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.     

两微极磁热弹性固体分界面上波的反射与折射

给出了磁场、热场和弹性场多场耦合作用下微极广义热弹性固体的一般控制方程.该方 程既包含了磁场、热场和弹性场的耦合作用,又在其广义热传导方程中涵盖了耦合热弹理论 (C-D)及其5类推广(L-S理论,G-L理论,G-N(II,III)理论和C-T理论).运用该微极广义磁热 弹性控制方程,研究了在定常磁场作用下, 具有均匀初始温度的两理想接触微极弹性介质平面分界面上磁热弹性波的反射和折射现象.给出了分别在缺少磁场、热场作用或不同广义热传 导理论下反射或折射热波、纵向位移波、耦合横向和微旋转波与入射纵向位移波的振幅比随 入射角变化的关系曲线.对缺少磁、热和微极性以及热松弛时间时对应的反射、折射系数进 行了对比.结果表明磁、热和微极性以及热松弛时间对振幅比均有不同程度的影 响,与磁、热和微极性一样,热松弛时间对不同类型波的影响能力差别明显,但对同 一类型的反射波和折射波的影响相似.     

Rayleigh–Lamb Waves in a Compound Wave Guide. Reflection from and Transmission Through the Vertical Interface between Media with Different Impedances

The method of superposition is used to study the first normal wave reflecting from and transmitting through the interface in a compound waveguide consisting of two rigidly joined elastic half-strips with equal width and different mechanical properties. We study how the impedances of the contacting media influence the transformation of the energy of the incident wave to those of the reflected and transmitted waves. Two cases are considered — propagating waves of higher orders appear in the reflected wavefield earlier than in the transmitted wavefield and propagating waves of higher orders appear in the transmitted wavefield earlier than in the reflected wave field. For both cases, the impedances vary so that the incident wave can propagate in both more rigid and softer media. It is shown that by increasing the impedances of the contacting media, the interface can be made more transparent     

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.     

Reflection and refraction of plane quasi-P waves at a corrugated interface between distinct triclinic elastic half spaces

The reflection and refraction pattern of elastic waves at a corrugated interface between two triclinic half-spaces is discussed. The incident wave is taken to be the cause of the interface disturbance and the reflected and refracted waves are effects. This leads to the causality requirement that the reflected and refracted waves must propagate away from the interface. Closed form expressions of reflection and transmission coefficients are derived using Rayleigh’s method of approximation. The formulae of reflection and transmission coefficients are derived in closed form for the first-order approximation of the corrugation. The analytical expressions of all the three phase velocities of qP, qSV and qSH waves have been derived. The variation of reflection and refraction coefficients with the angle of incidence and also with the corrugation parameter is shown. In this paper we have developed Graphical User Interface (GUI) Software in MATLAB which shows the variation of reflection and refraction coefficients with respect to incident angle and corrugation parameter. This software can be generalized to show the variation of reflection and refraction coefficients. Numerical computations are performed for a scientific model and the results obtained are shown graphically.     

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.     

波在双孔隙与单孔隙介质界面上的反射与透射

基于Biot理论和双重孔隙介质理论研究了弹性波在双重孔隙介质与流体饱和单一孔隙介质 界面的反射和透射问题,在界面上假定裂缝孔隙流体相对于固体骨架的位移为零,推导了反 射系数和透射系数的计算公式,数值讨论了反射系数和透射系数随入射角和频率的变化关 系. 同时,讨论了双重孔隙介质中3种压缩波(P-1, P-2和P-3波)和一种剪切波(S波) 的频散和衰减特性.     

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.     

Internal wave reflections and transmissions arising from a non-uniform mesh. Part I: An analysis for the Crank–Nicolson linear finite element scheme

This is the first of a series of three related papers dealing with some of the consequences of non-uniform meshes in a numerical model. In this paper the accuracy of the Crank–Nicolson linear finite element scheme, which is applied to the linear shallow water equations, is examined in the context of a single abrupt change in nodal spacing. The (in)accuracy is quantified in terms of reflection and transmission coefficients. An incident wave impinging on the interface between two regions with different nodal spacings is shown to give rise to no reflected waves and two transmitted waves. The analysis is verified using three different wavelengths (2Δx, 4Δx 8Δx) in three ‘hot-start’ numerical experiments with a mesh expansion factor of 2 and three experiments with a mesh contraction factor of 1/2. An energy flux analysis based on the concept of group velocity shows that energy is conserved across the interface.     

