Calculation of surface wave motions due to a subsurface point force: an application of elastodynamic reciprocity

The elastodynamic reciprocity theorem for two time-harmonic elastodynamic states of the same body is used to determine the surface wave motions generated by a subsurface point load of arbitrary direction in an isotropic, homogeneous, elastic half-space. The actual surface wave motions due to the point load expressed in a suitably general form represent one of the states, while the other one is an appropriately selected auxiliary solution consisting of incoming and outgoing surface waves. A direct application of the reciprocity theorem yields the desired information on the generated surface wave motion.     

Laser-ultrasonic generation of Lamb waves in the reaction force range

The Laser-ultrasonic generation of Lamb waves in an elastic plate is investigated theoretically and experimentally for a laser source whose intensity is high enough to create reaction forces (normal tractions) on the illuminated surface of the specimen. The analytical solutions for transient waves are derived using the integral transform method first by considering an arbitrary source shape and time excitation function, and then specifically for circular and line source shapes. The simulation study allows us not only to predict the behavior of individual wave modes but also to construct the overall responses; thus it helps us better understand the wave excitation mechanisms. The dispersive and multi-modal nature of laser-generated Lamb waves is presented by showing the spatiotemporal Fourier transform of displacements obtained by the simulation study. The transform, displayed in the frequency-wave number domain, enunciates the characteristics of the propagating individual Lamb wave modes. The simulation results are then compared with the 2-D Fourier transform of a set of experimental data obtained by scanning an aluminum plate specimen.     

Time-reversal imaging of objects near rough surfaces based on surface flattening transform

This paper considers the imaging of objects located close to rough surfaces such as ocean or terrain. If transmitters and receivers are also located close to rough surfaces, incident wave is no longer a plane wave nor a spherical wave in free space and it is necessary to consider Green’s functions with the point source located close to the surface, similar to the Sommerfeld dipole problem. This paper considers the near-surface imaging by making use of time-reversal imaging and surface flattening transform. Surface flattening transform converts the rough surface problem into flat surface with inhomogeneous random medium. Mutual coherence function is obtained and used to obtain imaging of point target near rough surface, making use of the multi-static data matrix, time-reversal matrix, the eigenvectors, and the steering matrix. Numerical examples are given. An important point is that integration of stochastic wave propagation and signal processing is necessary to obtain imaging through complex clutter environment. Surface flattening transform is related to the transformation electromagnetics which attracted much interest because of cloaking possibilities. This paper includes some discussions on the relations between surface flattening and transformation electromagnetics.     

Axisymmetric wave propagation of thermoelastic transversely isotropic half-space under buried loading using potential functions

By virtue of a new scalar potential function and Hankel integral transforms, the wave propagation analysis of a thermoelastic transversely isotropic half-space is presented under buried loading and heat flux. The governing equations of the problem are the differential equations of motion and the energy equation of the coupled thermoelasticity theory. Using a scalar potential function, these coupled equations have been uncoupled and a six-order partial differential equation governing the potential function is received. The displacements, temperature, and stress components are obtained in terms of this potential function in cylindrical coordinate system. Applying the Hankel integral transform to suppress the radial variable, the governing equation for potential function is reduced to a six-order ordinary differential equation with respect to z. Solving that equation, the potential function and therefore displacements, temperature, and stresses are derived in the Hankel transformed domain for two regions. Using inversion of Hankel transform, these functions can be obtained in the real domain. The integrals of inversion Hankel transform are calculated numerically via Mathematica software. Our numerical results for displacement and temperature are calculated for surface excitations and compared with the results reported in the literature and a very good agreement is achieved.     

平行柱体对平面波/高斯波束电磁散射

基于等效原理和互易性定理，研究了N个相互平行二维柱体对平面波/高斯波束的电磁散射特性，给出了求解N阶散射场公式.一阶散射可通过求解单个柱体的散射场得到，但对于高阶散射场而言，由于耦合散射的复杂性，很难给出精确的解析解.为了解决这一问题，借助等效原理和互易性定理给出了求解N阶散射场的面积分公式.只要给出柱体的i-1阶散射场及相关目标表面上的等效电流和(或)等效磁流，就可应用此公式求解i阶散射场.应用该近似方法计算了相互平行非均匀等离子体涂层导体圆柱的单/双站散射宽度，讨论了束腰半径、等离子体涂层厚度、电子密度、碰撞频率及雷达频率等对散射结果的影响.     

