Turbulent Dynamics of a Critically Reflecting Internal Gravity Wave

Results from computational fluid dynamics experiments of internal wave reflection from sloping boundaries are presented. In these experiments the incident wave lies in a plane normal to the slope. When the angle of wave energy propagation is close to the bottom slope the reflection causes wave breakdown into a quasi-periodic, turbulent boundary layer. Boundary layer energetics and vorticity dynamics are examined and indicate the importance of the three-dimensional turbulence. The boundary layer exhibits intermittent turbulence: approximately every 1.2 wave periods the boundary layer mixes energetically for a duration of about one-third of a wave period, and then it restratifies until the next mixing event. Throughout the wave cycle a strong thermal front is observed to move upslope at the phase speed of the incident waves. Simulations demonstrate that the net effects of turbulent mixing are not confined to the boundary layer, but are communicated to the interior stratified fluid by motions induced by buoyancy effects and by the wave field, resulting in progressive weakening of the background density gradient. Transition to turbulence is determined to occur at Reynolds numbers of approximately 1500, based upon the wavelength and maximum current velocity of the oncoming wave train. The boundary layer thickness depends on the Reynolds number for low Richardson numbers, with a characteristic depth of approximately one-half of the vertical wavelength of the oncoming wave. Received 21 May 1997 and accepted 14 October 1997     

Laminated piezoelectric plate for active underwater acoustic control

The pressure reflected from a bi-laminated piezoelectric plate has been determined using the Thomson-Haskell matrix method. The plate is composed of a piezoelectric layer with grounded vacuum and an elastic layer in contact with the fluid. An incident plane wave in the fluid medium strikes the plate at different angles. The required electric potential across the piezoelectric layer to cancel the reflection from the fluid/elastic boundary has been determined for the piezoelectric material PZT-5 at various thicknes parameters and incident frequencies. Project supported by the National Natural Science Foundation of China (No. 10172039).     

Longitudinal waves at a micropolar fluid/solid interface

The possibility of plane wave propagation in a micropolar fluid of infinite extent has been explored. The reflection and transmission of longitudinal elastic wave at a plane interface between a homogeneous micropolar fluid half-space and a micropolar solid half-space has also been investigated. It is found that there can exist four plane waves propagating with distinct phase speeds in an infinite micropolar fluid. All the four waves are found to be dispersive and attenuated. The reflection and transmission coefficients are found to be the functions of the angle of incidence, the elastic properties of the half-spaces and the frequency of the incident wave. The expressions of energy ratios have also been obtained in explicit form. Frequency equation for the Stoneley wave at micropolar solid/fluid interface has also been derived in the form of sixth-order determinantal expression, which is found in full agreement with the corresponding result of inviscid liquid/elastic solid interface. Numerical computations have been performed for a specific model. The dispersion curves and attenuation of the existed waves in micropolar fluid have been computed and depicted graphically. The variations of various amplitudes and energy ratios are also shown against the angle of incidence. Results of some earlier workers have been deduced from the present formulation.     

Plane waves at an imperfectly bonded interface of two orthotropic generalized thermoelastic rotating half-spaces with two relaxation times

The present paper concentrates on the study of reflection and refraction characteristics of plane waves at an imperfectly bonded interface of two orthotropic generalized thermoelastic rotating half-spaces with different elastic and thermal properties. In this type of problem of orthotropic thermoelastic rotating medium, there are three types of plane waves quasi longitudinal (QL-) wave, quasi thermal (T-mode) wave and quasi transverse (QT-) wave, whose velocities depend on the angle of incidence, imperfection and rotation. The amplitude ratios of reflected waves to that of incident one in each case have been derived. Some special cases of boundaries, i.e. normal stiffness, transverse stiffness, thermal contact conductance, slip boundary and welded contact boundary have been deduced from an imperfect one. Effect of rotation on the amplitude ratios of different reflected and refracted waves with respect to incident QL-wave at different boundaries have been studied graphically. It is observed that thermal properties, imperfect boundary and rotation have significant effect on the propagation of waves.     

