Scattering of scalar waves from a rough interface using a single integral equation

Using Green's function methods we consider the problem of scattering from a rough interface separating two semi-infinite homogenous media. We derive a single coordinate-space integral equation of the first kind for the generalized reflection coefficient R. A second integral equation of the first kind is derived for the generalized transmission coefficient T. The two equations are new results. In the limiting cases corresponding to Dirichlet and Neumann boundary conditions, we recover the usual boundary integral equations from the R-equation. In the flat surface limit, R and T reduce to the usual Fresnel reflection and transmission coefficients. The latter are derived from these more general results rather than assumed as in perturbation methods.     

Reflection operator methods for elastic waves I-irregular interfaces and regions

A wide range of reflection and transmission problems for elastic waves can be represented formally in terms of reflection and transmission operators. The action of these operators on the incident field is to produce the corresponding reflected and transmitted displacement fields. For interface problems, free surface reflections and reflections from a region, the operators can be represented in terms of a surface integral of the product of the Green's tensor and a force system which may be determined by the solution of a single integral equation.     

Wave front analysis of a plane compressional pulse scattered by a cylindrical elastic inclusion

The plane-strain problem of a stress pulse striking an elastic circular cylindrical inclusion embedded in an infinite elastic medium is treated. The method used determines dominant stress singularities that arise at wave fronts from the focusing of waves refracted into the interior. It is found that a necessary and sufficient condition for the existence of a propagating stress singularity is that the incident pulse has a step discontinuity at its front. The asymptotic wave front behavior of the first few P and SV waves to focus are determined explicitly and it is shown that the contribution from other waves are less important. In the exterior, it is found that in most composite materials the reflected waves have a singularity at their wave front which depends on the angle of reflection. Also the wave front behavior of the first few singular transmitted waves is given explicitly.The analysis is based on the use of a Watson-type lemma, developed here, and Friedlander's method [5]. The lemma relates the asymptotic behavior of the solution at the wave front to the asymptotic behavior of its Fourier transform on time for large values of the transform parameter. Friedlander's method is used to represent the solution in terms of angularly propagating wave forms. This method employs integral transforms on both time and θ, the circumferential coordinate. The θ inversion integral is asymptotically evaluated for large values of the time transform parameter by use of appropriate asymptotics for Bessel and Hankel functions and the method of stationary phase. The Watson-type lemma is then used to determine the behavior of the solution at singular wave fronts.The Watson-type lemma is generally applicable to problems which involve singular loadings or focusing in which wave front behavior is important. It yields the behavior of singular wave fronts whether or not the singular wave is the first to arrive. This application extends Friedlander's method to an interior region and physically interprets the resulting representation in terms of ray theory.     

On the T-matrix for water-wave scattering problems

A rigid cylinder of infinite length is floating in the free surface of deep water. The cylinder is held fixed and a given time-harmonic wave of small amplitude is incident upon it. The corresponding linear two-dimensional boundary-value problem for a velocity potential φ is treated using the null-field method, and an expression for the T-matrix is obtained. (The T-matrix connects the diffraction potential away from the cylinder to the given incident potential.) Fundamental properties of the T-matrix are derived from considerations of energy and reciprocity. For regular wavetrains incident from the right or from the left, there are well-known relations between the corresponding reflection and transmission coefficients; these relations are recovered by specialising the equations satisfied by the T-matrix. Two extensions to water of constant finite depth are described: one uses multipole potentials whilst the other uses Havelock wavemaker functions; this second approach also leads to a new method for treating the problem of waves in a semi-infinite channel with an end-wall of arbitrary shape.     