Free wave propagation in two-dimensional periodic plates

The propagation of flexural waves in a two-dimensional periodic plate which rests on an orthogonal array of equi-spaced simple line supports has been investigated. A type of plane wave motion has been considered. An energy method has been developed to predict the frequency of wave propagation in terms of the propagation constants. A Galerkin type of analysis has been used, incorporating assumed complex modes of wave motion for the identical rectangular elements of the periodic plate. Expressions for the frequency have been obtained firstly by using simple polynomial modes for the plate displacements, and then (alternatively) by using characteristics beam function modes. The use of these different modes has first been demonstrated by applying them to the analysis of wave propagation in periodic beams. A single polynomial mode which satisfies the geometric and wave-boundary conditions of the periodic plate element leads to an elegant expression relating the frequency and the wave propagation constants in the first propagation band. The frequencies so obtained compare well with those found from a multi-mode, characteristic beam function analysis. The latter involves much more algebra, is solved as an eigenvalue problem, and yields the frequencies in as many propagation bands as are desired. The bounding frequencies and corresponding wave motions in the first and higher propagation bands have been identified, and it has been shown that the propagation bands can overlap. Consideration has been given to one-dimensional “strip” structures which are equivalent to the two-dimensional plate when a plane wave in a general direction is propagating. Furthermore, it is shown that the natural frequencies of finite rectangular periodic plates can be obtained very simply from the results of the wave propagation analysis.     

Simulating regular wave propagation over a submerged bar by boundary element method model and Boussinesq equation model

Numerical models based on the boundary element method and Boussinesq equation are used to simulate the wave transform over a submerged bar for regular waves.In the boundary-element-method model the linear element is used,and the integrals are computed by analytical formulas.The Boussinesq-equation model is the well-known FUNWAVE from the University of Delaware.We compare the numerical free surface displacements with the laboratory data on both gentle slope and steep slope,and find that both the models simulate the wave transform well.We further compute the agreement indexes between the numerical result and laboratory data,and the results support that the boundary-element-method model has a stable good performance,which is due to the fact that its governing equation has no restriction on nonlinearity and dispersion as compared with Boussinesq equation.     

Exact and approximate analysis of surface acoustic waves in an infinite elastic plate with a thin metal layer

For an accurate approximation of the effect of a thin layer over a finite substrate, we consider the displacements are continuous across the interface, while the stress components are obtained from derivatives of displacements. As a result, the stress boundary conditions are transformed to a relationship between stresses in the vicinity of the interface of two layers and density of the metal layer. Through this approximation, we eventually have four equations to solve for the surface acoustic wave dispersion relation of the two-layer structure. The approximate and accurate results are compared with good agreement for small thickness of the metal layer. These results are intended for periodic structures with separate solutions for electroded and unelectroded regions, which can be connected by the continuity boundary conditions for the analysis of the complete structure of typical surface acoustic wave resonators.     

基于声压场描述的扩散声场多自由度互易原理研究

应用于复杂结构中频声振分析的扩散场多自由度互易原理采用位移变量描述系统,实质为弹性波场互易原理,应用于声波场时会造成模型自由度数不必要的增加。建立基于声压描述的扩散声场受挡模型,利用声辐射模态描述扩散声场中结构的表面受挡声压;据此提出基于声压描述的扩散声场多自由度互易原理,发现扩散声场中结构表面受挡声压的互谱矩阵与该结构在自由空间中振动辐射声波的声阻矩阵成正比。该互易原理与传统的单自由度互易原理表达形式相似,但适用于任意自由度结构。该互易原理可用于扩散声场中复杂结构的表面受挡声压的自谱及相关分析,仿真研究表明当边界元网格尺寸小于声波波长的1/6(线性单元)或1/3(二次单元)时,数值解与理论解完全吻合。      

Acoustic radiation of cylindrical elastic shells subjected to a point source: investigation in terms of helical acoustic rays.