Reflection and transmission coefficients for three-dimensional plane waves in elastic media

Problems for transient line and point load sources in a multilayered elastic medium may be treated by the method of generalized ray. In this method an integral representation of the Laplace-transformed multiply reflected and/or transmitted cylindrical/spherical wave, known as a ray integral, is constructed by linear superposition of the Laplace-transformed plane waves. The inverse Laplace transform of the ray integral can be found in closed form by applying the Cagniard method. For problems in the Cartesian coordinates for line load sources emitting cylindrical waves consistent with either the plane strain conditions or the antiplane strain conditions and for problems in the cylindrical coordinates for axisymmetric and asymmetric point load sources emanating spherical waves, it is well known that: (1) the system of incident, reflected, and transmitted cylindrical/spherical waves at an interface separating two dissimilar media can be divided into two independent of each other, if both present, parts: the coupled P and SV waves, and the SH waves, (2) the reflected and transmitted ray integrals representing the Laplace-transformed reflected and transmitted cylindrical/spherical waves can be constructed by linear superposition of the Laplace-transformed plane P and SV waves, or the plane SH waves, and (3) the potential reflection and transmission coefficients for the plane P, SV, and S H waves are basic to such a superposition. In the present paper we treat the asymmetric three-dimensional problem in the Cartesian coordinates for an arbitrary oriented point force radiating the spherical P and S waves. For this problem all four functions representing the displacement potentials are coupled in the boundary conditions at the interface, the total wave motion at the interface is composed of the coupled spherical P and S waves, and the Laplace-transformed reflected and transmitted spherical waves are therefore constructed by linear superposition of the three-dimensional coupled plane P and S waves. Since such a superposition requires the knowledge of the potential reflection and transmission coefficients for the three-dimensional coupled plane P and S waves, the purpose of the present paper is to derive systematically these coefficient formulas.     

Full field analysis of an anti-plane interface crack subjected to dynamic body forces

In this study, the transient full field response of an interface crack between two different media subjected to dynamic body force at one material is investigated. For time t < 0, the bimaterial medium is stress free and at rest. At t = 0, a concentrated anti-plane dynamic point loading is applied at the medium as shown in Fig. 1. The total wave field is due to the effect of this point loading and the scattering of the incident waves by the interface crack. An alternative methodology that is different from the conventional superposition method is used to construct the reflected, refracted and diffracted wave fields. A useful fundamental solution is proposed in this study and the full field solution is determined by superposition of the fundamental solution in the Laplace transform domain. The proposed fundamental problem is the problem of applying an exponentially distributed traction (in the Laplace transform domain) on the interfacial crack faces. The Cagniard–de Hoop method of Laplace inversion is used to obtain the transient solution in time domain. Exact transient closed form solutions for stresses and stress intensity factors are obtained. Numerical results for the time history of stresses and stress intensity factors during the transient process are discussed in detail.     

Quasi-P-waves at a corrugated interface between two dissimilar monoclinic elastic half-spaces

In this paper we have derived reflection and transmission coefficients of qP-waves at a corrugated interface between two different elastic half-spaces of monoclinic type. Using Rayleigh’s method, the expressions for reflection and transmission coefficients are derived in closed form for a specific interface and for the first order approximation of the corrugation. Numerical computations are performed for a specific model and the results obtained have been shown graphically. The variation of the modulus of reflection and transmission coefficients with the angle of incidence, frequency and corrugation of the interface are shown separately. These coefficients are found to be strongly influenced by the angle of incidence, frequency, elastic parameters and amplitude of the corrugation of the interface. It is found that (i) the modulus of reflection and transmission coefficients at the plane interface are independent of corrugation of the interface and that of frequency of the incident wave, (ii) the reflection and transmission coefficients of regularly reflected and transmitted waves are found to be greater than that of irregularly reflected and transmitted waves, (iii) the coefficients of irregularly reflected and transmitted waves are found to increase and decrease with increase of corrugation and frequency parameters respectively. The results of Singh and Khurana [Singh, S.J., Khurana, S., 2001. Reflection and transmission of P- and SV-waves at the interface two between monoclinic elastic half-spaces. Proc. Natl. Acad. Sci. India 71(A) (IV), 305–319] have been reduced from the present problem.     

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.     

