Wave reflection at a free interface in an anisotropic pyroelectric medium with nonclassical thermoelasticity

In this paper, the well-established two-dimensional mathematical model for linear pyroelectric materials is employed to investigate the reflection of waves at the boundary between a vacuum and an elastic, transversely isotropic, pyroelectric material. A comparative study between the solutions of (a) classical thermoelasticity, (b) Cattaneo–Lord–Shulman theory and (c) Green–Lindsay theory equations, characterised by none, one and two relaxation times, respectively, is presented. Suitable boundary conditions are considered in order to determine the reflection coefficients when incident elasto–electro–thermal waves impinge the free interface. It is established that, in the quasi-electrostatic approximation, three different classes of waves: (1) two principally elastic waves, namely a quasi-longitudinal Primary (qP) wave and a quasi-transverse Secondary (qS) wave; and (2) a mainly thermal (qT) wave. The observed electrical effects are, on the other hand, a direct consequence of mechanical and thermal phenomena due to pyroelectric coupling. The computed reflection coefficients of plane qP waves are found to depend upon the angle of incidence, the elastic, electric and thermal parameters of the medium, as well as the thermal relaxation times. The special cases of normal and grazing incidence are also derived and discussed. Finally, the reflection coefficients are computed for cadmium selenide observing the influence of (1) the anisotropy of the material, (2) the electrical potential and (3) temperature variations and (4) the thermal relaxation times on the reflection coefficients.     

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.     

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 Transmission of Plane Waves at a Water–Porous Sediment Interface with a Double-Porosity Substrate

This paper investigates the wave propagation at the interface between the ocean and the ocean floor. The ocean floor is assumed to be composed of covered porous sediment with an underlying double-porosity substrate. For this purpose, plane wave reflection and transmission in the coupled water–porous sediment–double-porosity substrate system are analytically solved in terms of displacement potentials. Using numerical examples, the effects of the material properties of the underlying double-porosity substrate on the reflection coefficients are discussed in detail. Variations in pore and fracture fluid, fracture volume fraction, and permeability coefficients are considered. In addition, two cases of boundary conditions at the porous sediment–double-porosity substrate interface, i.e., sealed-pore boundary and open-pore boundary, are compared in the numerical calculations. Results show that material property variations in the double-porosity substrate may significantly affect the reflected wave in the overlying water if the sandwiched sediment depth is less than the critical value.     

Explicit formulas for the reflection and transmission coefficients of one-component waves through a stack of an arbitrary number of layers

The transmission and reflection of one-component elastic, acoustic, optical waves on a stack of arbitrary number of different homogeneous layers have been intensively studied in the literature. However, all obtained formulas for the reflection and transmission coefficients are in implicit form. In this paper, we provide the explicit formulas for them. From these formulas we immediately arrive at the explicit formulas for the reflection and transmission coefficients of one-component waves through an FGM layer. Based on the obtained exact formulas, approximate formulas for the reflection and transmission coefficients are established for a stack of thin layers and for a thin FGM layer. It is numerically shown that they are good approximations. Since the obtained formulas are totally explicit, they are useful in evaluating, not only numerically but also analytically, the transmission and reflection coefficients of one-component waves.     

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.     

Propagation of plane P-waves at interface between elastic solid and unsaturated poroelastic medium

A linear viscoporoelastic model is developed to describe the problem of reflection and transmission of an obliquely incident plane P-wave at the interface between an elastic solid and an unsaturated poroelastic medium, in which the solid matrix is filled with two weakly coupled fluids (liquid and gas). The expressions for the amplitude reflection coefficients and the amplitude transmission coefficients are derived by using the potential method. The present derivation is subsequently applied to study the energy conversions among the incident, reflected, and transmitted wave modes. It is found that the reflection and transmission coefficients in the forms of amplitude ratios and energy ratios are functions of the incident angle, the liquid saturation, the frequency of the incident wave, and the elastic constants of the upper and lower media. Numerical results are presented graphically. The effects of the incident angle, the frequency, and the liquid saturation on the amplitude and the energy reflection and transmission coefficients are discussed. It is verified that in the transmission process, there is no energy dissipation at the interface.     

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.     

