Acoustic resonance scattering from a multilayered cylindrical shell with imperfect bonding

The method of wave function expansion is adopted to study the three dimensional scattering of a time-harmonic plane progressive sound field obliquely incident upon a multi-layered hollow cylinder with interlaminar bonding imperfection. For the generality of solution, each layer is assumed to be cylindrically orthotropic. An approximate laminate model in the context of the modal state equations with variable coefficients along with the classical T-matrix solution technique is set up for each layer to solve for the unknown modal scattering and transmission coefficients. A linear spring model is used to describe the interlaminar adhesive bonding whose effects are incorporated into the global transfer matrix by introduction of proper interfacial transfer matrices. Following the classic acoustic resonance scattering theory (RST), the scattered field and response to surface waves are determined by constructing the partial waves and obtaining the non-resonance (backgrounds) and resonance components. The solution is first used to investigate the effect of interlayer imperfection of an air-filled and water submerged bilaminate aluminium cylindrical shell on the resonances associated with various modes of wave propagation (i.e., symmetric/asymmetric Lamb waves, fluid-borne A-type waves, Rayleigh and Whispering Gallery waves) appearing in the backscattered spectrum, according to their polarization and state of stress. An illustrative numerical example is also given for a multi-layered (five-layered) cylindrical shell for which the stiffness of the adhesive interlayers is artificially varied. The sensitivity of resonance frequencies associated with higher mode numbers to the stiffness coefficients is demonstrated to be a good measure of the bonding strength. Limiting cases are considered and fair agreements with solutions available in the literature are established.     

Three-Dimensional Scattering of an Incident Plane Shear Horizontal Guided Wave by a Partly through-Thickness Hole in a Plate

We investigate the three-dimensional(3D) scattering problem of an incident plane shear horizontal wave by a partly through-thickness hole in an isotropic plate,in which the Lamb wave modes are also included due to the mode conversions by the scattering obstacle in the 3D problem.An analytical model is presented such that the wave fields are expanded in all of propagating and evanescent SH modes and Lamb modes,and the scattered far-fields of three fundamental guided wave modes are analyzed numerically for different sizes of the holes and frequencies.The numerical results are verified by comparing with those obtained by using the approximate Poisson/Mindlin plate model for small hole radius and low frequency.It is also found that the scattering patterns are different from those of the SO wave incidence.Our work is useful for quantitative evaluation of the plate-like structure by ultrasonic guided waves.     

Reflection of Lamb waves obliquely incident on the free edge of a plate

The reflection of obliquely incident symmetric and anti-symmetric Lamb wave modes at the edge of a plate is studied. Both in-plane and Shear-Horizontal (SH) reflected wave modes are spawned by an obliquely incident in-plane Lamb wave mode. Energy reflection coefficients are calculated for the reflected wave modes as a function of frequency and angle of incidence. This is done by using the method of orthogonal mode decomposition and by enforcing traction free conditions at the plate edge using the method of collocation. A PZT sensor network, affixed to an Aluminum plate, is used to experimentally verify the predictions of the analysis. Experimental results provide support for the analytically determined results.     

非线性兰姆波在阶梯板中传播的有限元仿真

在非线性兰姆波检测中,累积二次谐波是重要的检测信号.该文利用有限元法仿真了非线性兰姆波在阶梯板中的产生和传播过程,通过二维离散傅里叶变换分析了兰姆波的传播模态和各个模态通过波导横截面的能流大小,给出了各个模态在阶梯板中的反射、透射系数,总结了其随阶梯板厚度的变化规律.研究发现,随着台阶增高,散射效应使得兰姆波更难入射波...     

Low-frequency band gaps in one-dimensional thin phononic crystal plate with periodic stubbed surface

Using supercell plane wave expansion method, the Lamb wave band structure of one-dimensional thin plate with periodic stubs is investigated. The numerical results show that flat bands will appear and band gap can exist in a low-frequency domain. The position of the flat bands and width of the low-frequency Lamb wave band gap can be tuned by the stub height, plate thickness and filling fraction. The band gap is obtained by opening the folding points of the same plate modes not the crossing point of different plate mode when the stub height is small.     

