Reflection and transmission of elastic waves at five types of possible interfaces between two dipolar gradient elastic half-spaces

Reflection and transmission of an incident plane wave at five types of possible interfaces between two dipo-lar gradient elastic solids are studied in this paper. First, the explicit expressions of monopolar tractions and dipolar trac-tions are derived from the postulated function of strain energy density. Then, the displacements, the normal derivative of displacements, monopolar tractions, and dipolar tractions are used to create the nontraditional interface conditions. There are five types of possible interfaces based on all possible combinations of the displacements and the normal derivative of displacements. These interfacial conditions with consid-eration of microstructure effects are used to determine the amplitude ratio of the reflection and transmission waves with respect to the incident wave. Further, the energy ratios of the reflection and transmission waves with respect to the incident wave are calculated. Some numerical results of the reflection and transmission coefficients are given in terms of energy flux ratio for five types of possible interfaces. The influences of the five types of possible interfaces on the energy parti-tion between the refection waves and the transmission waves are discussed, and the concept of double channels of energy transfer is first proposed to explain the different influences of five types of interfaces.     

Propagation of Thermo-Elastic Waves at Several Typical Interfaces Based on the Theory of Dipolar Gradient Elasticity

The reflection and transmission properties of thermo-elastic waves at five possible interfaces between two different strain gradient thermo-elastic solids are investigated based on the generalized thermo-elastic theory without energy dissipation(the GN theory). First, the function of free energy density is postulated and the constitutive relations are defined. Then,the temperature field and the displacement field are obtained from the motion equation in the form of displacement and the thermal transport equation without energy dissipation in the strain gradient thermo-elastic solid. Finally, the five types of thermo-elastic interfacial conditions are used to calculate the amplitude ratios of the reflection and transmission waves with respect to the incident wave. Further, the reflection and transmission coefficients in terms of energy flux ratio are calculated and the numerical results are validated by the energy conservation along the normal direction. It is found that there are five types of dispersive waves, namely the coupled longitudinal wave(the CP wave), the coupled thermal wave(the CT wave), the shear wave, and two evanescent waves(the coupled SP wave and SS wave), that become the surface waves at an interface. The mechanical interfacial conditions mainly influence the coupled CP waves, SV waves, and surface waves, while the thermal interfacial conditions mainly influence the coupled CT waves.     

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.     

An Analysis of Turbulent Motions In and Around a Differentially Forced Density Interface

The Rapid-Distortion-Theory-based analysis proposed by Fernando and Hunt [1] is extended to study the nature of turbulence in and around a density interface sandwiched between turbulent layers with dissimilar properties. It is shown that interfacial motions consist of low-frequency, resonantly excited, nonlinear internal waves and high-frequency, linear internal waves driven by background turbulence. Based on the assumptions that (i) all resonant waves and some nonresonant waves having frequencies close to the resonant frequencies grow rapidly, break, and cause interfacial mixing, (ii) the spectral amplitude of the vertical velocity in the wave-breaking regime is constant, and (iii) kinetic energy is equipartitioned between linear and nonlinear breaking wave regimes, the r.m.s. vertical velocity at the interface and the turbulent kinetic energy flux into the interface are calculated. The migration velocity of the interface is calculated using the additional assumption that the buoyancy flux into a given turbulent layer is a fixed fraction of the turbulent kinetic energy flux supplied to the interface by the same layer. The calculations are found to be in good agreement with the entrainment data obtained in previous laboratory experiments in the parameter regime where the interface is dominated by internal wave dynamics. Received 23 July 1997 and accepted 8 January 1999     

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.     

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 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.     

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 and refraction of acoustic waves at the interface between a gas and a disperse systems

The reflection and refraction of acoustic waves at different angles of incidence on the interface between a vapor-gas-droplet system and air are studied. From an analysis of analytical solutions, it has been found that in the case of incidence on the interface from the side of the vapor-gas-droplet medium, there is a critical angle of incidence at which the wave is completely reflected from the boundary, i.e., total internal reflection takes place. It is shown that for a certain angle of incidence on the interface both from the air side and from the mixture side and for a certain volume fraction of water in the disperse system, complete transmission of the acoustic wave through the medium is observed.     

