SH-wave propagation and scattering in periodically layered composites with a damaged layer

Plane SH-wave propagation in periodically layered elastic composites with a damaged layer is investigated. Two different models are developed to approximate the damaged layer, namely, a periodic array of cracks and continuously distributed springs in the layer. In the first model, the total wave field in the elastic stack of layers with cracks is described as a sum of incident wave field modeled by the transfer matrix method and the scattered wave field governed by an integral representation in terms of the crack-opening-displacements on the crack-faces. The integral equation derived from the boundary conditions on the crack-faces is solved numerically by a Galerkin method. By using Bloch–Floquet theorem the crack-opening-displacements for a periodic array of cracks are expressed by the crack-opening-displacement on a reference crack. In the spring model, the spring constant is estimated by the material properties and the crack density and the modified transfer matrix method is used to compute the wave reflection and transmission coefficients. Numerical results obtained by both models are presented and discussed. Special attention of the analysis is devoted to wave transmissions and reflections, band gaps, wave localization and resonance phenomena due to damages. The influences of the damage types (periodic cracks and stochastic cracks approximated by distributed springs) on the wave field pattern and the band gaps are analyzed.     

Acoustic scattering by a modified Werner method

A modified integral Werner method is used to calculate pressure scattered by an axisymmetric body immersed in a perfect and compressible fluid subject to a harmonic acoustic field. This integral representation is built as the sum of a potential of a simple layer and a potential of volume. It is equivalent to the exterior Helmholtz problem with Neumann boundary condition for all real wave numbers of the incident acoustic field. For elastic structure scattering problems, the modified Werner method is coupled with an elastodynamic integral formulation in order to account for the elastic contribution of the displacement field at the fluid/structure interface. The resulting system of integral equations is solved by the collocation method with a quadratic interpolation. The introduction of a weighting factor in the modified Werner method decreases the number of volume elements necessary for a good convergence of results. This approach becomes very competitive when it is compared with other integral methods that are valid for all wave numbers. A numerical comparison with an experiment on a tungsten carbide end-capped cylinder allows a glimpse of the interesting possibilities for using the coupling of the modified Werner method and the integral elastodynamic equation used in this research.     

Scattering of surface waves by a sub-surface crack

The scattering of Rayleigh waves by a two-dimensional sub-surface crack, which is perpendicular to the free surface of an elastic half-space, is investigated. The boundary-value problem for the scattered field is stated, and reduced to an uncoupled system of integral equations which are solved numerically. At large distances from the crack the scattered field is shown to consist of outgoing Rayleigh waves and cylindrical body waves. Graphs of the variation of the amplitude and phase of the forward and backward scattered Rayleigh waves with the frequency and the geometrical parameters of the crack are presented. The stress intensity factors at both crack tips have also been obtained.     

Elastic waves excited by a plane source on the surface of a multilayered medium

Elastic waves excited by a plane piezoelectric source with an arbitrary shape on the surface of a multilayered medium have been studied for the first time in this paper. On the basis of Abzo-zena [Geophys. J. R. Astron. Soc. 58, 91-105 (1979)] and Menke [Geophys. J. R. Astron. Soc. 59, 315-323 (1979)], the propagator matrix for the elastic wave field in multilayered medium is extended from two- to three-dimensional (3D) space. 3D elastic wave field is investigated and the displacement-stress response for the boundary conditions is obtained. The propagation of elastic wave in multilayered media is analyzed in 3D space in the frequency domain. The P-SV and SH modes corresponding to the poles are studied. The excitation and propagation of the modes are analyzed further. It is found that the propagation velocities of the P-SV and SH modes do not depend on the propagation azimuth theta in the plane parallel to the free surface of the multilayered medium while the displacement amplitudes are strongly dependent on the azimuth theta. The directional distribution functions of the modes are independent of the medium parameters and the modes and dependent on the shape and excitation fashion of the source. Finally, as an example, the displacement fields of the P-SV and SH modes excited by a rectangle source are analyzed. The displacement representation and numerical results of the directivity distribution functions for the P-SV and SH modes are obtained.     

Diffraction of elastic waves by a sub-surface crack (anti-plane motion)

A rigorous theory of the diffraction of SH-waves by a stress-free crack embedded in a semi-infinite elastic medium is presented. The incident time-harmonic SH-wave is taken to be either a uniform plane wave or a cylindrical wave originating from a surface line-source. The resulting boundary-value problem for the unknown jump in the particle displacement across the crack is solved by employing an integral equation approach. The unknown quantity is expanded in a complete sequence of Chebyshev polynomials. By writing the Green function as a Fourier integral, an infinite system of linear, algebraic equations for the expansion coefficients is obtained. Numerical results are presented for the particle displacement at the surface of the half-space, the far field radiation characteristic, the scattering cross-section of the crack and the dynamic stress intensity factor at the crack tips, for a range of geometrical parameters.     

