Verification of surface wave solutions obtained by the reciprocity theorem

Surface wave motions generated by a time-harmonic point load applied at the surface of an isotropic linearly elastic half-space are conventionally solved by the use of integral transform techniques. The inverse transforms, are often complicated and will not always yield closed-form solutions. In this paper expressions for the displacements for surface wave motions radiated from point-load excitation are determined in a simple manner by the use of the elastodynamic reciprocity theorem. It is shown that the radiated amplitudes of the surface displacements obtained by the reciprocity approach are identical to the corresponding results obtained by the use of Hankel transform and by Lamb in his classical paper.     

Retrieving the elastodynamic Green's function of an arbitrary inhomogeneous medium by cross correlation

A correlation-type reciprocity theorem is used to show that the elastodynamic Green's function of any inhomogeneous medium (random or deterministic) can be retrieved from the cross correlation of two recordings of a wave field at different receiver locations at the free surface. Unlike in other derivations, which apply to diffuse wave fields in random media or irregular finite bodies, no assumptions are made about the diffusivity of the wave field. In a second version, it is assumed that the wave field is diffuse due to many uncorrelated sources inside the medium.     

Mackey Theorem and Two-Electron Wave Function of a Multi-Centre System

The two-electron wave function in a system of many equivalent atoms is investigated group-theoretically. It is shown that the classification of different types of two-electron (two-hole) localizations can be made by the double-coset decomposition of the symmetry group with respect to the local subgroup, and that the group appearing in the Mackey theorem can be used for the additional classification of states. The Mackey theorem on symmetrized squares and the generalized Frobenius reciprocity theorem are applied to the construction of two-electron states in octahedral symmetry. Received October 23, 1995; revised June 21, 1996; accepted for publication July 1, 1996     

A METHOD TO PREDICT ACOUSTIC RADIATION FROM AN ENCLOSED MULTICAVITY STRUCTURE

The acoustical reciprocity theorem can be used to solve the problem of vibroacoustic coupling. However, the theorem can be used only on the presupposition that the scattered sound field of the elastic surface concerned is known. This is the key point and the most difficult point for many complicated surfaces, such as a multicavity structure. A new method, covering-domain method, which transforms the calculation of scattered sound field of an arbitrary-shaped closed shell into that of a series of simply closed spherical shells, is applied in this paper to calculate the scattered sound field of a multicavity structure with elastic surfaces. So the radiated sound pressure of an elastic multicavity structure excited by an external force can be predicted by using the acoustical reciprocity theorem. It is verified to be correct by a corresponding test in this paper.     

Modeling of wave propagation in inhomogeneous media

We present a methodology providing a new perspective on modeling and inversion of wave propagation satisfying time-reversal invariance and reciprocity in generally inhomogeneous media. The approach relies on a representation theorem of the wave equation to express the Green function between points in the interior as an integral over the response in those points due to sources on a surface surrounding the medium. Following a predictable initial computational effort, Green's functions between arbitrary points in the medium can be computed as needed using a simple cross-correlation algorithm.     

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。     

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.     

Detection and location of cracks using loss of reciprocity in ultrasonic waves propagation

The reciprocity theorem is a general statement valid for elastic media, and it has been applied to the solution of elastic wave equations, transducers calibration, time reversal acoustics, etc. However, localized nonlinear scatterers are expected to break reciprocity even though the effect is, in several cases, negligible. Here the dependence of the reciprocity break on the presence of a localized damage and the influence of its relative position has been experimentally investigated. It will be shown that the break of reciprocity, usually considered a disadvantage, can be exploited as an imaging tool for localized cracks detection.     

The Density of Surface States as the Total Time Delay

For a scattering problem of tight-binding Bloch electrons by a weak random surface potential, a generalized Levinson theorem is put forward showing the equality of the total density of surface states and the density of the total time delay. The proof uses explicit formulas for the wave operators in the new rescaled energy and interaction representation, as well as an index theorem for adequate associated operator algebras.     