The mathematical model of reflection and refraction of longitudinal waves in thermo-piezoelectric materials

In this paper, the basic equations of motion, of Gauss and of heat conduction, together with constitutive relations for pyro- and piezoelectric media, are presented. Three thermoelastic theories are considered: classical dynamical coupled theory, the Lord–Shulman theory with one relaxation time and Green and Lindsay theory with two relaxation times. For incident elastic longitudinal, potential electric and thermal waves, referred to as qP, φ-mode and T-mode waves, which impinge upon the interface between two different transversal isotropic media, reflection and refraction coefficients are obtained by solving a set of linear algebraic equations. A case study is investigated: a system formed by two semi-infinite, hexagonal symmetric, pyroelectric–piezoelectric media, namely Cadmium Selenide (CdSe) and Barium Titanate (BaTiO₃). Numerical results for the reflection and refraction coefficients are obtained, and their behavior versus the incidence angle is analyzed. The interaction with the interface give rises to different kinds of reflected and refracted waves: (i) two reflected elastic waves in the first medium, one longitudinal (qP-wave) and the other transversal (qSV-wave), and a similar situation for the refracted waves in the second medium; (ii) two reflected potential electric waves and a similar situation for the refracted waves; (iii) two reflected thermal waves and a similar situation for the refracted waves. The amplitudes of the reflected and refracted waves are functions of the incident angle, of the thermal relaxation times and of the media elastic, electric, thermal constants. This study is relevant to signal processing, sound systems, wireless communications, surface acoustic wave devices and military defense equipment.     

Scattering and excitation of SH-surface waves by a protrusion at the mass-loaded boundary of an elastic half-space

A rigorous theory of the scattering and excitation of SH-surface waves by a protrusion at the mass-loaded boundary of an elastic half-space is presented. The boundary value problem (which is of the third kind) is solved by employing two suitably chosen Green functions. One of them is represented as a Fourier type of integral, the other is taken to be the Bessel function of the second kind and order zero. The procedure leads to a system of three, coupled, integral equations. This system is solved numerically. In case of an incident bulk wave, the amplitude of the launched surface wave is computed; in case of an incident surface wave, its transmission and reflection factor are computed. For both cases, an expression for the far-field radiation pattern of the scattered bulk wave is derived. A reciprocity relation is shown to exist between the amplitude of the launched surface wave and the far-field bulk wave radiation pattern. Numerical results are presented for a triangularly-prismatic protrusion; they are compared with the results pertaining to a corresponding indentation in the mass-loaded boundary, that have been obtained in a previous paper.     

DYNAMIC BEHAVIOR OF TWO COLLINEAR PERMEABLE CRACKS IN A PIEZOELECTRIC LAYER BONDED TO TWO HALF SPACES

IntroductionIn the fracture mechanics studies for piezoelectric materials,differently electricboundary conditions at the crack surfaces have been proposed by many researchers.Forexample,for the sake of analytical simplification,the assumption that the cra…     

The integral operator representation of propagation invariants for elastic waves in irregularly layered media

The reflection and transmission of elastic waves in irregularly layered media can be represented in terms of a set of propagation invariants for elastic wavefields. There are a number of such invariants depending on different integral operators acting on the displacement and traction fields. By exploiting different classes of invariants, representations for reflection and transmission operators at an irregular interface can be derived which are easier to compute than previous forms.

In a similar way, a simplified representation for the surface amplification factor for incident waves on a free surface can be constructed. This is particularly useful for calculating theoretical seismograms on the surface of a body.     

Reflection and transmission of antiplane surface waves by a surface-breaking crack in a layered elastic solid

The reflection and transmission of antiplane surface waves (Love waves) by a surface-breaking crack in a layered elastic solid is investigated. The crack is normal to the free surface, and breaks into the lower half-space solid. The formulation of the problem is reduced to a singular integral equation of the Cauchy type. In this equation, the unknown function, which is the slope of the crack-face displacement, is discontinuous at the interface between the two solids. It is shown that the magnitude of the discontinuity is related to the ratio of the shear moduli. A Gaussian numerical method is used to obtain the solution of the singular integral equation. At some distance from the plane of the crack, the wave motion is the superposition of a finite number of Love-wave modes. The amplitudes of these modes are readily evaluated in terms of the slope of the crack-face displacement. Curves are presented for the reflection coefficients corresponding to the first three modes and for the transmission coefficient as functions of the dimensionless frequency.     