Scattering of sound by an elastic spherical shell immersed in a waveguide with a fluid bottom

The paper is devoted to modeling an acoustic field scattered by an elastic spherical shell. The shell is immersed in a waveguide with a fluid attenuating bottom. Modal analysis is applied to calculations of both the point source field in a free waveguide and the scattering coefficients of the shell. The occurring integrals along a branch cut are expressed via the probability integral. Cases are analyzed when the source frequency differs from the critical frequencies of normal waves and when it coincides with one of them. The method is applied to estimate the effect of the field scattered by a thin elastic shell filled with air with transmission loss of the total acoustic field with distance and its depth distribution.     

The dynamic analysis of a cracked Timoshenko beam by the spectral element method

The aim of this paper is to introduce a new finite spectral element of a cracked Timoshenko beam for modal and elastic wave propagation analysis. The proposed approach deals with the spectral element method. This method is suitable for analyzing wave propagation problems as well as for calculating modal parameters of the structure. In the paper, the results of the change in modal parameters due to crack appearance are presented. The influence of the crack parameters, especially of the changing location of the crack, on the wave propagation is examined. Responses obtained at different points of the beam are presented. Proper analysis of these responses allows one to indicate the crack location in a very precise way. This fact is very promising for the future work in the damage detection field.     

DYNAMIC BEHAVIOUR OF A CRACKED SOLDERED JOINT

The two-dimensional “in-plane” time-harmonic elastodynamic problem for a multi-layered cracked soldered joint system is studied. This problem is solved by using a hybrid of both displacement and hyper-singular traction boundary integral equation method. The proposed method directly accounts for the effect of the outer boundary of a finite multi-layered body and the interaction between the internal and interface cracks. The open fracture model is used to present the interface crack. Numerical results are shown and discussed to reveal the effect of the existence and sizes of cracks, the crack interaction, the debonding effect, the influence of the wave frequency and the type of the material combination on the crack-tip fracture parameters and the displacement scattered far field.     

Scattering of a Rayleigh surface acoustic wave by a small-size inhomogeneity in a solid half-space

We use the Born approximation of the perturbation method to solve the problem of scattering of a harmonic Rayleigh surface acoustic wave by a weak-contrast inhomogeneity that is small compared with the wavelength and is located in a solid half-space near its boundary. The material of the inhomogeneity differs from the material of the half-space only in its density. The Rayleigh wave incident on the inhomogeneity is excited by a monochromatic surface force source acting normally to the half-space boundary. We derive expressions for the displacement fields in the scattered spherical compressional and shear (SV- and SH-polarized) waves. Scattering of the Rayleigh wave into a Rayleigh wave is studied in detail. We find expressions for the vertical and horizontal components of the displacement vector in the scattered Rayleigh wave as well as its radiated power. It is shown that the field of the scattered surface wave is mainly formed by vertical oscillations of the inhomogeneity in the field of the incident wave. In this case, the radiated power for the scattered Rayleigh wave formed by vertical motion of the inhomogeneity in the incident-wave field depends on the depth of the inhomogeneity as the fourth power of the function describing the well-known depth dependence of the vertical displacements in the Rayleigh surface wave. Correspondingly, the dependence of the radiated power for the scattered Rayleigh wave formed by horizontal motion of the inhomogeneity depends on its location depth as the fourth power of the depth dependence of the horizontal displacements in the Rayleigh surface wave. We perform calculations of the ratio between the powers of the scattered and incident Rayleigh waves for different ratios between the velocities of the compressional and shear waves in a solid. It is shown that the radiated power for the scattered surface wave decreases sharply with increasing depth of the subsurface-inhomogeneity location. Thus, the scattering of a Rayleigh wave into a Rayleigh wave is fairly efficient only when the location depth of the inhomogeneity does not exceed about one-third of the wavelength of the shear wave in an elastic medium.     

