Reflection and refraction of plane waves at interface between two piezoelectric media

This paper analyses reflection and refraction of plane waves at a perfect interface between two anisotropic piezoelectric media. The equations of elastic waves, quasi-static electric field, and constitutive relationships for the piezoelectric media are derived. A solution based on the inhomogeneous wave theory is developed to address the inconsistency between the numbers of independent wave modes in the media and the numbers of interfacial boundary conditions to obtain accurate reflection and refraction coefficients in case of strong piezoelectric media, where all the elastic and electric continuity conditions across the interface are satisfied simultaneously. The study shows that there exist independent and zero energy wave modes satisfying the general Snell’s law and propagating along the interface for any incident wave angle. These waves can be treated as pseudo surface waves. It is further found that all the reflection/refraction waves including the pseudo surface waves obey the energy conservation law at the interface boundary. In addition, the analysis also reveals that the reflection and refraction elastic waves can turn into pseudo surface waves at some critical incident angles.     

Thickness measurement of a thin-film layer on an anisotropic substrate by phase-sensitive acoustic microscope

Complex V(z) curves for single thin-film layers on anisotropic substrates are studied both experimentally and theoretically, and the application of V(z) measurement to the determination of film thickness on anisotropic substrates is discussed. Complex V(z) curves for aluminum layers (with thicknesses between 0.5 and 2 mum) on a silicon wafer have been calculated. The inverse Fourier transform of the V(z) curves, which corresponds to the reflection coefficient, shows sharp changes at critical angles of pseudosurface waves, pseudo-Sezawa waves, and Rayleigh surface waves. These critical angles strongly depend on the thickness. Complex V(z) curves for these specimens have been measured using a phase-sensitive acoustic microscope with a point focus lens at 400 MHz. The critical angles of the surface waves obtained from the measured V(z) curves are in good agreement with those obtained from the calculated V(z) curves. On the basis of this result, it is shown that the V(z) measurement is applicable to the determination of film thickness on an anisotropic substrate.     

Time-domain analysis of ultrasonic reflection from imperfect interfaces

The specular reflection of ultrasound from defect covered bond planes is analyzed in the time-domain using the independent scattering model for incident plane waves. We hypothesize that the early-time asymptotics of the reflected wave are given exactly by the independent scattering model for waves that are normally incident on the bond plane. For non-normal incidence, a more restricted result is available for reflected longitudinal waves. We present theoretical arguments for the plausibility of the hypothesis. Experimental measurements made on two sets of model bond planes test and support the hypothesis. Our motivations are as follows. An effort is underway to develop nondestructive methods for estimating the integrity of metal-metal bonds. These methods primarily focus on the reflection of ultrasound in the long wavelength limit, where one can estimate an effective elastic constant. People have characterized the overall quality of the bond in terms of this elastic constant. However, one cannot in this way infer more detailed information such as the average size of the defects or their area fraction. Higher frequency probes, which do provide more detailed information, have been studied in the independent scattering model. Consequently, we have (1) extended the independent scattering model to the time-domain, and (2) shown that it is asymptotically correct for early-time reflections from a defect covered plane (the bond plane). We expect that time-domain methods, based on the analysis presented in this paper, will form the basis for a more rigorous technique for determining the area fraction and the average defect size.     

Acoustic reflection from the boundary of anisotropic thermoviscoelastic solid with fluid

