Reflection and refraction of plane waves at the interface between piezoelectric and piezomagnetic media

The paper analyzes the reflection and refraction of a plane wave incidence obliquely at the interface between piezoelectric and piezomagnetic media. The materials are assumed to be transversely isotropic. Numerical calculations are performed for BaTiO₃/CoFe₂O₄ material combination. Four cases, incidence of the coupled quasi-pressure (QP) and quasi-shear vertical (QSV) wave from BaTiO₃ or CoFe₂O₄ media, are discussed. The reflection and transmission coefficients and energy coefficients varying with the incident angle are examined. Calculated results are verified by considering the energy conservation. Results show that the reflected and transmitted wave fields in the sagittal plane consist of six kinds of waves, i.e. the coupled QP and QSV waves, evanescent electroacoustic (EA) and magnetic potential (MP) waves in the piezoelectric medium (BaTiO₃), evanescent magnetoacoustic (MA) and electric potential (EP) waves in the piezomagnetic medium (CoFe₂O₄), among which the EA, MA, MP and EP waves propagate along the interface. The most amount of the incident energy goes with the waves that are the same type as the incident wave, while the energy arising from wave mode conversion occupies a less part of the incident energy. The electric energy in BaTiO₃ is higher than the magnetic energy in CoFe₂O₄; they both attain their maximum values at/before the critical angle. Critical angles have little effect on evanescent waves except when the total reflection takes place. These results would provide useful complementary information for magnetoelectric composite materials.     

S波由饱和土入射于弹性土时在界面上的反射与透射

从地震工程实际出发，借助Biot多孔介质中的波动方程，根据各种界面条件导出了S波从饱和土入射于弹性土时在交界面上反射与透射的一般计算公式。作为算例，数值计算分析了S波从饱和土入射于饱和土与弹性土交界面时，饱和土中P1、P2和S波的反射系数以及弹性土中P波、SV射系数与界面排水条件、入射角以及频率之间的关系。结果表明：各种波的反射、透射系数与入射角、入射频率以及界面排水条件有关系。     

Acoustic reflection from the boundary of anisotropic thermoviscoelastic solid with fluid

Vertical slownesses of waves at a boundary of an anisotropic thermoviscoelastic medium are calculated as roots of a polynomial equation of degree eight. Out of the corresponding eight waves, the four, which travel towards the boundary are identified as upgoing waves. Remaining four waves travel away from the boundary and are termed as downgoing waves. Reflection and refraction of plane harmonic acoustic waves are studied at a plane boundary between anisotropic thermoviscoelastic solid and a non-viscous fluid. At this fluid-solid interface, an incident acoustic wave through the fluid reflects back as an attenuated acoustic wave and refracts as four attenuating waves into the anisotropic base. Slowness vectors of all the waves in two media differ only in vertical components. Complex values of vertical slowness define inhomogeneous refracted waves with a fixed direction of attenuation, i.e. perpendicular to the interface. Energy partition is calculated at the interface to find energy shares of reflected and refracted waves. A part of incident energy dissipates due to interaction among the attenuated refracted waves. Numerical examples are considered to study the variations in energy shares with the direction of incident wave. For each incidence, the conservation of incident energy is verified in the presence of interaction energy. Energy partition at the interface seems to be changing very slightly with the azimuthal variations of the incident direction. Effects of anisotropy, elastic relaxation and thermal parameters on the variations in energy partition are discussed. The acoustic wave reflected from isothermal interface is much significant for incidence around some critical directions, which are analogous to the critical angles in a non-dissipative medium. The changes in thermal relaxation times and uniform temperature of the thermoviscoelastic medium do not show any significant effect on the reflected energy.     

