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 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.     

Radon Transform for Solving an Inverse Scattering Problem in a Planar Layered Acoustic Medium

A two-dimensional inverse scattering problem in a layered acoustic medium occupying a half-plane is considered. Data is the scattered wavefield from a surface point source measured on the boundary of the half-plane. On the basis of the Radon transform, an algorithm is constructed that recovers the velocity and the acoustic impedance of the medium from the scattering data. An analytical solution is presented for an inverse scattering problem, and several inverse scattering problems are solved numerically.     

Scattering by Rough Surfaces: the Dirichlet Problem for the Helmholtz Equation in a Non-locally Perturbed Half-plane

We consider the Dirichlet boundary value problem for the Helmholtz equation in a non-locally perturbed half-plane, this problem arising in electromagnetic scattering by one-dimensional rough, perfectly conducting surfaces. We propose a new boundary integral equation formulation for this problem, utilizing the Green's function for an impedance half-plane in place of the standard fundamental solution. We show, at least for surfaces not differing too much from the flat boundary, that the integral equation is uniquely solvable in the space of bounded and continuous functions, and hence that, for a variety of incident fields including an incident plane wave, the boundary value problem for the scattered field has a unique solution satisfying the limiting absorption principle. Finally, a result of continuous dependence of the solution on the boundary shape is obtained.     

Reciprocity relations and completeness of far-field pattern vectors for obstacle scattering of acoustic wave in a stratified medium

Under a generalized Sommerfeld radiation condition, we proved the uniqueness and existence of the direct obstacle scattering problem of time-harmonic acoustic waves in a stratified medium [8]. In this paper, we study the asymptotic behaviour of the scattered waves and prove three reciprocity relations among the free-wave far-field patterns and the guided-wave far-field pattern vectors corresponding to incident distorted plane waves and normal mode waves. Then we prove conditions under which a set of far-field patterns is complete in a Hilbert space based on the reciprocity relation. These properties are important in investigating the inverse obstacle scattering problems.     

Double-knife EDGE problems with elastic/plastic type screens

A problem of diffraction of a wave by a pair of semi-infinite screens is considered. The screens are lined with two different wave bearing materials that can support surface waves. This type of problem arises in the propagation and, scattering of acoustic and electromagnetic waves by surface wave guides. To be specific, we shall couch our problem in terms of acoustics. These diffraction problems for two parallel wave bearing screens lead to boundary value problems which are governed by the Helmholtz equation, and some specific third kind boundary conditions. Such problems are shown to be well-posed for finite energy space solutions. Their representation is given by means of the canonical factorization of a non-rational matrix function.This work was supported by DFG grant KO 634/32-1     

Electromagnetic Diffraction in an Anisotropic Medium

The problem considered is that of the diffraction of an electromagneticplane by a half plane in an uniaxially anisotropic medium. Itis shown that the solution may be expressed formally in termsof the solutions of the Sommerfeld half-plane problems.     

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.     

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．     

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．     

A Yoffe-type moving crack in one-dimensional hexagonal piezoelectric quasicrystals

A Yoffe-type moving crack in one-dimensional hexagonal piezoelectric quasicrystals is considered. The Fourier transform technique is used to solve a moving crack problem under the action of antiplane shear and inplane electric field. Full elastic stresses of phonon and phason fields and electric fields are derived for a crack running with constant speed in the periodic plane. Obtained results show that the coupled elastic fields inside piezoelectric quasicrystals depend on the speed of crack propagation, and exhibit the usual square-root singularity at the moving crack tip. Electric field and phason stresses do not have singularity and electric displacement and phonon stresses have the inverse square-root singularity at the crack tip for a permeable crack. The field intensity factors and energy release rates are obtained in closed form. The crack velocity does not affect the field intensity factors, but alters the dynamic energy release rate. Bifurcation angle of a moving crack in a 1D hexagonal piezoelectric quasicrystal is evaluated from the viewpoint of energy balance. Obtained results are helpful to better understanding crack advance in piezoelectric quasicrystals.     

Finite difference method in the problem of diffraction of a plane acoustic wave in a half-plane with a cut

In the numerical solution of the diffraction problem for an acoustic plane wave in a half-plane with a cut, boundary conditions that are equivalent to the radiation conditions at infinity are set in a neighborhood of the points of the cut. Joining the physical boundary conditions on the cut, a closing set of equations of order 4N, where N is the number of grid points on the cut, is obtained. The so-called Green’s grid function for the half-plane is used, which makes it possible to pass from one grid layer to another one for the solution satisfying certain conditions at infinity.     

