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.     

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.     

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.     

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

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

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.     

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.     

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.     

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.     

Relection of elastic waves at the elastically supported boundary of a couple stress elastic half-space

The relection elastic waves at the elastically supported boundary of a couple stress elastic half-space are studied in this paper. Different from the classical elastic solid, there are three kinds of elastic waves in the couple stress elastic solid, and two of them are dispersive. The boundary conditions of a couple stress elastic half-space include the couple stress vector and the rotation vector which disappear in the classical elastic solids. These boundary conditions are used to obtain a linear algebraic equation set, from which the amplitude ratios of relection waves to the incident wave can be determined. Then, the relection coeficients in terms of energy lux ratios are calculated numerically, and the normal energy lux conservation is used to validate the numerical results. Based on these numerical results,the inluences of the boundary parameters, which relect the mechanical behavior of elastic support, on the relection energy partition are discussed. Both the incident longitudinal wave(the P wave) and incident transverse wave(the SV wave) are considered.     

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.     

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 elastic waves at an interface with consideration of couple stress and thermal wave effects

The reflection and transmission of the thermo-elastic coupled waves at an interface of two different couple stress elastic solids are studied in this paper. Based on the Green-Lindsay theory, the governing equations and the constitutive equations are derived. Different from the classic elastic solid, the interface conditions include the surface couple, the rotation angle, the heat flux and the temperature change. The interface conditions are used to obtain the linear algebraic equations set from which the amplitude ratios of reflection and transmission waves to the incident wave can be determined. Then, the normal energy flux conservation is used to validate the numerical results. At last, the influences of two characteristic relaxation times and the five kinds of thermally and micromechanically interface conditions are discussed based on the numerical results. It is found that the thermal wave effects affect only the longitudinal wave while the couple stress effects affect only the transverse waves. The thermo-elastic coupling makes the longitudinal wave and the thermal wave not only dispersive but also attenuated.     

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 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.     

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.     

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.     

Interesting effects in harmonic generation by plane elastic waves

The harmonics of plane longitudinal and trans-verse waves in nonlinear elastic solids with up to cubic nonlinearity in a one-dimensional setting are investigated in this paper. It is shown that due to quadratic nonlinearity, a transverse wave generates a second longitudinal harmonic. This propagates with the velocity of transverse waves, as well as resonant transverse first and third harmonics due to the cubic and quadratic nonlinearities. A longitudinal wave generates a resonant longitudinal second harmonic, as well as first and third harmonics with amplitudes that increase linearly and quadratically with distance propagated. In a second investigation, incidence from the linear side of a pri-mary wave on an interface between a linear and a nonlinear elastic solid is considered. The incident wave crosses the interface and generates a harmonic with interface conditions that are equilibrated by compensatory waves propagating in two directions away from the interface. The back-propagated compensatory wave provides information on the nonlinear elastic constants of the material behind the interface. It is shown that the amplitudes of the compensatory waves can be increased by mixing two incident longitudinal waves of appropriate frequencies.     

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.     

Response of loose bonding on reflection and transmission of elastic waves at interface between elastic solid and micropolar porous cubic crystal

The problem of reflection and transmission of plane periodic waves incident on the interface between the loosely bonded elastic solid and micropolar porous cubic crystal half spaces is investigated. This is done by assuming that the interface behaves like a dislocation, which preserves the continuity of traction while allowing a finite amount of slip. Amplitude ratios of various reflected and transmitted waves have been depicted graphically. Some special cases of interest have been deduced from the present investigation.     

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.