Vertical slownesses of waves at a boundary of an anisotropic thermoviscoelastic medium are calculated as roots of a polynomial equation of degree eight. Out of the corresponding eight waves, the four, which travel towards the boundary are identified as upgoing waves. Remaining four waves travel away from the boundary and are termed as downgoing waves. Reflection and refraction of plane harmonic acoustic waves are studied at a plane boundary between anisotropic thermoviscoelastic solid and a non-viscous fluid. At this fluid-solid interface, an incident acoustic wave through the fluid reflects back as an attenuated acoustic wave and refracts as four attenuating waves into the anisotropic base. Slowness vectors of all the waves in two media differ only in vertical components. Complex values of vertical slowness define inhomogeneous refracted waves with a fixed direction of attenuation, i.e. perpendicular to the interface. Energy partition is calculated at the interface to find energy shares of reflected and refracted waves. A part of incident energy dissipates due to interaction among the attenuated refracted waves. Numerical examples are considered to study the variations in energy shares with the direction of incident wave. For each incidence, the conservation of incident energy is verified in the presence of interaction energy. Energy partition at the interface seems to be changing very slightly with the azimuthal variations of the incident direction. Effects of anisotropy, elastic relaxation and thermal parameters on the variations in energy partition are discussed. The acoustic wave reflected from isothermal interface is much significant for incidence around some critical directions, which are analogous to the critical angles in a non-dissipative medium. The changes in thermal relaxation times and uniform temperature of the thermoviscoelastic medium do not show any significant effect on the reflected energy.     

S波由饱和土入射于弹性土时在界面上的反射与透射

从地震工程实际出发，借助Biot多孔介质中的波动方程，根据各种界面条件导出了S波从饱和土入射于弹性土时在交界面上反射与透射的一般计算公式。作为算例，数值计算分析了S波从饱和土入射于饱和土与弹性土交界面时，饱和土中P1、P2和S波的反射系数以及弹性土中P波、SV射系数与界面排水条件、入射角以及频率之间的关系。结果表明：各种波的反射、透射系数与入射角、入射频率以及界面排水条件有关系。     

Reflection of PlaneWaves from Electro-magneto-thermoelastic Half-space with a Dual-Phase-Lag Model

The aim of this paper is to study the reflection of plane harmonic waves from a semi-infinite elastic solid under the effect of magnetic field in a vacuum. The expressions for the reflection coefficients, which are the relations of the amplitudes of the reflected waves to the amplitude of the incident waves, are obtained. Similarly, the reflection coefficient ratio variations with the angle of incident under different conditions are shown graphically. Comparisons are made with the results predicted by the dual-phase-lag model and Lord-Shulman theory in the presence and absence of magnetic field.     

Phase constraint for the waves diffracted by lossless symmetrical gratings at Littrow mount

The energy conservation of grating diffraction is analyzed in a particular condition of incidence in which two incident waves reach a symmetrical grating from the two sides of the grating normal at the first-order Littrow mounting. In such a situation the incident waves generate an interference pattern with the same period as the grating. Thus in each direction of diffraction, interference occurs between two consecutive diffractive orders of the symmetrical incident waves. By applying only energy conservation and the geometrical symmetry of the grating profile to this problem it is possible to establish a general constraint for the phases and amplitudes of the diffracted orders of the same incident wave. Experimental and theoretical results are presented confirming the obtained relations.     

Increased efficiency of surface wave stimulation on the inaccessible side of a thick isotropic plate with superimposed periodicity

Because of the growing number of applications of phononic crystals and other periodic structures, there is a renewed and growing interest in understanding the interaction of ultrasound with periodically corrugated surfaces. This paper presents a theoretical investigation of the transformation of ultrasound incident from the solid side onto a solid-liquid periodically corrugated interface. It is shown that it is possible to tailor the shape of a corrugated surface with given periodicity such that there is a significant amount of energy transformed into Scholte-Stoneley waves than if pure saw-tooth or sine-shaped surfaces were used. This permits the fabrication of periodic structures that can be patched on or engraved in body parts of a construction and enables efficient generation of Scholte-Stoneley waves. The study is performed for incident homogeneous plane waves as well as for bounded beams. Incident longitudinal waves are studied as well as incident shear waves.     