Scattering matrices of Lamb waves at irregular surface and void defects

Time-harmonic solution of Lamb wave scattering in a plane-strain waveguide with irregular thickness is investigated based on stair-step discretization and stepwise mode matching. The transfer relations of the transmission matrices and reflection matrices are derived in both directions of the waveguide. With these, an explicit expression of the scattering matrix is derived. When the scattering region of an inner irregular defect is geometrically divided into several parts composed of sub-waveguides with variable thicknesses and void regions with vertical free edges corresponding to the plate surfaces, the scattering matrix of the whole region could then be derived by modal matching along the artificial boundaries, as explicit functions of all the scattering matrices of the sub-waveguides and reflection matrices of the free edges. The effectiveness of the formulation is examined by numerical examples; the calculated scattering coefficients are in good accordance with those obtained from numerical simulation models.     

Analysis of Second-Harmonic Generation of Low-Frequency Dilatational Lamb Waves in a Two-Layered Composite Plate

We analyze the effect of second-harmonic generation(SHG) of primary Lamb wave propagation in a two-layered composite plate, and then investigate the influence of interfacial properties on the said effect at low frequency. It is found that changes in the interfacial properties essentially affect the dispersion relation and then the maximum cumulative distance of the double-frequency Lamb wave generated. This will remarkably influence the efficiency of SHG. To overcome the complications arising from the inherent dispersion and multimode natures in analyzing the SHG effect of Lamb waves, the present work focuses on the analysis of the SHG effect of low-frequency dilatational Lamb wave propagation. Both the numerical analysis and finite element simulation indicate that the SHG effect of low-frequency dilatational Lamb wave propagation is found to be much more sensitive to changes in the interfacial properties than primary Lamb waves. The potential of using the SHG effect of low-frequency dilatational Lamb waves to characterize a minor change in the interfacial properties is analyzed.     

三层介质超声谐振模式随材料和界面粘接性能变化的演变规律

在用超声波谐振对粘接材料的粘接强度进行无损评估时, 不同模式对粘接强度的敏感程度受到众多因素和参数的影响, 对检测结果的可靠性至关重要. 基于多层介质中声传播和界面弱粘接边界条件的理论模型, 将一个上下非对称的金属-粘接剂-金属三层结构的平面波反射系数函数中的谐振模式看作是上下铝金属层各自的Lamb波频散模式通过夹心粘接剂层相互耦合后叠加组成. 改变影响结构粘接强度的因素, 即粘接剂的性能参数(声阻抗、密度、厚度)和界面切向劲度系数k_t来分析三层结构谐振模式耦合方式的变化,得出结论: 粘接结构粘接性能的变化基本上不改变与被粘铝层相关的固有部分的Lamb波模式, 而它们的耦合模式则在谐振频率上产生平移并会与固有模式进行交换和替代; 不同参数的变化引起的模式演变有各自的规律, 大多可彼此区分.     

The band gaps of Lamb waves in a ribbed plate: A semi-analytical calculation approach

In this paper a semi-analytical method is proposed to investigate the transmission band gaps of Lamb waves through repetitive structures in a waveguide. For a unit cell the scattering matrix is obtained by the Lamb mode matching technique at each artificially sliced interface and then substituted into the Bloch theorem to solve the eigenproblem incurred by cell repetition. The method is implemented on a ribbed plate. The band gaps in the dispersion curves are confirmed by the dips of Lamb wave transmission coefficients, and attributed to the coupled vibrations of Lamb modes in different sliced sections of the unit cell.     

On the reflection of coupled Rayleigh-like waves at surface defects in plates

The reflection of coupled Rayleigh-like waves from surface defects in elastic plates is investigated experimentally and analyzed on the basis of an analytical model and finite difference simulations. The propagation of Rayleigh-like waves in plates is characterized by an energy transfer to the opposite plate side and back over a distance called the beat length. Experimental results clearly show this beating effect and its dependency on the frequency-thickness product, and excellent agreement is obtained with existing analytical predictions. The propagation and scattering are modeled separately for the fundamental A(0) and S(0) Lamb modes that constitute the incident Rayleigh-like wave. The reflection coefficients from surface slots are investigated using finite difference simulations and the reflected Rayleigh-like wave is obtained by superposition. The theoretical model reveals strong dependencies of the reflected field on the ratio between excitation distance and beat length and on the cutoff frequencies of specific higher Lamb modes. Standard pulse-echo measurements allow for the detection of small defects from a remote transducer location. Good agreement is obtained between the predicted and measured amplitude spectra of the reflected Rayleigh-like wave. The developed model allows for the evaluation of defect location and damaged plate side using a combination of time-of-flight and frequency measurements.     

