On acoustic and electric waves diffraction in piezoelectric medium by a permeable half-plane crack

The problem of electric and acoustic waves diffraction by a half-plane crack in a transversal isotropic piezoelectric medium is investigated. The crack is assumed to be electric permeable and free of tractions. The so-called “quasi-hyperbolic approximation” [15] is adopted. Applying Laplace transformations and Wiener–Hopf technique a closed form solution is obtained. By the means of Cagniard–de Hoop method a detailed dynamic full electroacoustic wavefield’s investigation is conducted. Mode conversion between electric and acoustic waves, effect of electroacoustic head wave, Bleustein–Gulyaev surface wave and the wavefield structure depending on the type of the incident wave (acoustic or electric) and its angle of incidence are analyzed in details. The dynamic field intensity factors at the crack tip depending on the angle of incidence and on time are derived explicitly. Numerical analysis is presented.     

On diffraction in a piezoelectric medium by a half-plane: The Sommerfeld problem

This paper is concerned with the diffraction problem in a transversely isotropic piezoelectric medium by a half-plane. The half-plane obstacle considered here is a semi-infinite slit, or a crack; both its surfaces are traction free and electric absorbent screens. In a generalized sense, we are dealing with the Sommerfeld problem in a piezoelectric medium.¶The coupled diffraction fields between acoustic wave and electric wave are excited by both incident acoustic wave as well as incident electric wave; and the sound soft and electric "blackness" conditions on the screens are characterized by a system of simultaneous Wiener-Hopf equations. Closed form solutions are sought by employing special techniques. Some interesting results have been obtained, such as mode conversions between acoustic wave and electric wave, novel diffraction patterns in the scattering fields, and the effect of electroacoustic head wave, as well as of surface wave-Bleustein-Gulyaev wave.¶Unlike the classical Sommerfeld problem, in which the only concern is the scattering field of electric wave, the strength of material, e.g. material toughness, is another concern here. From this perspective, relevant dynamic field intensity factors at the crack tip are derived explicitly.     

Local solutions to a model of piezoelectric materials

A local existence theorem is proved for a non‐linear coupled system modelling the electromechanical motion of a one‐dimensional piezoelectric body with domain switching. The system is composed by a heat equation describing the behaviour of the number of electric dipoles and by a wave equation governing the dynamic of the electric displacement. The main coupling in the system appears in the time‐dependent velocity of the waves depending on the number of electric dipoles. The proof of the result relies on a time decay estimate satisfied by the number of electric dipoles and an uniform estimate of the solution of the regularized wave equation. Copyright © 2004 John Wiley & Sons, Ltd.     

压电材料中两平行不相等界面裂纹的动态特性研究

利用Schmidt方法,研究了压电材料中两个平行不相等的可导通界面裂纹对简谐反平面剪切波的散射问题．利用Fourier变换,使问题的求解转换为对两对以裂纹面张开位移为未知变量的对偶积分方程的求解．数值计算结果表明,动态应力强度因子及电位移强度因子受裂纹的几何参数、入射波频率的影响．在特殊情况下,与已有结果进行了比较分析．同时,电位移强度因子远小于不可导通电边界条件下相应问题的结果．     

Artificial boundary conditions for two-dimensional equations of fluid dynamics. 1. Convective wave equation

The article models external flow problems on artificially bounded regions. In the linear approximation we examine the reflection of acoustic waves in a moving medium, incident at various angles on a fixed boundary. We consider the construction of various boundary conditions and estimate their reflecting properties for plane waves and waves from point sources. The plane wave approximation is justified theoretically. A method is proposed for estimating the integral reflection coefficient for waves with a whole range of incidence angles. __________ Translated from Prikladnaya Matematika i Informatika, No. 24, pp. 76–110, 2006.     

压电压磁复合材料中界面裂纹对弹性波的散射

利用Schmidt方法分析了压电压磁复合材料中可导通界面裂纹对反平面简谐波的散射问题．经过富里叶变换得到了以裂纹面上的间断位移为未知变量的对偶积分方程A·D2在求解对偶积分方程的过程中,裂纹面上的间断位移被展开成雅可比多项式的形式．数值模拟分析了裂纹长度、波速和入射波频率对应力强度因子、电位移强度因子、磁通量强度因子的影响A·D2从结果中可以看出,压电压磁复合材料中可导通界面裂纹的反平面问题的应力奇异性形式与一般弹性材料中的反平面问题应力奇异性形式相同．     

Scattering of a plane acoustic wave on a periodic array of piston radiators embedded into a rigid screen

The stationary scattering of a plane acoustic wave on a periodic or a doubly periodic set of piston radiators elastically connected with a rigid screen is considered. The screen separates two half-spaces filled with media possessing different acoustic properties. The model considered takes into account energy losses in the elastic ties between the piston radiators and rigid screen. The problem solution in the case where only one half-space is filled with an acoustic medium is found as a special case. The results of numerically analyzing the reflection coefficients of a plane wave as functions either of the frequency of the incident wave or of the angle of incidence of the plane acoustic wave for different values of the parameters involved are presented. Bibliography: 4 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 230, 1995, pp. 125–137. Translated by G. L. Nikitin.     