基于两向设计地震动的二维自由场构建

针对在设计地震动基础上构建平面半空间自由场的问题,在综合考虑水平向与竖直向设计地震动基础上获得平面半空间自由场,实现了将设计地震动与以任意角度组合入射的P波、SV波构成的自由场相联系。首先在已有成果的基础上,研究了以任意角度入射情况下P波及其反射波、SV波及其反射波对水平向和竖直向设计地震动的贡献;然后在假设入射P波、SV波与两向设计地震动具有相同频率组成的基础上,获得了以特定角度组合入射条件下P波、SV波的频谱和相位信息;进而通过傅里叶逆变换获得入射P波、SV波的时间序列;最后依据P波、SV波及其反射波在平面半空间的传播情况获得自由场条件下平面半空间内任意点的水平向和竖直向地震动。该文所采用的波场构建方法可实现对包含任意频谱和相位信息的两向设计地震动所对应的平面半空间自由场的构建,所获得的斜入射P波和SV波也具有各自的频谱和相位信息。最终得到的平面半空间自由场任意点水平向和竖直向地震动时间序列具有非一致性;根据入射P波、SV波结合地基边界条件确定的空间点地震动时间序列可用于结构的非一致地震动输入。     

斜入射SV波对地铁车站地震响应的影响

在显式有限元法结合黏弹性人工边界的时域波动方法的基础上,建立了三维平面SV波斜入射的输入方法。半空间算例说明了该文方法具有较好的精度,并基于所建立的斜入射方法研究了地震波斜入射对北宫门地铁车站地震响应的影响。该算例结果表明：在地震波斜入射情况下,地铁车站的地震响应规律与垂直入射时的情况具有明显差异;斜入射角度对水平加速度响应并不敏感,但对竖向加速度影响较大;斜入射使得车站柱子构件的剪力和轴力明显改变,柱子轴力随着入射角增加而明显增大;边墙控制点的应力状态也受入射角的影响较大,各控制点的第一、第三主应力都出现了随着入射角度增加而增大的规律。在地铁车站等地下结构抗震研究中,应考虑地震波斜入射的影响。     

饱和土体中多排桩屏障对压缩快波的隔离

假定桩的长度远大于直径, 将多排弹性桩组成的非连续屏障对平面压缩快波(简称P₁波)的隔离简化为弹性波多重散射的二维平面问题, 运用复变函数的保角映射方法和波场势函数展开法, 根据桩-土界面处应力和位移连续的边界条件, 得到散射波场势函数展开式中待定复系数的理论解。通过分析屏障后的位移比值(某点由入射和散射弹性波产生的总位移与入射弹性波单独产生的位移之比)的变化规律, 对比了单排、双排和三排桩屏障的隔离效果, 研究结果表明：1) 多排桩屏障对高频入射P₁波的隔离效果好于低频入射P₁波;2) 随着桩排数的增多, 多排桩屏障的隔离效果明显提高, 有效隔离区域明显增大;3) 当桩排数达到三排时, 屏障后一定范围内的隔离效果提高到了70%, 隔离效果比较理想。     

板状粘接结构中对称和反对称纵波与界面的相互作用

为了激发粘接结构中的导波或界面波,通常需要将声波从两半无限介质同相位或反相位地同时入射多层系统。针对此问题,基于矩阵方法,推导了界面处于理想连接的情况下,对称或反对称纵波从上下半无限空间入射时,三层板状粘接结构中纵、横波的反射与透射系数表达式。分析了入射角度、粘接层厚度以及基体材料等对声波反射(或透射)特性的影响。结果表明,对称或反对称纵波垂直入射时不发生波型转换。粘接结构中声波的反射(或透射)特性与入射角度、频率以及粘接层厚度等参数密切相关。在相同的粘接层厚度(或频率)范围内,随着声波频率(或粘接层厚度)的增加,谐振频率曲线向低频漂移。该方法可作为粘接结构中体波或导波传播特性研究的重要理论基础。     

Wave forces on steeply-sloping sea walls: oblique incidence

Previous work, [3], studying the forces exerted by non-breaking, normally-incident water waves of small amplitude on a sloping sea wall is here extended to the case of oblique incidence. The range of applicability of the Galerkin solutions is increased by means of the Shanks transform. Results are presented for a planar, outward-sloping sea wall. In shallow water, the total normal wave force per unit span is found to decrease as the wall slope increases, except for extremely obliquely incident waves. In deep water, it increases. Regarded as a function of the angle of incidence , the wave force in shallow water is virtually independent of , except for very oblique waves. In deep water, by contrast, the force first increases with and then decreases. In this case, the maximum wave force does not occur for normally incident waves.     