The paper deals with the acoustic radiation of a cylindrical elastic shell with no internal loading surrounded by a fluid medium when its external surface is subjected to a point source. The problem is addressed via the use of the spatial Fourier transform. An expression is obtained for the radiated pressure that is evaluated for the far field using both the stationary phase method and the fast Fourier transform (FFT). The acoustic field calculated from the FFT is much more complicated than that obtained by using only the stationary phase method. In agreement with the geometrical theory of diffraction (GTD), alternative interpretations of the radiated field in terms of helical acoustic rays allows one to understand the reason for this result. The outstanding phenomenon underlined by the use of the FFT is the emergence of an infinite number of spatial dispersion curves associated with each leaky wave propagating in shells when excited by a point source.     

Second-order random wave solutions for interfacial internal waves in N-layer density-stratified fluid

This paper studies the random internal wave equations describing the density interface displacements and the velocity potentials of N-layer stratified fluid contained between two rigid walls at the top and bottom. The density interface displacements and the velocity potentials were solved to the second-order by an expansion approach used by Longuet-Higgins （1963） and Dean （1979） in the study of random surface waves and by Song （2004） in the study of second- order random wave solutions for internal waves in a two-layer fluid. The obtained results indicate that the first-order solutions are a linear superposition of many wave components with different amplitudes, wave numbers and frequencies, and that the amplitudes of first-order wave components with the same wave numbers and frequencies between the adjacent density interfaces are modulated by each other. They also show that the second-order solutions consist of two parts： the first one is the first-order solutions, and the second one is the solutions of the second-order asymptotic equations, which describe the second-order nonlinear modification and the second-order wave-wave interactions not only among the wave components on same density interfaces but also among the wave components between the adjacent density interfaces. Both the first-order and second-order solutions depend on the density and depth of each layer. It is also deduced that the results of the present work include those derived by Song （2004） for second-order random wave solutions for internal waves in a two-layer fluid as a particular case.     

Propagation of surface elastic waves in the Cosserat medium

The problem of surface elastic wave propagation in the Cosserat medium (half-space) is considered. The strained state is characterized by two independent vectors: displacement and rotation. Solutions to the equations of motion are sought in the form of wave packets specified by an arbitrary Fourier spectrum. It is shown that, if the solution is sought in the form of a three-component displacement vector and a three-component rotation vector dependent on time, depth, and longitudinal coordinate, the initial system splits into two systems, one of which describes the Rayleigh wave and the other corresponds to a transverse wave decaying with depth. For both waves, analytical solutions in terms of displacements are obtained. It should be particularly noted that, unlike the Rayleigh wave, the solution for the transverse surface wave has no analogues in the classical elasticity theory. The transverse wave solution is numerically compared with the Rayleigh wave solution.     

Defect detection and localization in orthotropic wood slabs by inversion of dynamic surface displacements

The nondestructive evaluation inversion and generalized force-mapping techniques developed and demonstrated for isotropic thin plates by Bucaro et al. [(2004). "Detection and localization of inclusions in plates using inversion of point actuated surface displacements," J. Acoust. Soc. Am. 115, 201-206] are extended to the case of orthotropic plates. The extended techniques are applied to a finite-element generated numerical database for point excited wooden slabs with and without an internal defect at 5 and 10 kHz. Operation of the original isotropic algorithms on the wood surface displacements is shown to fail in recovering the uniform elastic parameters or in detecting and locating the defect. The new algorithms based on the wave equation for a thin, orthotropic plate successfully convert the surface displacements on the uniform wooden slab to elastic parameter maps which serve to detect and localize the defect in the flawed plate. The results, particularly at the higher frequency, indicate that the onset of failure in the thin plate approximation impacts both the inversion and the generalized force-mapping accuracy. However, in this case use of the inversion algorithm to obtain modified wave equation coefficients followed by operation of the force-mapping algorithm with these new parameters inserted is shown to successfully mitigate this effect.     