Reflection and Transmission of Elastic Waves from the Interface of a Fluid-saturated Porous Solid and a Double Porosity Solid

The reflection and transmission characteristics of an incident plane P1 wave from the interface of a fluid-saturated single porous solid and a fluid-saturated double porosity solid are investigated. The fluid-saturated porous solid is modeled with the classic Biot’s theory and the double porosity medium is described by an extended Biot’s theory. In a double-porosity model with dual-permeability there exist three compressional waves and a shear wave. The effects of the incident angle and frequency on amplitude ratios of the reflected and transmitted waves to the incident wave are discussed. Two boundary conditions are discussed in detail: (a) Open-pore boundary and (b) Sealed-pore boundary. Numerical results reveal that the characteristics of the reflection and transmission coefficients to the incident angle and the frequency are quite different for the two cases of boundary conditions. Properties of the bulk waves existing in the fluid-saturated porous solid and the double porosity medium are also studied.     

Reflection and transmission of plane waves at an imperfectly bonded interface between piezoelectric and piezomagnetic media

The paper analyzes the reflection and transmission of plane waves at an imperfect interface between piezoelectric (PE) and piezomagnetic (PM) media. The materials are assumed to be transversely isotropic. The linear spring model is used to describe the imperfection of bonding behavior at the interface. According to this model, the properties of the interface can be characterized by the normal and tangential interfacial stiffnesses. Numerial examples are performed for BaTiO₃/CoFe₂O₄ material combination. Four cases, a perfect, slip, normal weak bonding and unbonding interfaces for the coupled quasi-pressure (QP) wave incidence from BaTiO₃ medium are compared in detail. Numerical results of the reflection and transmission coefficients (RTCs) varying with incident angle for different interfacial stiffnesses are presented. Results show that the transmitted QP/reflected QSV waves are the strongest in the perfect/unbonded cases, respectively. The scattered waves in the slip and normal weak bonding cases are between those of the perfect and unbonded cases generally. Critial angles have noticeable effect on the RTCs and energy coefficients for the coupled scattered waves of the perfect and slip cases but have a little/no effect on those of the normal weak bonding/unbonded cases. It is found that the sum of the energy carried by the transmitted/reflected QP and QSV waves is less than unit for the imperfect bonding between BaTiO₃/CoFe₂O₄ solids. These results may provide some useful reference datum for the imperfection measurment at the PE and PM interface.     

Reflection and transmission of electromagnetic waves from a nonlinear anisotropic slab between two linear isotropic media

Summary In this paper, the author has studied the reflection from and transmission through a homogenous nonlinear anisotropic slab (anisotropy being due to an external magnetic field) bounded by two linear isotropic media. Nonlinear equations describing the growth of the two modes of propagation of an electromagnetic wave in the direction of the magnetic field, in an anisotropic nonlinear medium, have been set up and solved; the solutions have been used to obtain expressions for the reflected and transmitted components of the incident wave. The simpler problem of the reflection and transmission from an isotropic nonlinear slab has also been discussed as a special case.     

Influence of magnetic field on wave propagation at liquid-microstretch solid interface

The reflection and refraction of a longitudinal wave at an interface between a perfectly conducting nonviscous liquid half-space and a perfectly conducting microstretch elastic solid half-space are studied. The appropriate solutions to the governing equations are obtained in both the half-spaces satisfying the required boundary conditions at the interface to obtain a system of five non-homogeneous equations in the amplitude ratios of various reflected and transmitted waves. The system is solved by a Fortran program of the Gauss elimination method for a particular example of an interface between water and aluminum-epoxy composite. Numerical values of the amplitude ratios are computed for a certain range of the incidence angle both in the presence and absence of an impressed transverse magnetic field. The effects of the presence of the transverse magnetic field on the amplitude ratios of various reflected and transmitted waves are shown graphically.     

Propagation of plane elastic waves at a plane interface between two electro-microelastic solid half-spaces

This work is concerned with the wave propagation and their reflection and transmission from a plane interface between two different electro-microelastic solid half-spaces in perfect contact. It is found that there exist five basic waves in an infinite electro-microelastic solid, namely an independent longitudinal micro-rotational wave, two sets of coupled longitudinal waves influenced by the electric effect, and two sets of coupled transverse waves. The existence of the two sets of coupled longitudinal waves is new. In the absence of microstretch and electric effects, these two coupled longitudinal waves reduce to a longitudinal displacement wave of micropolar elasticity. Amplitude and energy ratios of various reflected and transmitted waves are presented when (i) a set of coupled longitudinal wave is made incident and (ii) a set of coupled transverse wave is made incident. Numerical computations have been performed for a particular model and the variations of amplitude and energy ratios are obtained against the angle of incidence. The results obtained are depicted graphically. It has been verified that the sum of energy ratios is equal to unity at the interface and the amplitude ratios of reflected and transmitted waves depend upon the angle of incidence, frequency and elastic properties of the media. Results of some earlier workers have also been reduced from the present formulation.     

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.