平面S波在非饱和土自由边界上的反射问题研究

运用非饱和孔隙介质理论阐述了弹性波在非饱和土中的传播特性,分析了平面S波在非饱和土层自由边界上的反射问题。根据边界条件,分别导出了在非饱和土自由边界上的四种反射波：反射P1波、反射P2波、反射P3波及反射S波的振幅反射率及能量反射率的理论表达式,并在此基础上进行了数值计算。算例中讨论了四种反射波的振幅反射率及能量反射率受平面S波入射角度及土层饱和度变化的影响情况。计算结果表明：各反射波的振幅反射率及能量反射率不仅与入射角有关,也受到饱和度变化的影响,这些结论对土动力学的理论研究以及相关工程地震勘探具有一定指导意义。     

The laws of plane defect wave propagation across the interface between two viscoplastic media

Relations determining the laws of reflection and refraction of a plane wave in the defect field at an interface between two viscoplastic media are obtained based on the dynamic equations of the continuum theory of defects. The reflection and transmission coefficients relating the amplitudes of reflected and transmitted waves to the incident wave amplitude are determined. The obtained relations are applied to a particular case of media with weakly decaying waves.     

P1波在非饱和土地基表面的反射特性

非饱和土是地表天然土体中最常见的一种形式。与饱和土不同,非饱和土内可传播四种体波:一种剪切波(S波)和三种压缩波(P1波、P2波、P3波)。其中P1传播特性与饱和土中的快压缩波相似:传播速度最快,且衰减系数最低。因此,该文运用非饱和孔隙介质理论阐述了弹性波在非饱和土中的传播特性,分析了平面P1波在非饱和土介质自由边界上的反射问题。根据边界条件,分别导出了在非饱和土自由边界上的四种反射波:反射P1波、反射P2波、反射P3波及反射S波的振幅反射系数及能量反射系数的理论表达式,并且在此基础上进行了数值计算。算例中讨论了四种反射波的振幅反射系数及能量反射系数受P1波入射角及地基土体饱和度变化的影响情况。结果表明:各反射波的振幅反射系数及能量反射系数不仅与入射角有关,也受到饱和度变化的影响。这些结论对土动力学的理论研究以及工程地震勘探具有一定指导意义。     

Some exact solutions for the propagation of transient electroacoustic waves. II: Plane interface between two piezoelectric media

Electroacoustic waves are coupled horizontal-shear (SH) and electromagnetic disturbances that propagate in certain types of piezoelectric media. At the interface between two semi-infinite piezoelectric media, a new type of electroacoustic surface wave, called a Maerfeld-Tournois wave, can exist that has no purely elastic wave counterpart. This article obtains exact transient solutions for the Maerfeld-Tournois and body waves generated by a dipole source on such an interface. These solutions are obtained for both conducting and nonconducting interfaces using a modification of the Lamb-Cagniard-Pekeris technique. In the quasistatic approximation, explicit results for all the waves present are given and the separated body and surface waves at the interface are discussed.     

Reflection and Refraction of Micropolar Magneto-thermoviscoelastic Waves at the Interface between Two Micropolar Viscoelastic Media

Using micropolar generalized thermoviscoelastic theories, problems of reflection and refraction of magneto-thermoeviscoelastic waves at the interface between two viscoelastic media are studied when a uniform magnetic field permeates the media. Coefficient ratios of reflection and refraction are obtained using continuous boundary conditions. Some special cases are considered, i.e., the absence of micropolar and viscous effects. By numerical calculations, variations of the amplitude ratios of reflection and refraction coefficients with the angle of incidence are shown graphically for incident rotational and dilatational waves at the interface between two media (one medium is aluminium-epoxy micropolar iscoelastic material, and the other is magnesium crystal micropolar viscoelastic material). Comparing the generalized thermoelastic theories developed by Lord and Shulman (LS) and by Green and Lindsay (GL) in this paper to conventional dynamics (CD) theory the effects of a magnetic field and viscosity are shown numerically in this paper.     

