Scattering coefficient reconstruction in plates using focused acoustic beams

A method has been developed and proven using highly focused acoustic beams that allows for the rapid reconstruction of scattering coefficients of a thin anisotropic plate immersed in liquid. In a single bistatic coordinate scan, nearly the entire range of wave number within the spatial and temporal frequency bandwidth of the transducer can be reconstructed. This paper also reports the development of a multiple-source complex transducer point model that includes all extrinsic factors and permits prediction of the wave number-frequency (k–f) domain result obtained from a scan of focused transducers in a pitch-catch reflection or transmission arrangement. Extensive experiments have been performed to test the method and the model and to demonstrate transducer beam effects on the k–f domain results, leading to a very efficient method for mapping major portions of the guided wave dispersion spectrum in thin-plate media. As a demonstration of the technique, an estimate of material elastic properties in an isotropic and a transversely-isotropic plate is reported, making only minimal use of the highly redundant dispersion data. Acoustic velocities inferred from these experiments show a disparity of less than 3% from contact acoustic estimates of the same parameters in either plate.     

Scattering of water waves by a submerged thin vertical elastic plate

The problem of water wave scattering by a thin vertical elastic plate submerged in infinitely deep water is investigated here assuming linear theory. The boundary condition on the elastic plate is derived from the Bernoulli–Euler equation of motion satisfied by the plate. This is converted into the condition that the normal velocity of the plate is prescribed in terms of an integral involving the difference in velocity potentials (unknown) across the plate multiplied by an appropriate Green’s function. The reflection and transmission coefficients are obtained in terms of integrals involving combinations of the unknown velocity potential on the two sides of the plate and its normal derivative on the plate, which satisfy three simultaneous integral equations, solved numerically. These coefficients are computed numerically for various values of different parameters and are depicted graphically against the wave number for different situations. The energy identity relating these coefficients is also derived analytically by employing Green’s integral theorem. Results for a rigid plate are recovered when the parameters characterizing the elastic plate are chosen negligibly small.     

Diffraction of surface waves on an inhomogeneous elastic plate

The oblique incidence of small-amplitude waves on an elastic semi-infinite composite plate floating on the free surface of finite-depth water is studied. The front part of a constant-width plate is highed to the basic part and has characteristics different from those of the basic part. The reflection and transmission coefficients of the waves and the vertical displacements of the plate are determined. It is shown that the heterogeneity of the plate material exerts a strong effect on surface-wave diffraction. Methods for decreasing the elastic deformations of the basic part of the plate are proposed. Lavrent'ev Institute of Hydrodynamics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 41, No. 4, pp. 42–48, July–August, 2000.     

Wave propagation in an orthotropic micropolar elastic solid

Two-dimensional plane wave propagation in an orthotropic micropolar elastic solid is studied. There exist three types of coupled waves in xy-plane, whose velocities depend upon the angle of propagation and material parameters. A problem on reflection of these plane waves from a stress-free boundary is considered. The reflection coefficients of various reflected waves are computed numerically for a particular model of the solid. The effects of anisotropy upon the velocities and reflection coefficients are depicted graphically for different angles of propagation.     

Interfacial imperfection on reflection and transmission of plane waves in anisotropic micropolar media

This study is concerned with the reflection and transmission of plane waves at an imperfectly bonded interface between two orthotropic micropolar elastic half-spaces with different elastic and micropolar properties. There exist three types of coupled waves in xy-plane. The reflection and transmission coefficients of quasi-longitudinal (QLD) wave, quasi-coupled transverse microrotational (QCTM) wave and quasi-coupled transverse displacement (QCTD) wave have been derived for different incidence waves and deduced for normal force stiffness, transverse force stiffness, transverse couple stiffness and perfect bonding. The numerical values of modules of the reflection and transmission coefficients are presented graphically with the angle of incidence for orthotropic micropolar medium (MOS) and isotropic micrpolar medium (MIS). Some particular cases of interest have been deduced from the present investigation.     

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.     