Scattering of harmonic anti-plane shear waves by an interface crack in magneto-electro-elastic composites

IntroductionCompositematerialconsistingofapiezoelectricphaseandapiezomagneticphasehasdrawnsignificantinterestinrecentyears,duetotherapiddevelopmentinadaptivematerialsystems .Itshowsaremarkablylargemagnetoelectriccoefficient,thecouplingcoefficientbetweenst…     

A note on the diffraction of an obliquely incident surface wave by a partially immersed fixed vertical barrier

The problem of the diffraction of surface waves, obliquely incident on a partially immersed fixed vertical barrier in deep water, is solved approximately by reducing it to the solution of an integral equation, for small angle of incidence of the incident wave. The corrections to the reflection and transmission coefficients over their normal incidence values for small angle of incidence are obtained and presented graphically for some intermediate values of wave numbers.     

The scattering of general SH plane wave by interface crack between two dissimilar viscoelastic bodies

The scattering of general SH plane wave by an interface crack between two dissimilar viscoelastic bodies is studied and the dynamic stress intensity factor at the crack-tip is computed. The scattering problem can be decomposed into two problems: one is the reflection and refraction problem of general SH plane waves at perfect interface (with no crack); another is the scattering problem due to the existence of crack. For the first problem, the viscoelastic wave equation, displacement and stress continuity conditions across the interface are used to obtain the shear stress distribution at the interface. For the second problem, the integral transformation method is used to reduce the scattering problem into dual integral equations. Then, the dual integral equations are transformed into the Cauchy singular integral equation of first kind by introduction of the crack dislocation density function. Finally, the singular integral equation is solved by Kurtz's piecewise continuous function method. As a consequence, the crack opening displacement and dynamic stress intensity factor are obtained. At the end of the paper, a numerical example is given. The effects of incident angle, incident frequency and viscoelastic material parameters are analyzed. It is found that there is a frequency region for viscoelastic material within which the viscoelastic effects cannot be ignored. This work was supported by the National Natural Science Foundation of China (No.19772064) and by the project of CAS KJ 951-1-20     

Reflection and Transmission of Elastic Waves from the Interface of a Fluid-saturated Porous Solid and a Double Porosity Solid

The reflection and transmission characteristics of an incident plane P1 wave from the interface of a fluid-saturated single porous solid and a fluid-saturated double porosity solid are investigated. The fluid-saturated porous solid is modeled with the classic Biot’s theory and the double porosity medium is described by an extended Biot’s theory. In a double-porosity model with dual-permeability there exist three compressional waves and a shear wave. The effects of the incident angle and frequency on amplitude ratios of the reflected and transmitted waves to the incident wave are discussed. Two boundary conditions are discussed in detail: (a) Open-pore boundary and (b) Sealed-pore boundary. Numerical results reveal that the characteristics of the reflection and transmission coefficients to the incident angle and the frequency are quite different for the two cases of boundary conditions. Properties of the bulk waves existing in the fluid-saturated porous solid and the double porosity medium are also studied.     

Scattering of elastic waves by a surface-breaking crack

Scattering of incident surface waves and incident body waves by a surface-breaking crack is investigated in a two-dimensional geometry. By decomposing the scattered fields into symmetric and antisymmetric fields with respect to the plane of the crack, two boundary value problems for a quarter-plane have been obtained. The formulation of each boundary-value problem has been reduced to a singular integral equation which has been solved numerically. For incident surface waves the back-scattered and forward-scattered surface waves have been plotted versus the dimensionless frequency. Curves are also presented for the scattered displacement fields in the interior of the body generated by incident body waves, both versus the angle of incidence and versus the dimensionless frequency.     