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．     

Reconstruction of cracks of different types from far‐field measurements

In this paper, we deal with the acoustic inverse scattering problem for reconstructing cracks of possibly different types from the far‐field map. The scattering problem models the diffraction of waves by thin two‐sided cylindrical screens. The cracks are characterized by their shapes, the type of boundary conditions and the boundary coefficients (surface impedance). We give explicit formulas of the indicator function of the probe method, which can be used to reconstruct the shape of the cracks, distinguish their types of boundary conditions, the two faces of each of them and reconstruct the possible material coefficients on them by using the far‐field map. To test the validity of these formulas, we present some numerical implementations for a single crack, which show the efficiency of the proposed method for suitably distributed surface impedances. The difficulties for numerically recovering the properties of the crack in the concave side as well as near the tips are presented and some explanations are given. Copyright © 2009 John Wiley & Sons, Ltd.     

Diffraction of impulsive elastic waves by a fluid cylinder

We consider the diffraction of impulsive SV waves by a fluid circular cylinder. The cylinder is embedded in an unbounded isotropic homogeneous elastic medium and it is filled with some acoustic fluid. The line source, generating the incident pulse, is situated outside the cylinder parallel to its axis. We investigate the problem by the method of dual integral transformation as developed by Friedlander. The resulting integrals are evaluated approximately to obtain the short-time estimate of the motion near the wave front in the shadow zone of the elastic medium. We also interpret the approximate solution in terms of Keller’s geometrical theory of diffraction.     

Numerical solution of integral equations of first and second kind in scalar and vector problems of diffraction on a cube

We consider the problem of numerical simulation of the scattering of acoustic and electromagnetic waves on a cube whose edge ha s length up to 8 wave lengths of the incident wave. We describe a scheme using a representation of the boundary integral equation in the form of an operator convolution equation on the symmetry group of the cube. We compare the results of numerical solution of integral equations of first and second kind for scalar and vector problems of diffraction of a plane wave on a cube. Translated fromProblemy Matematicheskoi Fiziki, 1998, pp. 36–45.     

Diffraction of Creeping Waves by a Line of Jump of Curvature (a Three-Dimensional Acoustic Medium)

The problem of diffraction of creeping waves by a line of jump of curvature in a three-dimensional acoustic medium is studied. Moreover, a sufficiently ``oblique' incidence is taken into account. Two cases of the curvature jump on the conjunction line of surfaces are considered: (i) the curvature does not change sign, but changes value, (ii) the surface of positive curvature is joined with a half-plane. Bibliography: 4 titles.     

The linear sampling method for anisotropic media

We consider the inverse scattering problem of determining the support of an anisotropic inhomogeneous medium from a knowledge of the incident and scattered time harmonic acoustic wave at fixed frequency. To this end, we extend the linear sampling method from the isotropic case to the case of anisotropic medium. In the case when the coefficients are real we also show that the set of transmission eigenvalues forms a discrete set.     

A decomposition scheme for acoustic obstacle scattering in a multilayered medium

We consider the acoustic wave scattering by an impenetrable obstacle embedded in a multilayered background medium, which is modelled by a linear system constituted by the Helmholtz equations with different wave numbers and the transmission conditions across the interfaces. The aim of this article is to construct an efficient computing scheme for the scattered waves for this complex scattering process, with a rigorous mathematical analysis. First, we construct a set of functions by a series of coupled transmission problems, which are proven to be well-defined. Then, the solution to our complex scattering in each layer is decomposed as the summation in terms of these functions, which are essentially the contributions from two interfaces enclosing this layer. These contributions physically correspond to the scattered fields for simple scattering problems, which do not involve the multiple scattering and are coupled via the boundary conditions. Finally, we propose an iteration scheme to compute the wave field in each layer decoupling the multiple scattering effects, with the advantage that only the solvers for the well-known transmission problems and an obstacle scattering problem in a homogeneous background medium are applied. The convergence property of this iteration scheme is proven.     

Sommerfeld's solution as the limiting amplitude and asymptotics for narrow wedges

We analyze the Sommerfeld solution to the stationary diffraction by a half‐plane. We prove that this solution is the limiting amplitude for time‐dependent scattering of incident plane waves with a broad class of the profile functions. We also show that this solution is the asymptotics of the limiting amplitudes of solutions to time‐dependent scattering problem with narrow wedges when the angle of the wedge tends to zero.