Reflection and Transmission of Elastic Waves from the Interface of a Fluid-saturated Porous Solid and a Double Porosity Solid

The reflection and transmission characteristics of an incident plane P1 wave from the interface of a fluid-saturated single porous solid and a fluid-saturated double porosity solid are investigated. The fluid-saturated porous solid is modeled with the classic Biot’s theory and the double porosity medium is described by an extended Biot’s theory. In a double-porosity model with dual-permeability there exist three compressional waves and a shear wave. The effects of the incident angle and frequency on amplitude ratios of the reflected and transmitted waves to the incident wave are discussed. Two boundary conditions are discussed in detail: (a) Open-pore boundary and (b) Sealed-pore boundary. Numerical results reveal that the characteristics of the reflection and transmission coefficients to the incident angle and the frequency are quite different for the two cases of boundary conditions. Properties of the bulk waves existing in the fluid-saturated porous solid and the double porosity medium are also studied.     

Lamb wave scattering by a symmetric pair of surface-breaking cracks in a plate

The scattering problem of a Lamb wave incident on a symmetric pair of surface-breaking transverse cracks in a plate is considered. The Lamb wave is assumed to be obliquely incident on the crack plane. Since the cracks are part-through, the scattered field will contain reflected as well as transmitted waves. The energy of the incoming wave is partitioned into reflected and transmitted wave modes. Energy coefficients of the reflected and transmitted waves are calculated as a function of incident frequency and crack depth. The incidence angle of the incoming wave is also treated as a parameter. Both the reflected and transmitted wave fields are considered as linear superpositions of all real and complex wave modes in the plate. Decomposition of modes is achieved with the help of an orthogonality condition based on the principle of reciprocal work. Continuity of displacement and stress fields is imposed at the crack plane. Energy coefficients for reflection and transmission are obtained from the mode amplitudes. Energy coefficients are shown to be a strong function of incident frequency and crack depth. Experiments are conducted with a PZT transducer network interacting with a symmetric pair of machined cracks in an aluminum plate. Trends predicted by the analysis are reflected in the experimental results.     

Propagation of plane P-waves at interface between elastic solid and unsaturated poroelastic medium

A linear viscoporoelastic model is developed to describe the problem of reflection and transmission of an obliquely incident plane P-wave at the interface between an elastic solid and an unsaturated poroelastic medium, in which the solid matrix is filled with two weakly coupled fluids (liquid and gas). The expressions for the amplitude reflection coefficients and the amplitude transmission coefficients are derived by using the potential method. The present derivation is subsequently applied to study the energy conversions among the incident, reflected, and transmitted wave modes. It is found that the reflection and transmission coefficients in the forms of amplitude ratios and energy ratios are functions of the incident angle, the liquid saturation, the frequency of the incident wave, and the elastic constants of the upper and lower media. Numerical results are presented graphically. The effects of the incident angle, the frequency, and the liquid saturation on the amplitude and the energy reflection and transmission coefficients are discussed. It is verified that in the transmission process, there is no energy dissipation at the interface.     

Free surface Stokes flows obstructed by multiple obstacles

Gravity‐driven Stokes flow down an inclined plane over and around multiple obstacles is considered. The flow problem is formulated in terms of a boundary integral equation and solved using the boundary element method. A Hermitian radial basis function (RBF) is used for the interpolation of the free surface, generation of the unit normal and curvature, and to prescribe the far‐field conditions. For flow over an obstacle, hemispheres are taken. For flow around an obstacle, circular cylinders are modelled and the contact angle condition on the obstacle/free surface intersection specified using the RBF formulation. Explicit profiles are produced for flow over and around two obstacles placed in various locations relative to one another. Interaction due to two obstacles is given by comparisons made with the profiles for flow over and around individual obstacles. In general, when the obstacles are separated by a sufficiently large distance the flow profiles are identical to a single obstacle analysis. For flow over and around two obstacles in‐line with the incident flow, effects of the governing parameters are examined, with variations in plane inclination angle, Bond number, obstacle size, and in the case of obstacles intersecting the free surface, static contact angle is considered. Finally flows over and around three obstacles are modelled. Copyright © 2009 John Wiley & Sons, Ltd.     