位于弹性半空间上的理想流体层动力反应—平面P波入射

根据弹性介质与理想流体交界面的边界条件以及理想流体层自由表面的边界条件,基于位移势函数,推导了平面P波从弹性半空间入射到与理想流体层的交界面时,确定交界面波的透射与反射系数的理论方程。根据该方程,通过数值算例,分析了弹性半空间的弹性模量和密度、平面P波的入射角以及入射频率对理想流体层动压力的影响。     

位于弹性半空间上的理想流体层动力反应--平面SV波入射

根据弹性固体与理想流体动力学方程,导出了固-液介质交界面上波的透射与反射系数的理论公式,分析了平面SV波从弹性半空间入射到与理想流体层的交界面时,弹性半空间的弹性模量和密度,平面SV波的入射角和频率,以及流体的体积模量和密度对理想流体层动压力的影响。     

The Character of Deformations at the Interface of Elastic Media under the Slip Condition

Fresnel coefficients obtained when solving the problem of wave propagation through the interface of two elastic media and expressions for components of the elastic-distortion tensor allow one to study the character of dynamic deformations at the interface. Deformation modes different from zero at the interface of the elastic media under the slip-contact condition have been determined. Dependences of deformation amplitudes at the interface on the wave incidence angle and parameters of the adjacent media for incident longitudinal and transverse waves have been constructed and analyzed.     

Brewster angle with a negative-index material

The demonstration and confirmation of metamaterials with simultaneous negative permittivity and permeability, and thus a negative refractive index, has resulted in a surge of interest in the reflection and refraction phenomena at the interfaces of these so-called negative-index materials (NIMs). We present a systematic study of the Brewster angle, i.e., the angle of incidence at which no reflection occurs, for both TE and TM waves scattering at the interface between two semi-infinite planar media, one of which may be a NIM. Detailed physical explanations that account for the Brewster angle for a plane wave incident upon a NIM are provided under the framework of the Ewald-Oseen extinction theorem, considering the reemission of induced electric and magnetic dipoles. The conditions under which the Brewster angle exists are concisely summarized in a map of different material parameter regimes.     

Elastodynamic Doppler effects and wave energy partition by a sliding interface

Wave reflection and refraction at a sliding interface are discussed. The regular dynamic friction rule, Coulomb’s law, is assumed for the sliding interface. Doppler frequency shift is found, and it depends on the ratio of a sliding velocity to an apparent horizontal velocity of an incident wave. Wave energy partition by the sliding interface is also discussed. In the case of an incident P-wave, a transmitted/refracted P-wave gets the major part of the incident wave energy. On the other hand, in the case of an incident SV-wave, a reflected SV-wave gets most of incident wave energy. It is also found an energy supply or loss by the sliding interface. It strongly depends on the friction coefficient and on the sliding direction. Further, it is shown that there is no possibility for detecting the sliding velocity by means of Doppler frequency shift.     

Reflection of plane waves at the boundary of a pre-stressed elastic half-space subject to biaxial stretch and simple shear

This paper is concerned with the effect of finite pure homogeneous biaxial stretch together with simple shear deformation on the reflection from a plane boundary of elastic waves propagating in a half-space of incompressible isotropic elastic material. This generalizes the previous work in which, separately, pure homogeneous strain and simple shear were considered. For a special class of constitutive law, it is shown that an incident plane harmonic wave propagating in the considered plane gives rise to a surface wave in addition to a reflected wave for every angle of incidence. The amplitude of the surface wave may vanish at certain discrete angles depending on the state of stress, biaxial stretch and simple shear deformation and then specialized to recover results obtained previously. The amplitude of the reflected wave is independent of the pre-stress but does depend upon the magnitude of deformation under consideration. The dependence of the reflected and surface wave behavior on the angle of incidence, amount of shear strain, biaxial stretch and the state of stress is illustrated graphically.     