A solution for transient surface waves of the B-G type in a dissipative piezoelectric crystal

We derive a solution to the problem of shear horizontal electroacoustic surface waves in a piezoelectric half-space. We formulate a time-domain dynamic problem accounting for time dispersion for both electric and elastic fields and use a separation of variables to express the solution in terms of a wave propagator. Transient surface waves of the B-G type are found to propagate with a constant speed and exponentially decay in space. Their amplitude vanishes at large distances from the boundary as the reciprocal of the depth. Dispersive and non dispersive solutions are compared.     

Diffraction of a plane wave by a step discontinuity in an impedance plane

In the present paper, the high-frequency diffraction of a plane wave by a right-angle step discontinuity in an impedance plane is analyzed with the help of the uniform geometric theory of diffraction, beginning with the Maluzhenets solution. The principal term of an asymptotic solution of the problem, which is uniform with respect to the angle of incidence of the plane wave and the angle of observation, is derived. The excitation of primary and multiply diffracted fields radiated from the upper edge, as well as surface waves, is considered (the lower edge does not radiate cylindrical or surface waves owing to a right-angle step). For simplicity, the details of computation are given here for a right-angle step discontinuity. A similar procedure is applied to other examples with more complicated geometry. Bibliography: 7 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 250, 1998, pp. 97–108. Translated by V. V. Zalipaev.     

Scattering of a SH-wave by an elastic fiber of nonclassical cross section with an interface crack

The problem of interaction of a plane time-harmonic SH-wave with an elastic fiber of quasi-square or quasi-triangular cross section, when an interface crack is present between an infinite elastic matrix and the fiber, is considered. The modified null-field method taking into account the asymptotic behavior of the solution at crack tips is exploited for obtaining numerical results. The effects of fiber shape, fiber/matrix material combination, debonding (crack size), and direction of wave incidence on the scattering amplitude in the far zone are analyzed. Russian translation published in Mekhanika Kompozitnykh Materialov, Vol. 44, No. 2, pp. 245–254, March–April, 2008.     

夹层压电材料中垂直于界面的共线双裂纹动力学问题分析

采用Schmidt方法分析了在简谐反平面剪切波作用下,两个半空间夹层压电材料中的共线裂纹的动力学行为．压电材料层内裂纹垂直于界面,电边界条件假设为可导通．通过Fourier变换,使问题的求解转换为两对三重积分对偶方程．通过数值计算,给出了裂纹的几何尺寸、压电材料常数、入射波频率等对于应力强度因子的影响．结果表明,在不同的入射波频率范围,动力场将阻碍或促使压电材料内裂纹的扩展．与不可导通电边界条件相比,导通裂纹表面的电位移强度因子比不可导通裂纹的电位移强度因子要小许多．     

Propagation of axisymmetric longitudinal waves in a finitely prestrained circular cylinder imbedded in a finitely prestrained infinite elastic body

Within the framework of a piecewise homogeneous body model and with the use of the three-dimensional linearized theory of elastic waves in initially stressed bodies (TLTEWISB), the propagation of axisymmetric longitudinal waves in a finitely prestrained circular cylinder (fiber) imbedded in a finitely prestrained infinite elastic body (matrix) is investigated. It is assumed that the fiber and matrix materials have the same density and are in compressible. The stress-strain relations for them are given through the Treloar potential. Numerical results regarding the influence of initial strains in the fiber and matrix on wave dispersion are presented and discussed. These results are obtained for the following cases: the fiber and matrix are both without initial strains; only the fiber is prestretched; only the matrix is prestretched; the fiber and matrix are both prestretched simultaneously; the fiber and matrix are both precompressed simultaneously. Russian translation published in Mekhanika Kompozitnykh Materialov, Vol. 44, No. 5, pp. 665–684, September–October, 2008.     

Propagation of shear waves in piezoelectric “superlattice-substrate” structures

The results of a computer simulation of the dispersion relations for the propagation of shear waves in piezoelectric “superlattice-substrate” structures are analyzed. The superlattice consists of a finite number of layers and is made up of materials with 6mm symmetry. The dispersion relations are obtained using a formulation for periodic hamiltonian systems. This approach makes it possible to account for the anisotropy, the piezoelectric interaction of the mechanical and electric fields, and an arbitrary number of layers in the superlattice. Numerical results are presented for CdS-ZnO layers. Selective spatial localization of acoustic modes is demonstrated for different spectral regions. The effects of the ordering of the superlattice layers, of their number, and of the boundary conditions on the dispersion spectra and on the form of the shear wave motion are examined. Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev, and Institute of Metal Physics, National Academy of Sciences of Ukraine, Kiev. Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 30, pp. 148–156, 1999.     