直角平面内弹性圆夹杂对入射平面SH波的散射

利用复变函数法、多极坐标移动技术及傅立叶级数展开求解二维直角平面内圆形弹性夹杂对稳态入射平面SH波的散射问题。首先写出直角平面内不含夹杂时的入射波场和反射波场;其次建立直角平面内含夹杂时夹杂外的散射波解和夹杂内的驻波解,并利用叠加原理写出问题的总波场,借助夹杂边界处应力和位移的连续条件建立求解散射波解和驻波解中未知系数的无穷代数方程组并求解,通过算例具体讨论了直角平面水平边界点的位移幅度比和夹杂边界处径向应力集中系数随不同无量纲波数、入射角及圆孔位置的变化情况,结果表明了算法的有效实用性。     

Transverse wave at a plane interface between isotropic elastic and unsaturated porous elastic solid half-spaces

Based on the poroelasticity theory, this article investigates the reflection and transmission characteristics of an incident plane transverse wave at a plane interface between an isotropic elastic half-space and an unsaturated poroelastic solid half-space. For this purpose, the effect of the saturation degree and frequency on the properties of the four bulk waves in unsaturated porous medium, i.e., three longitudinal waves and one transverse wave, are discussed at first. Two general cases of mode conversion are considered: (i) The initial transverse wave is incident from an unsaturated poroelastic half-space to the interface, and (ii) the initial transverse wave is incident from an elastic solid half-space to the interface. The expressions for the partition of energy at the interface during transmission and reflection process of waves are presented in explicit forms. At last, numerical computations are performed for these two cases and the results obtained are depicted, respectively. The variation of the amplitude ratios and energy ratios with the saturation degree and incident angle is illustrated in detail. It is also verified that, at the interface, the sum of energy ratios is approximately equal to unity as expected.     

Reflection and transmission of plane waves at the interface of an elastic half-space and a fractional order thermoelastic half-space

The problem of reflection and transmission due to longitudinal and transverse waves incident obliquely at a plane interface between uniform elastic solid half-space and fractional order thermoelastic solid half-space has been studied. It is found that the amplitude ratios of various reflected and refracted waves are functions of angle of incidence and frequency of incident wave and are influenced by the fractional order thermoelastic properties of media. The expressions of amplitude ratios and energy ratios have been computed numerically for a particular model. The variation of amplitude and energy ratios with angle of incidence is shown graphically. The conservation of energy at the interface is verified.     

Reflection and Refraction of Anti-Plane Shear Waves from a Moving Phase Boundary

The reflection and refraction of anti-plane shear waves from an interface separating half-spaces with different moduli is well understood in the linear theory of elasticity. Namely, an oblique incident wave gives rise to a reflected wave that departs at the same angle and to a refracted wave that, after transmission through the interface, departs at a possibly different angle. Here we study similar issues for a material that admits mobile elastic phase boundaries in anti-plane shear. We consider an energy minimal equilibrium state in anti-plane shear involving a planar phase boundary that is perturbed due to an incident wave of small magnitude. The phase boundary is allowed to move under this perturbation. As in the linear theory, the perturbation gives rise to a reflected and a refracted wave. The orientation of these waves is independent of the phase boundary motion and determined as in the linear theory. However, the phase boundary motion affects the amplitudes of the departing waves. Perturbation analysis gives these amplitudes for general small phase boundary motion, and also permits the specification of the phase boundary motion on the basis of additional criteria such as a kinetic relation. A standard kinetic relation is studied to quantify the subsequent energy partitioning and dissipation on the basis of the properties of the incident wave.     