Reflection and Transmission of Elastic Waves at the Interface Between Water and a Double Porosity Solid

The reflection and transmission of an incident plane wave at an interface between water and a fluid-saturated double porosity solid are investigated. The properties of the three compressional waves and one shear wave in double porosity media are discussed in detail. The effect of the viscosity, permeability, and porosity on the phase velocity and attenuation of four bulk waves is presented. Comparison between the double porosity theory and the Biot theory reveals that there exists obvious difference in the phase velocity, attenuation and the reflection and transmission coefficients. Two cases of boundary conditions, i.e., the sealed-pore and the open-pore boundaries, are introduced in the numerical calculation. At last, the effect of the fracture permeability and porosity on the reflection and transmission coefficients considered. It is revealed that the amplitudes of the reflected and transmitted waves strongly depend the boundary condition, fracture permeability, and porosity.     

Waves in Nonlocal Elastic Solid with Voids

In this paper, the governing relations and equations are derived for nonlocal elastic solid with voids. The propagation of time harmonic plane waves is investigated in an infinite nonlocal elastic solid material with voids. It has been found that three basic waves consisting of two sets of coupled longitudinal waves and one independent transverse wave may travel with distinct speeds. The sets of coupled waves are found to be dispersive, attenuating and influenced by the presence of voids and nonlocality parameters in the medium. The transverse wave is dispersive but non-attenuating, influenced by the nonlocality and independent of void parameters. Furthermore, the transverse wave is found to face critical frequency, while the coupled waves may face critical frequencies conditionally. Beyond each critical frequency, the respective wave is no more a propagating wave. Reflection phenomenon of an incident coupled longitudinal waves from stress-free boundary surface of a nonlocal elastic solid half-space with voids has also been studied. Using appropriate boundary conditions, the formulae for various reflection coefficients and their respective energy ratios are presented. For a particular model, the effects of non-locality and dissipation parameter (\(\tau \)) have been depicted on phase speeds and attenuation coefficients of propagating waves. The effect of nonlocality on reflection coefficients has also been observed and shown graphically.     

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.     

Elastic waves in an electro-microelastic solid

The present study is concerned with the wave propagation in an electro-microelastic solid. The reflection phenomenon of plane elastic waves from a stress free plane boundary of an electro-microelastic solid half-space is studied. The condition and the range of frequency for the existence of elastic waves in an infinite electro-microelastic body are investigated. The constitutive relations and the field equations for an electro-microelastic solid are stemmed from the Eringen’s theory of microstretch elasticity with electromagnetic interactions. Amplitude ratios and energy ratios of various reflected waves are presented when an elastic wave is made incident obliquely at the stress free plane boundary of an electro-microelastic solid half-space. It has been verified that there is no dissipation of energy at the boundary surface during reflection. Numerical computations are performed for a specific model to calculate the phase speeds, amplitude ratios and energy ratios, and the results obtained are depicted graphically. The effect of elastic parameter corresponding to micro-stretch is noticed on reflection coefficients, in particular. Results of Parfitt and Eringen [Parfitt, V.R., Eringen, A.C., 1969. Reflection of plane waves from a flat boundary of a micropolar elastic half-space. J. Acoust. Soc. Am. 45, 1258–1272] have also been reduced as a special case from the present formulation.     

Transverse wave at a plane interface between isotropic elastic and unsaturated porous elastic solid half-spaces

Based on the poroelasticity theory, this article investigates the reflection and transmission characteristics of an incident plane transverse wave at a plane interface between an isotropic elastic half-space and an unsaturated poroelastic solid half-space. For this purpose, the effect of the saturation degree and frequency on the properties of the four bulk waves in unsaturated porous medium, i.e., three longitudinal waves and one transverse wave, are discussed at first. Two general cases of mode conversion are considered: (i) The initial transverse wave is incident from an unsaturated poroelastic half-space to the interface, and (ii) the initial transverse wave is incident from an elastic solid half-space to the interface. The expressions for the partition of energy at the interface during transmission and reflection process of waves are presented in explicit forms. At last, numerical computations are performed for these two cases and the results obtained are depicted, respectively. The variation of the amplitude ratios and energy ratios with the saturation degree and incident angle is illustrated in detail. It is also verified that, at the interface, the sum of energy ratios is approximately equal to unity as expected.     