多排柱腔构成的非连续屏障对SV波的隔离

运用复变函数的保角映射方法和波函数展开法,根据柱腔壁处土体完全自由,即应力为零的边界条件,得到了多排柱腔构成的屏障对入射平面SV波的多重散射的理论解,根据屏障后归一化位移(入射和散射SV波产生的总位移与入射SV波在同一点产生的位移之比)的变化规律,着重分析了多排(双排和三排)和单排柱腔对平面SV波的隔离效果,结果表明：(1) 双排和三排柱腔屏障的隔离效果明显好于单排柱腔屏障;(2) 当入射SV波的频率较低时,双排并列正对的柱腔屏障的隔离效果较好,而当频率较高时,双排交错的柱腔屏障的隔离效果较好;(3) 三排梅花桩布置的柱腔屏障的隔离效果好于双排柱腔,当入射SV波的频率较高时,屏障后有近3/4的面积的隔离效果都超过了60%;(4) 无论是单排还是多排柱腔屏障,当入射SV波频率较高时,靠近屏障边缘区域的隔离效果要好于中心区域。     

港湾振荡下港内低频波浪的数值研究

该文采用完全非线性Boussinesq模型模拟了由双色波群诱发的狭长型矩形港内的二阶长波共振现象。基于一个港内低频波浪的分离方法,系统研究了港湾处于最低的四个共振模态下入射短波的波长和波幅对港内锁相长波和自由长波的波幅以及它们的相对成分的影响。研究表明:在该文所研究的特定港口和短波频率、波幅范围内,锁相长波和自由长波波幅均随着短波波长的增大而增大,并且第一共振模态下的锁相长波与自由长波的振幅比往往要大于其他三个模态下的值。在各共振模态下,锁相长波与自由长波的波幅均随入射短波波幅成平方关系变化,但它们的振幅比却几乎不受入射短波波幅的影响。     

考虑地震波幅值衰减的斜入射二维自由场

本文将传统基于地震波垂直入射的自由场构建方案推广到地震波斜入射情况。在假设平面半空间自由场由以特定角度组合入射的平面P波、SV波及其反射波构成的前提下,通过引入介质品质因子、临界幅值比和子波影响距离等参数完成了均匀线弹性介质中基于地表地震记录、考虑地震波衰减影响的自由场构建。通过地基特征点处入射P波、SV波的时、频域特性,特征点位移时程研究了临界幅值比和地震波幅值衰减因素对自由场的影响。结果显示,对于水工结构等低频激励起主要影响的结构来说,构建自由场时临界幅值比的选择具有较大自由性;在岩体较完整、涉及范围较小时,地震波幅值衰减因素对自由场的影响较小,可忽略不计;但在一些极端情况下,地震波幅值衰减因素对自由场的影响可能达到20%左右。     

Dispersion of linearly polarized light propagating in a thin birefringent plate

We analyze theoretically the dispersion of linearly polarized light propagating in a uniaxial anisotropic medium where multibeam interference is present. Explicit expressions of the group-delay dispersion for transmitting waves are derived for the simplest situation, and the effect of dispersion on pulse broadening is analyzed for a few selected cases. Our results reveal that at normal incidence and in the situation where the optic axis is parallel to the surface of birefringent plate (in the x-y plane), the dispersion of the refracted wave decreases with the extent of birefringence. In particular, the dispersion for the electric field parallel to the polarization direction of the incident light changes with the rotation angle between the optic axis and the polarization direction of the incident field, whereas the dispersion for the refracted field whose direction is vertical to the polarization of incident light is independent of this angle. For oblique incidence, dispersion varies substantially for different incident angles. In the situation where the optic axis is in the x-z plane at either normal or oblique incidence, the dispersion increases in a periodically oscillating manner as a function of the relative thickness of the birefringent plate.     