水下掩埋目标的散射声场计算与实验

水下掩埋目标声散射问题是识别和探测掩埋目标的理论基础, 是声散射研究领域的热点问题. 本文基于射线声学推导了掩埋情况下目标声散射计算的格林函数近似式, 并在此基础上进一步给出了相应的远场积分公式. 在有限元方法的基础上, 将推导得到的公式写入有限元仿真软件, 对软件功能进行拓展, 构建二维轴对称目标的声散射模型, 并计算掩埋情况下弹性实心球在不同条件下的目标强度, 获得了其散射声场随频率、掩埋深度、沙层吸收系数等参数的变化规律. 开展实心球的自由空间和浅掩埋条件下水池声散射实验, 利用共振隔离技术处理实验数据, 提取目标声散射的纯弹性共振特征进行分析, 结果表明可将其用于掩埋目标识别和探测. 最后利用总散射声场与理论计算结果进行对比, 验证了理论仿真的正确性.     

On multidimensional ultrasonic scattering in an inhomogeneous elastic background.

This work is concerned with the modeling of elastic wave scattering by solid or fluid-filled objects embedded in an inhomogeneous elastic background. The medium is probed by a monochromatic force and the scattered field is computed (forward problem) or observed (inverse problem) at some known receiver locations. Based on vector integral equations for elastic scattering, a general framework is developed, independent of both the problem geometry and the transmitter-receiver characteristics. This framework encompasses both forward and inverse modeling. In the forward model, a Born approximation for an inhomogeneous background is applied to obtain a closed form expression for the scattered field. In the inverse model, this approximation is also invoked to linearize for the multiparameter characteristic of the object. Finally, an iterative inversion scheme alternating forward and inverse modeling is proposed to improve the resolution and accuracy of the reconstruction algorithm.     

Specular propagation over rough surfaces: numerical assessment of Uscinski and Stanek's mean Green's function technique

The purpose of this paper is to numerically evaluate the effectiveness and accuracy of Uscinski and Stanek's mean Green's function technique for computing the mean field of a wave scattered by a rough surface. We present here a direct comparison of this technique with a rigorous numerical method, the forward scattering integral equation method, and another analytical method, the first-order smoothing approximation. Furthermore, we compare the roughness generated equivalent admittance using the three methods. Numerical computations reveal that the scattered field calculated by this technique is not accurate particularly for the equivalent admittance at low grazing angles, even though the mean surface current density is recovered when the wave has traversed several correlation lengths on the surface.     

Scattering and diffraction of a plane wave by a randomly rough half-plane

The scattering and diffraction of a TE (transverse electric) plane wave by a randomly rough half-plane are studied by a combination of three techniques: the Wiener-Hopf technique, the small perturbation method and a probabilistic method based on the shift-invariance of a homogeneous random function. By use of the D^a-Fourier transformation based on the shift-invariance, it is shown that the scattered wave is written by an inverse Fourier transformation of a homogeneous random function with a complex parameter. For a small rough case, such a random function with a complex parameter is expanded in a perturbation series and then the first-order solution is obtained exactly in an integral form. The first-order solution involves two physical processes such that the edge-diffracted wave is scattered by the randomly rough plane and the scattered wave, due to roughness, is diffracted by the half-plane. The solution is transformed into a sum of the Fresnel integrals with complex arguments, an integral along the steepest descent path and a branch-cut integral, which are evaluated numerically. Then, intensities of the coherently scattered wave and incoherent wave are calculated in the region near the edge and illustrated in figures.     

Scattering and diffraction of a plane wave by a randomly rough half-plane*

Abstract

The scattering and diffraction of a TE (transverse electric) plane wave by a randomly rough half-plane are studied by a combination of three techniques: the Wiener-Hopf technique, the small perturbation method and a probabilistic method based on the shift-invariance of a homogeneous random function. By use of the D^a-Fourier transformation based on the shift-invariance, it is shown that the scattered wave is written by an inverse Fourier transformation of a homogeneous random function with a complex parameter. For a small rough case, such a random function with a complex parameter is expanded in a perturbation series and then the first-order solution is obtained exactly in an integral form. The first-order solution involves two physical processes such that the edge-diffracted wave is scattered by the randomly rough plane and the scattered wave, due to roughness, is diffracted by the half-plane. The solution is transformed into a sum of the Fresnel integrals with complex arguments, an integral along the steepest descent path and a branch-cut integral, which are evaluated numerically. Then, intensities of the coherently scattered wave and incoherent wave are calculated in the region near the edge and illustrated in figures.     

Elastic scattering amplitude of a scalar particle in a plane wave field

The elastic scattering amplitude of a scalar particle in an arbitrary plane wave electromagnetic field is obtained in the form of a double integral by the method of dispersion relations. Particular cases of giving the plane wave field are investigated. It is shown that the existence of scalar particle radiation in an arbitrary plane wave electromagnetic field results in elastic scattering, whose amplitude determines the change in particle mass in this field.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 5, pp. 32–37, May, 1990.     