Longitudinal waves at a micropolar fluid/solid interface

The possibility of plane wave propagation in a micropolar fluid of infinite extent has been explored. The reflection and transmission of longitudinal elastic wave at a plane interface between a homogeneous micropolar fluid half-space and a micropolar solid half-space has also been investigated. It is found that there can exist four plane waves propagating with distinct phase speeds in an infinite micropolar fluid. All the four waves are found to be dispersive and attenuated. The reflection and transmission coefficients are found to be the functions of the angle of incidence, the elastic properties of the half-spaces and the frequency of the incident wave. The expressions of energy ratios have also been obtained in explicit form. Frequency equation for the Stoneley wave at micropolar solid/fluid interface has also been derived in the form of sixth-order determinantal expression, which is found in full agreement with the corresponding result of inviscid liquid/elastic solid interface. Numerical computations have been performed for a specific model. The dispersion curves and attenuation of the existed waves in micropolar fluid have been computed and depicted graphically. The variations of various amplitudes and energy ratios are also shown against the angle of incidence. Results of some earlier workers have been deduced from the present formulation.     

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.     

The coupling between turbulent,penetrative convection and internal waves

Experiments aimed at exploring the coupling of penetrative convection with internal waves in the adjoining, stable layer were performed in a long convection cell. The experiments are motivated by preliminary theoretical results suggesting that an intrinsic phase instability may exist in the coupled system in which case long internal waves modulate the height and strength of convective plumes. Using a temperature-controlled, stably stratified experimental apparatus, measured temperature data reveal the presence of long internal wave modes that persist for many convective time scales. The frequencies of these waves increase linearly in time during the energy transfer between the convective and stratified regions as the depth of the stratified region diminishes and the depth of the mixed layer increases. Temporal variations in the heat flux, interface rise characteristics, and frequencies of internal wave motions are reported. A natural temporal modulation of the thickness of the transition layer separating the mixed layer from the stratified layer occurs following commencement of heating, with the amplitude and frequency of the modulation varying with the initial stratification. Temperature variance data suggest that a fairly strong interaction between convection and internal waves occurs, especially when the interface region is midway between the upper and lower boundaries of the cell and the no-slip boundary conditions play a less influential role on the dynamics of the coupling.     

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 transmission of elastic waves through a couple-stress elastic slab sandwiched between two half-spaces

The reflection and transmission of elastic waves through a couple-stress elastic slab that is sandwiched between two couple-stress elastic half-spaces are studied in this paper. Because of the couple-stress effects, there are three types of elastic waves in the couple-stress elastic solid, two of which are dispersive. The interface conditions between two couple-stress solids involve the surface couple and rotation apart from the surface traction and displacement. The nontraditional interface conditions between the slab and two solid half-spaces are used to obtain the linear algebraic equation sets from which the amplitude ratios of reflection and transmission waves to the incident wave can be determined. Then, the energy fluxes carried by the various reflection and transmission waves are calculated numerically and the normal energy flux conservation is used to validate the numerical results. The special case, couple-stress elastic slab sandwiched by the classical elastic half-spaces, is also studied and compared with the situation that the classical elastic slab sandwiched by the classical elastic half-spaces. Incident longitudinal wave (P wave) and incident transverse wave (SV wave) are both considered. The influences of the couple-stress are mainly discussed based on the numerical results. It is found that the couple-stress mainly influences the transverse modes of elastic waves.     

Frontal collision of internal solitary waves of first mode

The dynamics and energetics of a frontal collision of internal solitary waves (ISW) of first mode in a fluid with two homogeneous layers separated by a thin interfacial layer are studied numerically within the framework of the Navier–Stokes equations for stratified fluid. It was shown that the head-on collision of internal solitary waves of small and moderate amplitude results in a small phase shift and in the generation of dispersive wave train travelling behind the transmitted solitary wave. The phase shift grows as amplitudes of the interacting waves increase. The maximum run-up amplitude during the wave collision reaches a value larger than the sum of the amplitudes of the incident solitary waves. The excess of the maximum run-up amplitude over the sum of the amplitudes of the colliding waves grows with the increasing amplitude of interacting waves of small and moderate amplitudes whereas it decreases for colliding waves of large amplitude. Unlike the waves of small and moderate amplitudes collision of ISWs of large amplitude was accompanied by shear instability and the formation of Kelvin–Helmholtz (KH) vortices in the interface layer, however, subsequently waves again become stable. The loss of energy due to the KH instability does not exceed 5%–6%. An interaction of large amplitude ISW with even small amplitude ISW can trigger instability of larger wave and development of KH billows in larger wave. When smaller wave amplitude increases the wave interaction was accompanied by KH instability of both waves.