse displacement (QCTD) wave have been derived for different incidence waves and deduced for normal force stiffness, transverse force stiffness, transverse couple stiffness and perfect bonding. The numerical values of modules of the reflection and transmission coefficients are presented graphically with the angle of incidence for orthotropic micropolar medium (MOS) and isotropic micrpolar medium (MIS). Some particular cases of interest have been deduced from the present investigation.     

Reflection and refraction of plane waves on the interface of two anisotropic media

We use generalized functionally invariant solutions [1] of the equations of motion to obtain and analytically study solutions of the plane problem of reflection and refraction of plane waves on the interface of two anisotropic media with four elastic constants depending on the angles of incidence of primary waves for various relations between the elastic constants of the contacting media. For the primary waves, we find the ranges of incidence angles for which real and complex secondary waves are excited. We study all possible combinations of the distribution of phase velocities and reflection and refraction angles in detail and obtain conditions characterizing the directions of the energy flux vectors for the primary and secondary waves depending on the incidence angles of the primary waves for different relations between the elastic constants of the contacting media, which satisfy the necessary and sufficient conditions for the elastic energy form to be positive definite.     

The influence of the viscosity and the magnetic field on reflection and transmission of waves at interface between magneto-viscoelastic materials

The influence of the viscosity on reflection and refraction of plane shear elastic waves in two magnetized semi-infinite media is investigated. The numerical results for the absolute values of the reflection and refraction coefficients and their relative changes for a particular choice of the media are presented graphically. The relative changes of these coefficients are calculated for two special orientations of the magnetic field. It is found that the absolute values of these coefficients are not only functions of the angle of incidence but they are also functions of both the large primary magnetic field and the viscosity of the media. They also vary with the orientation of the magnetic field. Finally, we show that the results of earlier works could be obtained here as particular cases. Finally, this study is regarded an attempt to accommodate magnetic field in visco-elastic media where reflection and refraction of plane shear waves are considered. Results which are obtained in this investigation useful for practical applications or for understanding some aspects of physical acoustic.     

Discontinuous wave fronts propagation in anisotropic layered media

The problem of dynamic interaction of wave phase fronts with anisotropic elastic media interfaces is considered. A technique based on joint use of the ray theory, locally plane approach and theory of stereomechanical impact is elaborated. It is employed for the investigation of discontinuous waves propagation in anisotropic tectonic structures. The cases of interaction of quasi-longitudinal and quasi-shear discontinuous waves with the interfaces separating different anisotropic elastic media are treated. The issues are considered which are associated with the wave front surfaces bifurcations, generation of their singularities and caustics, as well as with stress concentration and formation of zones where the stresses tend to infinity.     

Reflection and transmission of elastic waves at an elastic/porous solid saturated by two immiscible fluids

Wave propagation in a porous elastic medium saturated by two immiscible fluids is investigated. It is shown that there exist three dilatational waves and one transverse wave propagating with different velocities. It is found that the velocities of all the three longitudinal waves are influenced by the capillary pressure, while the velocity of transverse wave does not at all. The problem of reflection and refraction phenomena due to longitudinal and transverse wave incident obliquely at a plane interface between uniform elastic solid half-space and porous elastic half-space saturated by two immiscible fluids has been analyzed. The amplitude ratios of various reflected and refracted waves are found to be continuous functions of the angle of incidence. Expression of energy ratios of various reflected and refracted waves are derived in closed form. The amplitude ratios and energy ratios have been computed numerically for a particular model and the results obtained are depicted graphically. It is verified that during transmission there is no dissipation of energy at the interface. Some particular cases have also been reduced from the present formulation.     

Effects of initial stress on the reflection and transmission waves at the interface between two piezoelectric half spaces

The effects of initial stress on the reflection and transmission waves at the interface between two piezoelectric half spaces are studied in this paper. First, the secular equations in the traverse isotropic piezoelectric half space are derived from the general dynamic equation with initial stress taken into consideration. Then, the interface conditions that displacement, stress, electric potential, and electric displacement are continuous across interface are required to be satisfied by three sets of coupled waves, namely, quasi-longitudinal wave, quasi-transverse wave and the electric–acoustic wave. The algebraic equations resulting from the interface conditions are solved to obtain the amplitude ratio of various waves and furthermore the energy reflection and transmission coefficients of various waves. The numerical results are shown graphically and the effects of initial stress are discussed.     

Reflection of three-dimensional plane waves at the free surface of a rotating triclinic half-space under the context of generalized thermoelasticity

The reflection of three-dimensional(3D) plane waves in a highly anisotropic(triclinic) medium under the context of generalized thermoelasticity is studied. The thermoelastic nature of the 3D plane waves in an anisotropic medium is investigated in the perspective of the three-phase-lag(TPL), dual-phase-lag(DPL), Green-Naghdi-III(GNIII), Lord-Shulman(LS), and classical coupled(CL) theories. The reflection coefficients and energy ratios for all the reflected waves are obtained in a mathematical form. The rotational effects on the reflection characteristics of the 3D waves are discussed under the context of generalized thermoelasticity. Comparative analyses for the reflection coefficients of the waves among these generalized thermoelastic theories are performed. The energy ratios for each of the reflected waves establish the energy conservation law in the reflection phenomena of the plane waves. The highly anisotropic materials along with the rotation may have a significant role in the phenomenon of the reflection behavior of the 3D waves. Numerical computations are performed for the graphical representation of the study.     