两个相邻目标对平面波、高斯波束的光散射

基于等效原理和互易性定理研究了两个靠近目标对平面波、高斯波束的光散射问题,给出了这一复合光散射模型的二阶散射结果。通常一阶散射结果容易求解,但由于耦合效应的复杂性,很难给出二阶散射结果的解析形式。为了解决这一问题,应用互易性定理给出了求解任意相邻介质目标二阶散射场的公式,同时借助等效原理将求解散射场公式中的体积分简化为面积分的形式,从而降低了求解难度。求解了两相邻球形粒子的复合散射场,并将求解结果与应用时域积分方程法求得的结果进行了比较。同时,还讨论了束腰半径、目标位置对散射截面及偏振度的影响。     

A study of time harmonic guided Lamb waves and their caustics in composite plates

Spatial steady-state Lamb wave propagation in an anisotropic composite plate excited by harmonic surface sources is modeled using a Green’s matrix representation in a frequency-wavenumber domain. An approach based on a residue integration technique for two dimensional wavenumber integrals for the computation of displacements outside an excitation source is presented in this paper. In the far-field zone of the excitation source, the method of stationary phase is used, which gives an asymptotic expansion of the displacement vector as a sum of cylindrical waves. Near caustic directions, a far-field solution is computed in terms of Airy functions. The results obtained applying residue integration technique and asymptotic expansion are found to be coinciding with the results of the computation by using the adaptive quadratures. Moreover, these approaches agree well with experimental data. Then, the advantages and disadvantages of the various methods applied for modeling of Lamb wave propagation are discussed in this paper. Focussing and other properties of Lamb waves are studied using numerical examples.     

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.     

Acoustic waves generated by a laser point pulse in a transversely isotropic cylinder

A three-dimensional (3D) model is presented to predict the acoustic waves generated by a laser point pulse in a transversely isotropic cylinder. The Fourier series expansion and the two-dimensional Fourier transform are introduced to calculate the 3D transient response under either the ablation or the thermoelastic generation. The presented physical model and the numerical inverse scheme are applied to a fiber reinforced composite cylinder with a strong anisotropy. Experimental radial displacements of the cylinder surface are detected by the laser ultrasonic technique and analyzed by the ray trajectories for both generation regimes. Corresponding theoretical displacements are obtained numerically and compared to the experimental signals. Good agreement is found between theoretical and experimental results. The focusing effects that anisotropy gives rise to are observed in both theory and experiment under either regime.     

基于层状模型的功能梯度板中兰姆波传播研究

功能梯度板中的兰姆波的传播在实际工程中有着非常广泛的应用。采用层状模型研究了兰姆波在材料特性沿厚度方向连续变化的功能梯度板中的传播特性。通过数值计算获得了层状板中兰姆波的色散关系,并与已有结果进行了比较,获得了材料属性沿厚度方向呈指数变化和多项式变化时功能梯度板中兰姆波的波速和位移解。当材料属性连续变化时,兰姆波各阶模态的波速与位移都将发生变化。相比于兰姆波的高阶模态,低阶模态的波速变化更加明显。本文的研究可为功能梯度板的设计提供参考。     

Measurement of the dispersion relation of guided non-axisymmetric waves in filament-wound cylindrical structures

To determine the dispersion relation, guided waves are excited in specimens over a broad frequency range. The surface displacements are measured over time and space. The recorded data are analysed using a quasi-three-dimensional spectrum estimation algorithm. In the time domain a fast Fourier transform is used to extract the frequencies. To obtain the wave numbers, in space a two-dimensional matrix-pencil approach is applied to the data set. Using a suitable constitutive model (transversely isotropic or orthotropic) dispersion curves are calculated. Good agreement was found between the experimental and the numerically calculated dispersion relations after adjusting the material parameters. Since the dispersion relation of a structure depends on the mechanical material properties frequency-dependent material parameters can be extracted from the above-mentioned relation between frequency and wave number.     

Interior dynamics of subharmonious surface wave in an idealized bidimensional granular layer

In this paper, high-speed photography was used to investigate the intrinsic dynamics of subharmonious surface wave in a vertical vibrated and idealized bidimensional granular layer. Using the high-speed photography, velocity fields of the granular layer at different stages through two cycles were obtained, which show the continuous particle motions during a cycle. From the velocity fields, a crystal structure in the wave-hollow was observed, which is reported for the first time. Furthermore, quantitative results of kinetic energy distribution in the layer were calculated, which shows temporal correspondence with the evolution of the wave pattern.