Prolonged acousto-optic interaction with Lamb waves in crystalline plates

The propagation and acousto-optic interaction of Lamb modes in an anisotropic plate of tellurium dioxide (TeO2) are studied numerically and analytically. In the case of a Y-cut X-propagating TeO2 plate, the very high elastic anisotropy of the crystal greatly modifies the dispersion curves, giving rise to their multiple oscillations. The existence ranges of backward Lamb modes increase with the mode order contrary to the case of isotropic plates. The quasi-collinear light scattering by Lamb waves is considered. Owing to the structure of Lamb wave field, a simultaneous light diffraction at two different optical frequencies can take place while Lamb waves are excited only at the single frequency. It is demonstrated with the Z-cut (110)-propagating plate that a small change in the acoustic frequency can result in a significant shift in the frequency of the scattered light.     

Simple expressions of the reflection and transmission coefficients of fundamental Lamb waves by a rectangular notch

The scattering of Lamb waves by a two-dimensional rectangular notch is investigated for rapid inspection of defects in a structure. To derive the reflection and transmission coefficients of the scattered waves in a simple way, the scattering caused by the notch is analyzed through the composition of individual scattering processes. Linear equations corresponding to the reflection and transmission coefficients are constructed along with scattering graphs. For an illustration of the efficacy of the presented method, the scattering of fundamental symmetric and anti-symmetric modes are inspected according to the depth and width of a notch in a plate. Validity of these expressions is demonstrated by the comparison of the theoretical analysis results with those from the finite element analysis.     

Anomalous propagation characteristics of evanescent waves

An analysis is done of the crossing of a forbidden region in a thin plate by a backward propagating Lamb wave: the refraction/reflection effects undergone by the coupled modes produced at each boundary of the forbidden region are taken into consideration, as well as the penetration of the backward wave as an evanescent wave. The outcome of the acoustic perturbation is analysed for a few angles of incidence and experiments are performed that confirm the theoretical predictions.     

Guided waves in elastic rectangular bars:A solution using partial plate mode decomposition

Similar to Auld's solution for Lamb waves,the wave modes in elastic rectangular bar are solved by partial wave decomposition method.The partial waves are composed of plate modes with the same wavenumber component in waveguide longitudinal direction,thus free boundary conditions on one pair of opposite surfaces are automatically satisfied.Based on completeness assumption and orthogonality of the plate modes,four independent eigenequations are eventually derived for dispersion curve and mode shape investigation.Numerical evaluation shows the calculated results are in consistent with the FEM results.It is then verified that the plate modes which obliquely bounced back and forth between the two opposite surfaces compose the guided modes traveling in the rectangular waveguides with certain wave numbers in transversely resonant cases.     

矩形截面固体波导中弹性导波模式的板波叠加解

参照板中兰姆波模式的子波叠加解求解了矩形截面固体波导中弹性导波的模式。其中子波设定为横向满足自由边界反射条件、纵向波数分量相同的各板波模式。假设声场的完备性,利用子波模式的正交性,结合矩形截面的几何对称性可以得到4组独立导波特征方程用于频散曲线以及导波模式计算,其计算结果与有限元法计算结果相符。研究以解析模型表明:矩形截面固体波导中的导波是其内部以一定纵向波数分量斜向传播的板波模式在两个自由侧面上经过反复反射与模式转化后,横向耦合谐振而形成的。      

Normal mode expansion method for laser-generated ultrasonic Lamb waves in orthotropic thin plates

A quantitative theory for modeling the laser-generated transient ultrasonic Lamb waves, which propagates along arbitrary directions in orthotropic plates, is presented by employing an expansion method of generalized Lamb wave modes. The displacement field is expressed by a summation of the symmetric and antisymmetric modes in the surface stress-free orthotropic plate, and therefore the theory is particularly appropriate for waveform analyses of Lamb waves in thin plates because one needs only to evaluate several lower modes. The transient waveforms excited by the thermoelastic expansion and the oil-coating evaporation are analyzed for a transversely isotropic thin plate. The results show that the theory provides a quantitative analysis to characterize anisotropic elastic stiffness properties of orthotropic plates by laser-generated Lamb wave detection.     