The scattering of electromagnetic waves in fractal media

The scattering of electromagnetic waves in fractal media is studied. The fractal dimension is naturally involved in the formulation of two physical problems studied in this paper. The general theory of multiple scattering of electromagnetic wave in fractal media is developed by modifying Twersky's theory. Statistical quantities, such as the average field and average intensity of the multiple scattered wave, are studied for a wave propagating in a fractal medium. The scattering cross section of the medium is deduced. The backscattering of electromagnetic waves is also studied. The results showing the range of dependence of the backscattered signals are in agreement with numerical simulations by Rastogi and Scheucher (1990). It also suggests a method of measuring the fractal dimension of the fractal embedded media using radar sounding. The theory developed in this paper can also be used for problems related to multiple scattering of other kinds of waves, such as acoustic waves, elastic waves etc, in fractal media.     

Seismic wave propagation in laterally inhomogeneous geological region via a new hybrid approach

2D seismic wave propagation in a local multilayered geological region rested in an inhomogeneous half-space with a seismic source is studied. Plane strain state is suggested. The vertical variation of the soil properties in the half-space is modelled by a set of horizontal flat isotropic, elastic and homogeneous layers. The finite local region is with non-parallel layers and free surface relief. Efficient hybrid wavenumber integration-boundary integral equation method (WNI-BIEM) is proposed, validated and applied for synthesis of seismic signals in the finite soil stratum. The numerical simulation reveals that the developed hybrid method is able to demonstrate the sensitivity of the obtained synthetic signals to the seismic source properties, to the heterogeneous character of the wave path and to the relief peculiarities of the local stratified geological deposit. The advantages and disadvantages of the proposed method are discussed.     

Specular propagation over rough surfaces: numerical assessment of Uscinski and Stanek's mean Green's function technique

The purpose of this paper is to numerically evaluate the effectiveness and accuracy of Uscinski and Stanek's mean Green's function technique for computing the mean field of a wave scattered by a rough surface. We present here a direct comparison of this technique with a rigorous numerical method, the forward scattering integral equation method, and another analytical method, the first-order smoothing approximation. Furthermore, we compare the roughness generated equivalent admittance using the three methods. Numerical computations reveal that the scattered field calculated by this technique is not accurate particularly for the equivalent admittance at low grazing angles, even though the mean surface current density is recovered when the wave has traversed several correlation lengths on the surface.     

TOMOGRAPHY FORMULA FOR BIOCHEMICAL IMAGING OF THIN TISSUE WITH DIFFUSE-PHOTON DENSITY WAVES

Using the transport theory to describe the near infrared light propagating in tissue with finite parallel-plane geometry, and taking the zero-boundary condition, we obtain the analytical expression of average photon density and Green's function incorporating the boundary effects in the homogeneous tissure. Making use of perturbation theory we also obtain the analytical expression of scattered wave induced by the heterogeneity, and present the 2-dimensional spatial transform of scattered wave with respect to transverse coordinate. If the information of heterogeneity on depth and thickness is available, diffraction tomography formula is presented to save the time of image reconstruction; if the information is unknown, we suggest to obtain the inhomogeneous function from the one-dimensional integral equation of 2-dimensional spatial transform of scattered wave applying the direct matrix method or iterative method for image reconstruction. This approach avoids directly solving three-dimensional integral equation of scattered wave. In our proposed approach the strong points of the direct matrix method, iterative method, and diffraction tomography are fully combined.     

Ultrasonic sizing of short surface cracks

This paper presents a method for ultrasonic sizing of surface cracks based on time domain and frequency domain Rayleigh wave near-field analysis. The procedure allows for the entire range of ratio of crack depth to Rayleigh wavelength a/λ to be covered with one single measurement. In the time domain the time-of-flight method was extended to cracks smaller than the wavelength by correlation of the time delay of the transmitted Rayleigh wave with the crack depth. In the frequency domain, the inverse scattering problem was solved by comparison of the measured scattering coefficients and central frequencies of the reflected and transmitted Rayleigh waves with theoretical curves. The sizing procedure was demonstrated experimentally with narrow slots and real fatigue cracks. The out-of-plane displacement component was measured pointwise in the scattered near field by means of laser interferometry. The determination of the scattering parameters in the near field was enabled by a procedure that allows for the Rayleigh wave to be separated from the other modes scattered at the defect. The experimental results showed good accuracy and repeatability down to the smallest available ratio of crack depth to Rayleigh wavelength a/λ = 0.15.