Elastic waves in an electro-microelastic solid

The present study is concerned with the wave propagation in an electro-microelastic solid. The reflection phenomenon of plane elastic waves from a stress free plane boundary of an electro-microelastic solid half-space is studied. The condition and the range of frequency for the existence of elastic waves in an infinite electro-microelastic body are investigated. The constitutive relations and the field equations for an electro-microelastic solid are stemmed from the Eringen’s theory of microstretch elasticity with electromagnetic interactions. Amplitude ratios and energy ratios of various reflected waves are presented when an elastic wave is made incident obliquely at the stress free plane boundary of an electro-microelastic solid half-space. It has been verified that there is no dissipation of energy at the boundary surface during reflection. Numerical computations are performed for a specific model to calculate the phase speeds, amplitude ratios and energy ratios, and the results obtained are depicted graphically. The effect of elastic parameter corresponding to micro-stretch is noticed on reflection coefficients, in particular. Results of Parfitt and Eringen [Parfitt, V.R., Eringen, A.C., 1969. Reflection of plane waves from a flat boundary of a micropolar elastic half-space. J. Acoust. Soc. Am. 45, 1258–1272] have also been reduced as a special case from the present formulation.     

Propagation of plane elastic waves at a plane interface between two electro-microelastic solid half-spaces

This work is concerned with the wave propagation and their reflection and transmission from a plane interface between two different electro-microelastic solid half-spaces in perfect contact. It is found that there exist five basic waves in an infinite electro-microelastic solid, namely an independent longitudinal micro-rotational wave, two sets of coupled longitudinal waves influenced by the electric effect, and two sets of coupled transverse waves. The existence of the two sets of coupled longitudinal waves is new. In the absence of microstretch and electric effects, these two coupled longitudinal waves reduce to a longitudinal displacement wave of micropolar elasticity. Amplitude and energy ratios of various reflected and transmitted waves are presented when (i) a set of coupled longitudinal wave is made incident and (ii) a set of coupled transverse wave is made incident. Numerical computations have been performed for a particular model and the variations of amplitude and energy ratios are obtained against the angle of incidence. The results obtained are depicted graphically. It has been verified that the sum of energy ratios is equal to unity at the interface and the amplitude ratios of reflected and transmitted waves depend upon the angle of incidence, frequency and elastic properties of the media. Results of some earlier workers have also been reduced from the present formulation.     

二维直角平面内固定圆形夹杂对稳态入射反平面剪切波的散射

利用复变函数法、多极坐标及傅立叶级数展开技术求解了二维直角平面内固定圆形夹杂对稳态入射反平面剪切（shearing horizontal, SH）波的散射问题。首先构造出介质内不存在夹杂时的入射波场和反射波场,然后建立介质内存在夹杂时由夹杂边界产生的能够自动满足直角边应力自由条件的散射波解,从而利用叠加原理写出介质内的总波场。利用夹杂边界处位移条件和傅立叶级数展开方法列出求解散射波中未知系数的无穷代数方程组,在满足计算精度的前提下通过有限项截断,得到相应有限代数方程组的解,最后通过算例具体讨论了二维直角平面水平边界点的位移幅度比和相位随量纲一波数、入射波入射角及夹杂位置的不同而变化的情况,结果表明了算法的有效实用性。     

Propagation,reflection, and transmission of SH-waves in slightly compressible,finitely deformed elastic media

The propagation, reflection, and transmission of SH waves in slightly compressible, finitely deformed elastic media are considered in this paper. The dispersion relation for SH-wave propagation in slightly compressible, finitely deformed layer overlying a slightly compressible, finitely deformed half-space is derived. The present paper also deals with the reflection and refraction (transmission) phenomena due to the SH wave incident at the plane interface between two distinct slightly compressible, finitely deformed elastic media. The closed form expressions for the amplitude ratios of reflection and refraction coefficients of the reflected and refracted SH waves are obtained from suitable boundary conditions. For the numerical discussions, we consider the Neo-Hookean form of a strain energy function. The phase speed curves, the variations of reflection, and transmission coefficients with the angle of incidence, and the plots of the slowness sections are presented by means of graphs.     