套管中模式波的响应特征*

在固井质量评价中主要利用套管中模式波的幅度或衰减变化反映水泥的胶结状况,不同测井仪器在套管中激发的模式波的类型不同,研究套管中各模式波的传播特征和影响因素可充分挖掘测量数据的潜在应用价值。CBL和SBT等测井仪器在套管中主要激发拉伸波,类同于平板中的零阶对称Lamb波,水泥环封隔测井仪器的斜入射模式在套管中主要激发套管弯曲波,类同于平板中的零阶反对称Lamb波,垂直入射模式激发套管共振波,类同于平板中的高阶对称Lamb波。该文重点分析了这些模式波的衰减特征及其对微环的响应,套管弯曲波在套后耦合轻质水泥时对微环不敏感,但在套后耦合常规水泥或重水泥时,其衰减明显高于胶结良好的状况;拉伸波对微环的存在最为敏感;套管共振波对微环不敏感。     

Rayleigh wave scattering from a vertical edge of isotropic substrates

The paper numerically studies the harmonic Rayleigh wave scattering at the 90-degree corner of isotropic substrate. The finite element method is used. The main attention is paid to two cases. The first one is the apex of the substrate corner is rounded off. The second one consists in that a layer of foreign material is deposited on the face which scatters the Rayleigh wave. The dependence of the reflection and the transmission coefficients on the Poisson ratio, the angle of incidence, the fillet radius, and the layer thickness are obtained. It is found that if the Rayleigh wave is incident perpendicularly to the substrate border, then the fillet of small radius as compared to the wavelength increases the reflection coefficient and decreases the transmission coefficient by factors 1.3–1.8. At normal incidence, the Poisson ratio does not change qualitatively the dependence of the reflection and transmission coefficients on the fillet radius. But the Poisson ratio can substantially affect the angle dependence of these coefficients if the wave is incident obliquely on the corner rounded off. It is also found out that a layer can modify the conditions of scattering such that the incident wave is totally reflected without transmission and conversion into bulk waves in a wide interval of angle of incidence, although, in principle, the bulk wave generation is allowed within a part of this angle interval.     

斜入射分层线性各向异性等离子体电磁散射时域有限差分方法分析

提出了斜入射分层线性各向异性等离子体电磁散射的时域有限差分(FDTD)方法,通过将二维麦克斯韦方程等价地转换为一维麦克斯韦方程,避免了用二维时域有限差分方法分析该散射问题,极大地提高了计算效率.分析推导了TEz和TMz波斜入射线性分层各向异性等离子体电磁散射的FDTD方法,然后通过该方法计算不同入射角的各向异性等离子板的电磁波反射系数,并与其解析解进行比较,结果表明该方法的准确性和有效性.最后,将该算法应用于计算涂覆分层各向异性等离子体金属板在不同入射角下的反射系数,分析了不同入射角对反射系数的影响.     

Trapped rossby waves

The possibility of tidal dynamics at strictly imaginary Lamb parameters has been known for more than three decades. The present paper explores the prevailing physics in this parameter regime. To this end, basic features of the global circulation such as baroclinicity and geostrophy have to be incorporated into tidal dynamics. The tidal equations of the thermal wind are readily obtained in the framework of spherical bishallow water theory. Density surfaces of a circulation with available potential energy alter the spatial inhomogenities of the generic tidal problem. Wave dynamics in an inhomogeneous medium are characterized not only by a dispersion relation but also by a wave guide geography: significant wave amplitudes are trapped in specific regions of frequency-dependent width. As an inherently global issue, evaluation of the Rossby wave guide geography for a given circulation cannot rely on the familiar regional filters of tidal theory. On the global domain, the Rossby wave specification is given by the Margules filter. A thermal wind is stable against nondivergent Rossby wave disturbances. Rossby waves propagating with a geostrophic wind are governed by prolate dynamics (real Lamb parameters) while imaginary Lamb parameters emerge for the oblate dynamics of Rossby waves running against a geostrophic wind. Oblate Rossby wave dynamics include pole-centered wave guides and very low-frequency disturbances propagating eastward against a westward wind.