Reflection and transmission of plane waves at the interface of an elastic half-space and a fractional order thermoelastic half-space

The problem of reflection and transmission due to longitudinal and transverse waves incident obliquely at a plane interface between uniform elastic solid half-space and fractional order thermoelastic solid half-space has been studied. It is found that the amplitude ratios of various reflected and refracted waves are functions of angle of incidence and frequency of incident wave and are influenced by the fractional order thermoelastic properties of media. The expressions of amplitude ratios and energy ratios have been computed numerically for a particular model. The variation of amplitude and energy ratios with angle of incidence is shown graphically. The conservation of energy at the interface is verified.     

Seismic Reflection from an Interface Between an Elastic Solid and a Fractured Porous Medium with Partial Saturation

The theory of Tuncay and Corapcioglu (Transp Porous Media 23:237–258, 1996a) has been employed to investigate the possibility of plane wave propagation in a fractured porous medium containing two immiscible fluids. Solid phase of the porous medium is assumed to be linearly elastic, isotropic and the fractures are assumed to be distributed isotropically throughout the medium. It has been shown that there can exist four compressional waves and one rotational wave. The phase speeds of these waves are found to be affected by the presence of fractures, in general. Of the four compressional waves, one arises due to the presence of fractures in the medium and the remaining three are those encountered by Tuncay and Corapcioglu (J Appl Mech 64:313–319, 1997). Reflection and transmission phenomena at a plane interface between a uniform elastic half-space and a fractured porous half-space containing two immiscible fluids, are analyzed due to incidence of plane longitudinal/transverse wave from uniform elastic half-space. Variation of modulus of amplitude and energy ratios with the angle of incidence are computed numerically by taking the elastic half-space as granite and the fractured porous half-space as sandstone material containing non-viscous wetting and non-wetting fluid phases. The results obtained in case of porous half-space with fractures, are compared graphically with those in case of porous half-space without fractures. It is found that the presence of fractures in the porous half-space do affect the reflection/transmission of waves, which is responsible for raising the reflection and lowering the transmission coefficients.     

Reflection of plane waves under initial stresses at a free surface

The paper deals with the phenomenon of reflection of plane elastic waves in a free surface when the medium is initially stressed. It has been shown analytically that the reflected P and SV waves depend on initial stresses present in the medium. The numerical values of reflection coefficients for different initial stresses and the angle of incidence have been calculated by the Computer I.C.L. 1901-A and the results are given in the form of graphs. Many interesting results are found in the paper which are not seen in an initially stress-free medium.     

Propagation of plane waves at the imperfect boundary of elastic and electro-microstretch generalized thermoelastic solids

The problem of reflection and transmission of plane waves incident on the contact surface of an elastic solid and an electro-microstretch generalized thermoelastic solid is discussed. It is found that there exist five reflected waves, i.e., longitudinal displacement （LD） wave, thermal （T） wave, longitudinal microstretch （LM） wave and two coupled transverse displacement and microrotational （CD（I） and CD（II）） waves in the electro-microstretch generalized thermoelastic solid, and two transmitted waves, i.e., longitudinal （P） and transverse （SV） waves in the elastic solid. The amplitude ratios of different reflected and transmitted waves are obtained for an imperfect boundary and deduced for normal force stiffness, transverse force stiffness, and perfect bonding. The variations of amplitude ratios with incidence angles have been depicted graphically for the LD wave and the CD（I） wave. It is noticed that the amplitude ratios of reflected and transmitted waves are affected by the stiffness, electric field, stretch, and thermal properties of the media. Some particular interest cases have been deduced from the present investigations.     

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.     

The mathematical model of reflection and refraction of longitudinal waves in thermo-piezoelectric materials

In this paper, the basic equations of motion, of Gauss and of heat conduction, together with constitutive relations for pyro- and piezoelectric media, are presented. Three thermoelastic theories are considered: classical dynamical coupled theory, the Lord–Shulman theory with one relaxation time and Green and Lindsay theory with two relaxation times. For incident elastic longitudinal, potential electric and thermal waves, referred to as qP, φ-mode and T-mode waves, which impinge upon the interface between two different transversal isotropic media, reflection and refraction coefficients are obtained by solving a set of linear algebraic equations. A case study is investigated: a system formed by two semi-infinite, hexagonal symmetric, pyroelectric–piezoelectric media, namely Cadmium Selenide (CdSe) and Barium Titanate (BaTiO₃). Numerical results for the reflection and refraction coefficients are obtained, and their behavior versus the incidence angle is analyzed. The interaction with the interface give rises to different kinds of reflected and refracted waves: (i) two reflected elastic waves in the first medium, one longitudinal (qP-wave) and the other transversal (qSV-wave), and a similar situation for the refracted waves in the second medium; (ii) two reflected potential electric waves and a similar situation for the refracted waves; (iii) two reflected thermal waves and a similar situation for the refracted waves. The amplitudes of the reflected and refracted waves are functions of the incident angle, of the thermal relaxation times and of the media elastic, electric, thermal constants. This study is relevant to signal processing, sound systems, wireless communications, surface acoustic wave devices and military defense equipment.     

Wave propagation at the boundary surface of elastic and initially stressed viscothermoelastic diffusion with voids media

The present investigation is concerned with the wave propagation at the boundary surface of elastic half-space and initially stressed viscothermoelastic diffusion with voids half-space. The longitudinal and transverse waves are incident obliquely at the plane interface between uniform elastic half-space and initially stressed viscothermoelastic diffusion with voids half-space. It is found that the amplitude ratios of various reflected and transmitted waves are functions of angle of incidence, frequency of incident wave and are influenced by the initial stress, diffusion, voids, elastic and viscoelastic properties of media. The expressions of amplitude ratios and energy ratios are obtained in closed form and computed numerically for a specific model. The variations of energy ratios with angle of incidence are shown graphically. The conservation of energy at the interface is verified.     

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.