A Combined Method for Calculating the Field of a Point Source in a Waveguide (Slightly Curved Media)

The two-dimensional problem of propagating the waves generated by a point source in a depth slightly curved waveguide is investigated. The Dirichlet condition is given on the boundary of the waveguide, and also two consistency conditions are defined on the interface between the fluid and the bottom. The velocity of the acoustic wave is assumed to be arbitrarily dependent on the depth of the waveguide and weakly dependent on the trace coordinate. With the help of rather cumbersome transformations, the solution is expressed as a sum of geometric-optical waves, normal waves, and a remainder. Sufficient conditions on the total amount of detached normal and geometric-optical waves are obtained. Bibliography: 2 titles. Dedicated to P. V. Krauklis on the occasion of his seventieth birthday __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 308, 2004, pp. 225–234.     

Well-posedness of the poincar�� problem in a cylindrical domain for the higher-dimensional wave equation

It is known that waves (acoustic waves, radio waves, elastic waves, and electric waves) in cylindrical tubes are described by the wave equation. In the theory of hyperbolic-type partial differential equations, boundary-value problems with data on the whole boundary serve as examples of ill-posedness of the posed problems. In this work, it is shown that the Poincar´e problem in a cylindrical domain for the higher-dimensional wave equation is uniquely solvable. A uniqueness criterion for a regular solution is also obtained.     

横观各向同性电磁弹性介质中裂纹对SH波的散射

研究横观各向同性电磁弹性介质中裂纹和反平面剪切波之间的相互作用．根据电磁弹性介质的平衡运动微分方程、电位移和磁感应强度微分方程,得到SH波传播的控制场方程．引入线性变换,将控制场方程简化为Helmholtz方程和两个Laplace方程A·D2通过Fourier变换,并采用非电磁渗透型裂面边界条件,得到了柯西奇异积分方程组．利用Chebyshev多项式求解积分方程,得到应力场、电场和磁场以及动应力强度因子的表达,并给出了数值算例．     

Influence of a Weak Boundary on the Wave Field of an Approaching Source

The response of a weak interface inside an isotropic elastic medium to an approaching source is considered. It is shown that a strong shear wave arises in the wave field reflected at an angle greater than a critical one. Properties of this wave are studied, and theoretical seismograms describing the contribution of all reflected waves to the total field are presented. Bibliography: 19 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 308, 2004, pp. 89–100.     

On the excitation of a closely spaced array by a line source

An infinite row of periodically spaced, identical rigid circularcylinders is excited by an acoustic line source which is parallelto the generators of the cylinders. A method for calculatingthe scattered field accurately and efficiently is presented.When the cylinders are sufficiently close together, Rayleigh–Blochsurface waves that propagate energy to infinity along the arrayare excited. An expression is derived which enables the amplitudesof these surface waves to be computed without requiring thesolution to the full scattering problem.     

On the mechanisms of heat transport across vacuum gaps

Heat exchange between closely positioned bodies has become an important issue for many areas of modern technology including, but not limited to, integrated circuits, atomic force microscopy, and high-density magnetic recording, which deal with bodies separated by gaps as narrow as a few nanometers. It is now recognized that heat transport across a gap of sub-micron width does not follow the Stefan–Boltzmann law, which is based on a conventional theory developed for sufficiently wide gaps. This paper describes the structure of thermally excited electromagnetic fields in arbitrarily narrow gaps, and it also shows that heat can be carried across narrow vacuum gaps by acoustic waves. The structure of the acoustic wave fields is also described, and it is shown that they become the dominant heat carriers in gaps narrower than a certain critical width, which is estimated to be a few nanometers. For example, consider a vacuum gap between silicon half-spaces. When the gap’s width is below a critical value, which is about 7.5 nm, the contribution of acoustic waves must be taken into account. Assuming that the wavelength of thermally excited acoustic waves is of order 1 nm, it may be possible to estimate the contribution of acoustic waves to heat transport across gaps with 4 nm < h < 7.5 nm by the kinetic theory, but for narrower gaps with h < 4 nm, this approximation is not valid, and then the full wave theory must be used. Also for gaps narrower than about 2.5 nm, there is no need to take into account electromagnetic radiation because its contribution is negligible compared to that of acoustic waves.     

Three-dimensional Tollmien-Schlichting waves generated by sound in the boundary layer on an elastic surface at transonic free-stream velocities

Within the framework of the triple-deck theory, the effect of surface elasticity on three-dimensional packets of Tollmien-Schlichting waves generated by acoustic disturbances induced near the boundary layer at transonic free-stream velocities is investigated. It is shown that the elasticity of the surface considerably weakens the most unstable oblique waves but does not change the characteristic horseshoe shape of wave packets with two disturbance peaks propagating at an angle to the incoming flow.