半无限周期声子晶体的全反射隧穿能带

利用色散法研究了在入射角大于全反射角时弹性波在一维半无限周期声子晶体中的传输特性。结果发现当弹性波大于全反射角入射一维半无限周期声子晶体时出现了两级全反射隧穿导带;并得出了全反射隧穿导带随入射角和周期厚度的响应曲线,分析可知:导带的频率中心随入射角的增加而增加,随周期厚度的增加而降低;导带的频率宽度随入射角的增加而减小,随周期厚度的增加而降低。     

THE EFFECT OF DUAL-PHASE-LAG MODEL ON REFLECTION OF THERMOELASTIC WAVES IN A SOLID HALF SPACE WITH VARIABLE MATERIAL PROPERTIES

The present article represents an analysis of reflection of P-wave and SV-wave on the boundary of an isotropic and homogeneous generalized thermoelastic half-space when the boundary is stress-free as well as isothermal. The modulus of elasticity is taken as a linear function of reference temperature. The basic governing equations are applied under four theories of the generalized thermoelasticity： Lord-Shulman （L-S） theory with one relaxation time, Green-Naghdi （G-N） theory without energy dissipation and Tzou theory with dual-phase-lag （DPL）, as well as the coupled thermoelasticity （CTE） theory. It is shown that there exist three plane waves, namely, a thermal wave, a P-wave and an SV-wave. The reflection from an isothermal stress-free surface is studied to obtain the reflection amplitude ratios of the reflected waves for the incidence of P- and SV-waves. The amplitude ratios variations with the angle of incident are shown graphically. Also the effects of reference temperature of the modulus of elasticity and dual-phase lags on the reflection amplitude ratios are discussed numerically.     

流体饱和弹性半空间裂隙对平面P_I波的衍射

采用间接边界元法,求解了饱和半空间裂隙对平面PI波的二维衍射问题。基于单层位势理论,将边界离散并直接在边界单元上施加虚拟荷载(水平作用力、竖向作用力和流量源的叠加)以构造散射波场,并由边界条件确定虚拟荷载密度,总波场由自由波场和散射波场共同组成。通过参数分析研究了入射波频率、入射倾角、埋深、孔隙率、边界渗透条件等因素对饱和半空间中裂隙对平面PI波衍射的影响规律。结果表明:裂隙随埋深增大,地表位移谱振荡加剧,峰值有所降低;随着入射频率增加,孔隙率影响逐渐增大;垂直入射时,水平位移的放大区域主要分布在裂隙两端,斜入射时,主要集中在裂隙正上方地表附近;透水和不透水两种情况下的地表位移幅值和相位差别较小,但干土情况与饱和情况下的位移幅值相差较大。     

Critical angles and grazing incidence: the breakdown of regular shock reflection in solids

The regular pattern for shock reflection in a nonlinear hyperelastic solid is a centered array of shocks and simple wave fans. As the angle of incidence approaches grazing incidence or a critical angle, the reflection pattern overtakes the incident wave until finally the regular pattern can no longer be sustained. By expanding the reflection solution in powers of amplitude about the linear reflection solution, it is possible to develop a procedure to solve the reflection problem for weak but finite shocks for any material symmetry. Explicit solutions can be exhibited for isotropic materials, including relationships between amplitude and limiting angle for various boundary conditions and incident waves. Some problems require consideration of a nonlinear boundary condition even in the first approximation. Typically, these cases lead to considerable amplification in the leading reflected wave.