分析层状土介质中群桩对任意入射地震波响应的半解析法

本文舍弃桩基动力分析中的土介质平面应变假定，采用薄层内位移线性变化条件下的动力Ｇｒｅｅｎ函数形成桩－土－桩动力相互作用所需的土介质柔度矩阵，用梁单元模拟桩的运动，以地震波入射为初始运动输入，建立了适用于群桩基础对任意入射地震波响应的半解析分析模型，研究了层状土介质中群桩对不同角度入射ＳＨ、ＳＶ和Ｐ波的动力响应。结果表明，本文提供的方法不仅可以避免平面应变解的低频误差，具有较高精度，而且计算工作量较少。     

Reflection and refraction of plane waves at the interface between piezoelectric and piezomagnetic media

The paper analyzes the reflection and refraction of a plane wave incidence obliquely at the interface between piezoelectric and piezomagnetic media. The materials are assumed to be transversely isotropic. Numerical calculations are performed for BaTiO₃/CoFe₂O₄ material combination. Four cases, incidence of the coupled quasi-pressure (QP) and quasi-shear vertical (QSV) wave from BaTiO₃ or CoFe₂O₄ media, are discussed. The reflection and transmission coefficients and energy coefficients varying with the incident angle are examined. Calculated results are verified by considering the energy conservation. Results show that the reflected and transmitted wave fields in the sagittal plane consist of six kinds of waves, i.e. the coupled QP and QSV waves, evanescent electroacoustic (EA) and magnetic potential (MP) waves in the piezoelectric medium (BaTiO₃), evanescent magnetoacoustic (MA) and electric potential (EP) waves in the piezomagnetic medium (CoFe₂O₄), among which the EA, MA, MP and EP waves propagate along the interface. The most amount of the incident energy goes with the waves that are the same type as the incident wave, while the energy arising from wave mode conversion occupies a less part of the incident energy. The electric energy in BaTiO₃ is higher than the magnetic energy in CoFe₂O₄; they both attain their maximum values at/before the critical angle. Critical angles have little effect on evanescent waves except when the total reflection takes place. These results would provide useful complementary information for magnetoelectric composite materials.     

The propagation and reflection of finite amplitude cylindrically symmetric shear waves in a hyperelastic incompressible solid

Summary We consider axially symmetric shear waves propagating in an incompressible hyperelastic thick-walled cylindrical shell, whose strain energy function is expressible as a truncated power series in terms of the basic strain invariants. A continuous pulse is initiated at the interior boundary of the cylinder by surface tractions of finite duration. The pulse propagates away from the interior boundary, then reflects from the outer boundary, and subsequently reflects back and forth between the two boundaries of the cylinder. We analyze shock development of the first incident and first reflected wave. The incident pulse can break before it reaches the outer boundary. Using Whitham's nonlinearization technique, we determine conditions under which the incident wave breaks and which shock waves can subsequently occur. Similar calculations are carried out for the first reflection. The formulas obtained for the incident pulse provide accurate estimates of the breaking distance and time, and the location of the shock paths, for any incident shock waves that occur. Results obtained for the reflected wave cannot be used to make similar estimates, but they do reveal that once the pulse has completely left the outer boundary, the possible shock that can occur is the same as for the incident wave. Our analysis is carried out for axial shear waves. A similar analysis can be done for torsional shear waves, but not for combined axial and torsional shear wave propagation. We illustrate the conclusions of our shock analysis with numerical solutions obtained using a relaxation scheme for systems of conservation laws. Numerical results are obtained for axial shear and for combined axial and torsional shear. These results indicate that the shock behavior indicated by our analysis of axial shear is also valid for combined axial and torsional shear wave propagation.