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。     

Three-dimensional steady state Green function for a layered isotropic plate

The elastodynamic response of a layered isotropic plate to a source point load having an arbitrary direction is studied in this paper. A decomposition technique is developed within each homogeneous isotropic lamina to simplify the general three-dimensional plane-wave propagation problem as a separate plane-strain problem and an anti-plane-wave propagation problem. The accuracy of computation is assured by cross-checking the numerical results by different methods. Results are checked numerically for a vertical point load acting on a homogeneous and a layered plate by using a hybrid method. On the other hand, results are checked for a horizontal point load by using dynamic reciprocal identities. Results are presented for both a homogeneous as well as a layered plate.     

Surface properties of a network model of electrons in solids

Surface properties of a network model of electrons in solids are discussed. In the model, electrons are restricted to move along lines connecting node points of a periodic network. Each node point on the network corresponds to an atom. A potential well is introduced for each atom or node point so that as the depth of the potential becomes large, one has the tight binding model and, as the potential vanishes and the number of node points increases, one has the Sommerfeld free electron model. Exact wave functions and energy levels have been obtained elsewhere for the model when periodic boundary conditions are applied.When free boundaries are introduced, a band of surface states develops. Its depth of penetration into the crystal as a function of wave number is investigated. The edges of the surface energy bands are located and their density of states is calculated.     

Acoustic radiation generated by local excitation of submerged beams and strings

Free and forced motions of submerged one-dimensional waveguides have been investigated. Both the submerged beam and the submerged string can support a wave system which includes a pure surface wave in the fluid adjoining the waveguide, provided that certain conditions on the frequency (for the beam) or the physical parameters of the system (for the string) are satisfied. Dispersion curves are presented for steel beams and steel strings in water. Steady-state solutions have been derived for excitation by a concentrated time-harmonic load. The displacement responses at the point of application of the load and in the far field have been examined. Polar plots display and radiated far field in the fluid. At low frequencies the polar plots are circular in planes containing the waveguide, but the radiation pattern develops lobes as the frequency increases. Reflection, transmission and scattering into the fluid of an incident wave system by an elastic support of stiffness κ and mass m have also been examined, and the possibility of resonance phenomena has been investigated.     

Temporal analysis of tissue displacement induced by a transient ultrasound radiation force

One of the stress sources that can be used in dynamic elastography imaging methods is the acoustic radiation force. However, displacements of the medium induced by this stress field are generally not fully understood in terms of spatial distribution and temporal evolution. A model has been developed based on the elastodynamic Green's function describing the different acoustic waves generated by focused ultrasound. The function is composed of three terms: two far-field terms, which correspond to a purely longitudinal compression wave and a purely transverse shear wave, and a coupling near-field term which has a longitudinal component and a transverse component. For propagation distances in the shear wavelength range, the predominant term is the near field term. The displacement duration corresponds to the propagation duration of the shear wave between the farthest source point and the observation point. This time therefore depends on the source size and the local shear modulus of the tissue. Evolution of the displacement/time curve profile, which is directly linked to spatial and temporal source profiles, is computed at different radial distances, for different durations of force applications and different shear elastic coefficients. Experimental results performed with an optical interferometric method in a homogeneous tissue-mimicking phantom agreed with the theoretical profiles.     

金属薄膜亚波长微结构的光束集束器件设计

利用时域有限差分方法,模拟了P偏振光高斯光束入射到金属银亚波长细缝与光栅结构的透射情况.由于表面等离子激元波的影响,对称光栅结构透射光呈现对称出射,而非对称结构可以实现小角度定方向的光束集束现象.借助公式推导法及Helmholtz互易定理,可设计出平行入射P偏振光的光束集束器件.由于高斯光束本身的发散性及近场分布的需要,针对高斯光束的器件在结构参量上缩小了12%.在对其他结构参量的优化的基础上,实现了针对633 nm高斯光束的对称出射及光束集束器件的设计.     