Phase Conjugation of a Plane Electromagnetic Wave at an Interface

Boundary conditions for electromagnetic waves which undergo phase conjugation at an interface between two media of different refractive index are considered. Appropriate laws of reflection and refraction are derived, showing a certain symmetry with ordinary interaction at a boundary, and affecting the Fresnel equations accordingly.     

Stress waves in layered rocks

Summary In this paper two problems will be solved analytically: the reflection of plane stress waves at a free boundary and the reflection and refraction of plane stress waves at a welded interface separating two dissimilar materials. In particular, stress reflection and refraction coefficients will be derived and the superimposed state of stress in regions of wave interference will be calculated. The solutions are visualized in a series of examples by means of computer-generated isochromatic fringe patterns. They show the partition of stress between reflected and refracted waves, stress distribution in regions of wave interference and demonstrate stress wave propagation in laycred structures.     

Some exact solutions for the propagation of transient electroacoustic waves. I: Piezoelectric half-space

In piezoelectric materials, coupled electromagnetic and horizontally polarized shear surface wave disturbances can exist which have no purely elastic counterpart. The properties of these electroacoustic surface waves and similarly coupled body wave components have to date only been described by far-field approximations in the frequency domain. This article obtains exact transient solutions for the electroacoustic surface and body waves generated by a dipole source on a piezoelectric half-space. These solutions are obtained for both conducting and nonconducting surface boundary conditions using a modification of the Lamb-Cagniard-Pekeris technique previously applied to similar elastic and acoustic wave propagation problems. Explicit results for the separated surface and body wave contributions at the surface of the piezoelectric are given and discussed.     

Scattering of inhomogeneous wave by viscoelastic interface crack

Summary The reflection, refraction and scattering of inhomogeneous plane waves of SH type by an interface crack between two dissimilar viscoelastic bodies are investigated. The singular integral equation method is used to reduce the scattering problem into the Cauchy singular integral equation of first kind by introduction of the crack dislocation density function. Then, the singular integral equation is solved numerically by Kurtz's piecewise continous function method. The crack opening displacement and dynamic stress intensity factor characterizing the scattered near-field are estimated for various incident angles, frequencies and relaxation times. The differences on crack opening displacement and stress intensity factor between elastic and viscoelastic interface crack are contrasted. And the effects of incident angle, incident frequency and relaxation time of the viscoelastic material are analyzed and explained by the features of phase lag and energy dissipation of the viscoelastic wave.     

Wave scattering by a horizontal circular cylinder in a two-layer fluid with an ice-cover

In a two-layer fluid wherein the upper layer is of finite depth and bounded above by a thin but uniform layer of ice-cover modelled as a thin elastic sheet and the lower layer is infinitely deep below the interface, time-harmonic waves with a given frequency can propagate with two different wavenumbers. The wave of smaller wavenumber propagates along the ice-cover while wave of higher wavenumber propagates along the interface. In this paper problems of wave scattering by a horizontal circular cylinder submerged in either the lower or in the upper layer due to obliquely as well as normally incident wave trains of both the wave numbers are investigated by using the method of multipole expansions. The effect of the presence of ice-cover on the various reflection and transmission coefficients due to incident waves at the ice-cover and the interface is depicted graphically in a number of figures.     

Reflection and transmission of waves from a plane interface between two microstretch solid half-spaces

In this paper, we have investigated the wave propagation and their reflection and transmission from a plane interface between two different microstretch elastic solid half-spaces in perfect contact. It is shown that there exist five waves in a linear homogeneous isotropic microstretch elastic solid, one of them travel independently, while other waves are two sets of two coupled waves. It is also shown that these waves travel with different velocities, three of which disappear below a critical frequency. Amplitude ratios and energy ratios of various reflected and transmitted waves are presented when a set of coupled longitudinal waves and a set of coupled transverse waves is made incident. It is found that the amplitude ratios of reflected and transmitted waves are functions of angle of incidence, frequency and are affected by the elastic properties of the media. Some special cases have been reduced from the present formulation.