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 an incompressible transversely isotropic fibre-reinforced elastic media

The boundary conditions at free surface of an incompressible, transversely isotropic elastic half-space are satisfied to obtain the reflection coefficients for the case when outer slowness section is re-entrant. Two quasi-shear waves will be reflected for an angular range of direction of incident wave. The numerical illustrations of reflection coefficients are presented graphically for three arbitrary materials.     

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.     

Reflection for three-dimensional plane waves in triclinic crystalline medium

The propagation of three-dimensional plane waves at a traction free boundary of a half-space composed of triclinic crystalline material is discussed.A method has been developed to find the analytical expressions of all the three phase velocities of quasi-P (qP),quasi-SV(qSV)and quasi-SH(qSH)in three dimensions.Closed form expressions in three dimensions for the amplitude ratios of reflection coefficients of qP,qSV and qSH waves in a triclinic medium are obtained.These expressions are used for numerically studying the variation of the reflection coefficients with the angle of incidence.The graphs are drawn for different polar angle and azimuth.Numerical results presented indicate that the anisotropy affect the reflection coefficients significantly in the three dimensional case compared to the two-dimensional case.     

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.     

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.     

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.     

On theory of generalized thermoelastic solids with voids and diffusion

Governing equations of thermoelastic diffusion material with voids are modified with the help of Lord and Shulman theory of generalized thermoelasticity. These governing equations are then solved in two-dimension to show the existence of four coupled longitudinal waves and a shear wave. The complex absolute values of the speeds of the coupled longitudinal waves are computed numerically against the frequency for Magnesium material. The reflection of these plane waves from a stress free thermally insulated boundary is also studied, where the dependence of the reflection coefficients on angle of incidence is shown graphically for the incidence of coupled longitudinal wave only. The speeds and reflection coefficients of plane waves are also computed numerically in the absence of voids and diffusion parameters, which are shown graphically to observe the effects of voids and diffusion.     

Scattering of water waves by a submerged thin vertical elastic plate

The problem of water wave scattering by a thin vertical elastic plate submerged in infinitely deep water is investigated here assuming linear theory. The boundary condition on the elastic plate is derived from the Bernoulli–Euler equation of motion satisfied by the plate. This is converted into the condition that the normal velocity of the plate is prescribed in terms of an integral involving the difference in velocity potentials (unknown) across the plate multiplied by an appropriate Green’s function. The reflection and transmission coefficients are obtained in terms of integrals involving combinations of the unknown velocity potential on the two sides of the plate and its normal derivative on the plate, which satisfy three simultaneous integral equations, solved numerically. These coefficients are computed numerically for various values of different parameters and are depicted graphically against the wave number for different situations. The energy identity relating these coefficients is also derived analytically by employing Green’s integral theorem. Results for a rigid plate are recovered when the parameters characterizing the elastic plate are chosen negligibly small.     

Numerical estimation of the influence of joint stiffness on free vibrations of frame structures via the scattering of waves at elastic joints

In the structural design of mechanical products, natural frequencies must be controlled to reduce noise and vibration. In particular, the stiffness of the joints which assemble the structural components affects the natural frequencies. Therefore, it is important to predict the influence of joint stiffness on natural frequencies. Generally, these effects are determined by iterative finite element analyses of assembled structural models. Because this results in high computational costs, the sensitivity of natural frequencies to joint stiffness should be determined by a different approach to make the structural design process more efficient. Therefore, this paper proposes the use of reflection and transmission coefficients of elastic joints to predict the dependency of natural frequencies on joint stiffness. First, we formulate the reflection and transmission coefficients of joint stiffness, and then organize the coefficients using a ray tracing method. These formulations enable us to discuss the mechanisms which determine the natural frequency of a structure based on a wave approach using the phase-closure principle. Therefore, by applying the phase-closure principle to the frame structure, we investigate the formation of bending modes, which suggests that the effects of joint stiffness on natural frequencies correspond to the dependence of the reflection and transmission coefficients on joint stiffness. Therefore, these coefficients are useful indicators for estimating the influence of joint stiffness.     

Acoustic wave incidence on a multilayer medium containing a bubbly fluid layer

The problem of acoustic wave reflection and transmission through a multilayer medium containing a bubbly fluid layer is considered. For the water-water with air bubbles-water model the wave reflection and transmission coefficients are calculated and compared with the experimental data. The problem parameters, at which these coefficients take extremum values, are determined. The influence of vapor within the bubbles on the acoustic wave transmission through a layer of a fluid with the vapor-gas bubbles is shown.