互易定理计算分层半空间上方任意取向偶极子的远区场

利用互易定理推导了考虑直达波和界面反射波时半空间分层界面上任意取向偶极子在上半空间远区辐射场的解析表达式. 进一步考虑电偶极子距离分界面为某一高度情形, 在考虑直达波和反射波程差基础上导出相应辐射场公式. 辐射场解析表达式中分层半空间TM和TE波反射系数可以通过连分数方法或传播矩阵法计算获得. 本文推导过程物理概念清晰, 所得解析结果适于计算观察点远离界面情形下偶极子的远区辐射场. 数值计算结果表明, 利用本文结果可以快速分析半空间上方任意取向偶极子的远区辐射场.     

On the geometry of motions in one integrable problem of the rigid body dynamics

Due to Poinsot’s theorem, the motion of a rigid body about a fixed point is represented as rolling without slipping of the moving hodograph of the angular velocity over the fixed one. If the moving hodograph is a closed curve, visualization of motion is obtained by the method of P.V. Kharlamov. For an arbitrary motion in an integrable problem with an axially symmetric force field the moving hodograph densely fills some two-dimensional surface and the fixed one fills a three-dimensional surface. In this paper, we consider the irreducible integrable case in which both hodographs are two-frequency curves. We obtain the equations of bearing surfaces, illustrate the main types of these surfaces. We propose a method of the so-called non-straight geometric interpretation representing the motion of a body as a superposition of two periodic motions.     

On interference effects in the elastodynamic Green's functions G33(x, omega) of Si and GaAs

In this paper, we analyze interference effects present in the elastodynamic Green's functions G33(x,omega) of the cubic crystals Si and GaAs, which are associated with folded portions of the wave surface of the slow transverse (ST) acoustic mode. G33(x,omega) represents the three dimensional extension of the amplitude distribution imaged in the transmission acoustic microscopy of these crystals. The intensity contrast for oscillations of a particular wave vector k in the interference pattern is determined essentially by the 3D Fourier transform of G33(x,omega)G33*(x,omega). According to the Fourier autocorrelation theorem, that transform is equivalent to the autocorrelation function of the corresponding distribution G(33)(k,omega) in k-space. We show that due to the linear mapping between k-space and the slowness vector s-space, the interference phenomena discussed here are related to geometrical features of the slowness surface of the ST mode. We present calculations of these effects based on the angular spectrum technique.     

The form-invariance of wave equations without requiring a priori relations between field variables

According to the principle of relativity,the equations describing the laws of physics should have the same forms in all admissible frames of reference,i.e.,form-invariance is an intrinsic property of correct wave equations.However,so far in the design of metamaterials by transformation methods,the form-invariance is always proved by using certain relations between field variables before and after coordinate transformation.The main contribution of this paper is to give general proofs of form-invariance of electromagnetic,sound and elastic wave equations in the global Cartesian coordinate system without using any assumption of the relation between field variables.The results show that electromagnetic wave equations and sound wave equations are intrinsically form-invariant,but traditional elastodynamic equations are not.As a by-product,one can naturally obtain new elastodynamic equations in the time domain that are locally accurate to describe the elastic wave propagation in inhomogeneous media.The validity of these new equations is demonstrated by some numerical simulations of a perfect elastic wave rotator and an approximate elastic wave cloak.These findings are important for solving inverse scattering problems in many fields such as seismology,nondestructive evaluation and metamaterials.     

Narrow band laser-generated surface acoustic waves using a formed source in the ablative regime

A narrow band laser-generated acoustic signal was created using a 4-element lenticular array. This arrangement of the array produces an acoustical signal with frequency content that is compatible with the response of a noncontact and remote broadband receiver, such as a capacitive air-coupled transducer. To support the experimental observations, a simplified concept is presented to explain the effect of a line array source on the frequency content of a surface acoustic wave. The analytical model solution for the wave front shape is derived from the point load solution of Lamb's problem that represents the displacement of a surface acoustic wave generated by an ablative line array. The distribution function, which was used for the model to represent the laser light energy, was tailored to depict the actual energy distribution that illuminates the lenticular array. Filtering functions are applied to the resultant surface displacement function to retain frequencies similar to those detected by the broadband 50 kHz-2.25 MHz receiver. The theoretical model showed good agreement with experimental results.