金属材料SHPB实验数据处理的三波校核法

通过自编程序调整入射波、透射波和反射波的相对位置。当移动波相对位置时,程序界面显示2 幅图:一幅显示入射波、透射波、反射波以及透射波与反射波之差;另一幅显示试样的3条应力应变。这3条 曲线分别由第1幅图中的入射波与透射波、入射波与反射波以及三波法求得。根据SHPB实验的基本假定, 这3条应力应变曲线的塑性段应该一致,同时透射波与入射波之差应等于反射波,把这2个等价的特点作为 正确处理SHPB实验数据标准的方法称为三波校核法。     

T-stress evaluation for slightly curved crack using perturbation method

A general formulation for evaluating the T-stress at crack tips in a curved crack is introduced. In the formulation, a singular integral equation with the distribution of dislocation along the curve is suggested. For a slightly curved crack, a small parameter is generally assumed for the crack configuration. By using the assumption for the small parameter, the perturbation method is suggested and it reduces the singular integral equation into many successive singular integral equations. If the cracked plate has a remote loading and the curve configuration is a quadratic function, the mentioned successive singular integral equations can be solved in a closed form. Therefore, the solution for the T-stress in a closed form is obtained. The obtained results for T-stress are shown by figures. It is found that if the involved parameter is not too small, the influence of the curve configuration is significant. Comparison for T-stresses obtained from a quadratic-shaped curved crack and an arc crack is presented.     

Wave propagation and reflection-transmission in a stratified viscoelastic solid

The paper investigates time-harmonic wave propagation in continuously stratified solids and provides the results of a reflection-transmission process generated by a layer sandwiched between homogeneous half-spaces. The layer is continuously stratified and allows for jump discontinuities at a finite number of planes. The dissipative effects are accounted for through the classical Boltzmann law of viscoelasticity. By using displacement and traction as convenient vector variables, the governing equations are considered in a vector Volterra integral equation and the solution is determined by means of a matricant. Next the matricant is applied to determine the reflection and transmission coefficients of a layer, with a generic piecewise continuous profile of the material properties. The reflection-transmission process produced by an obliquely incident wave, is considered for horizontally-polarized waves. The low-frequency approximation is derived for the reflection and transmission coefficients. Next, the high-frequency approximation is investigated by a WKB-like procedure which involves a complex valued frequency-dependent shear modulus. The displacement solution is obtained for the forward- and the backward-propagating waves in the layer along with the reflection and transmission coefficients.     

分层流体中内孤立波在台阶上的反射和透射

基于匹配渐近展开和格林函数的方法,研究了两层流体系统中内孤立波在台阶地形上透射、 反射及其分裂的演化特征. 通过保角变换和求解奇异Fredholm积分方程,获得了反映地形 效应对Boussinesq方程影响的约化边界条件,藉此建立了KdV演化方程的``初值'问题, 根据散射反演理论获得了反射波和透射波的解析表达式. 分析结果表明：上下流体层的厚度 比、密度比以及台阶高度对于反射和透射波振幅及其分裂具有显著的影响. 尤其当上层流体 厚度小于下层厚度时,由于存在临界点,在其附近反射波的幅值随台阶高度的演化由单调增 变为单调减,透射波的幅值由单调减变为单调增;上台阶的反射波与入射波反相,其最大幅 值可达到入射波的数倍;此外,下台阶反射波也可发展为单支孤立波,它区别于单层流体中 反射波仅为衰减的振荡波列.     

层状介质中多个非共面Griffith裂纹的弹性波散射问题研究

本文利用积分变换的方法,研究了层状介质中多个非共面Griffith裂纹的弹性波散射问题,导出了当入射波分别为P(SV)波及SH波时的散射对偶积分方程,并以双覆盖层半空间中的双裂纹为例,对反平面剪切波的散射场进行了求解和较为详尽的分析。最后通过数值计算,得到了许多有关的动力学特性曲线,从而揭示了层状介质中多裂纹弹性波散射问题中的某些内在规律。     

Diffraction by a double grating

The reflection and transmission of a plane electromagnetic wave by a double grating is investigated. The double grating consists of two mutually parallel, planar arrays of perfectly conducting strips of vanishing thickness. Two types of polarization are investigated, viz.E- and H-polarization. For both types, integral equations for the unknown current densities in the two strips belonging to a single period of the grating, are derived. Subsequently, these integral equations are solved numerically, whereupon the reflection and transmission factors for the different spectral orders are computed.