Oblique incidence of surface waves on an elastic plate

Oblique incidence of small-amplitude waves on an elastic plate floating on a free liquid surface is studied. The reflection and transmission factors of the waves and the vertical displacements and strains of the plate are determined. It is shown that all these characteristics depend greatly on the incidence angle and frequency of the wave and the width of the plate. Approximate solutions for the reflection and transmission factors are obtained. Lavrent’ev Institute of Hydrodynamics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 4, pp. 62–68, July–August, 1999.     

Explicit formulas for the reflection and transmission coefficients of one-component waves through a stack of an arbitrary number of layers

The transmission and reflection of one-component elastic, acoustic, optical waves on a stack of arbitrary number of different homogeneous layers have been intensively studied in the literature. However, all obtained formulas for the reflection and transmission coefficients are in implicit form. In this paper, we provide the explicit formulas for them. From these formulas we immediately arrive at the explicit formulas for the reflection and transmission coefficients of one-component waves through an FGM layer. Based on the obtained exact formulas, approximate formulas for the reflection and transmission coefficients are established for a stack of thin layers and for a thin FGM layer. It is numerically shown that they are good approximations. Since the obtained formulas are totally explicit, they are useful in evaluating, not only numerically but also analytically, the transmission and reflection coefficients of one-component waves.     

Numerical simulation of the interaction of gravity waves with elastic ice floes

The self-consistent motion of a fluid and elastically oscillating plates partially covering the fluid is simulated numerically in the linear approximation. The problem is reduced to the simultaneous solution of the Laplace equation for the fluid and the equation of elastic plate oscillations for the ice. The numerical and analytical solutions, the latter obtained from an integral equation containing the Green’s function, are compared. To solve the problem numerically, the boundary element method for the Laplace equation and the finite element method for the equation describing the elastic plate are proposed. The coefficients of transmission and reflection of surface gravity waves from the floating plates are calculated. It is shown that the solution may be quasi-periodic with characteristics determined by the initial values of the wave and ice-floe parameters. The ice floes may exert a filtering effect on the surface wave spectrum, essentially reducing its most reflectable components. Sankt-Peterburg. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, pp. 123–131, May–June, 2000.     

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.     

Reflection and refraction of plane quasi-P waves at a corrugated interface between distinct triclinic elastic half spaces

The reflection and refraction pattern of elastic waves at a corrugated interface between two triclinic half-spaces is discussed. The incident wave is taken to be the cause of the interface disturbance and the reflected and refracted waves are effects. This leads to the causality requirement that the reflected and refracted waves must propagate away from the interface. Closed form expressions of reflection and transmission coefficients are derived using Rayleigh’s method of approximation. The formulae of reflection and transmission coefficients are derived in closed form for the first-order approximation of the corrugation. The analytical expressions of all the three phase velocities of qP, qSV and qSH waves have been derived. The variation of reflection and refraction coefficients with the angle of incidence and also with the corrugation parameter is shown. In this paper we have developed Graphical User Interface (GUI) Software in MATLAB which shows the variation of reflection and refraction coefficients with respect to incident angle and corrugation parameter. This software can be generalized to show the variation of reflection and refraction coefficients. Numerical computations are performed for a scientific model and the results obtained are shown graphically.     

Quasi-P-waves at a corrugated interface between two dissimilar monoclinic elastic half-spaces

In this paper we have derived reflection and transmission coefficients of qP-waves at a corrugated interface between two different elastic half-spaces of monoclinic type. Using Rayleigh’s method, the expressions for reflection and transmission coefficients are derived in closed form for a specific interface and for the first order approximation of the corrugation. Numerical computations are performed for a specific model and the results obtained have been shown graphically. The variation of the modulus of reflection and transmission coefficients with the angle of incidence, frequency and corrugation of the interface are shown separately. These coefficients are found to be strongly influenced by the angle of incidence, frequency, elastic parameters and amplitude of the corrugation of the interface. It is found that (i) the modulus of reflection and transmission coefficients at the plane interface are independent of corrugation of the interface and that of frequency of the incident wave, (ii) the reflection and transmission coefficients of regularly reflected and transmitted waves are found to be greater than that of irregularly reflected and transmitted waves, (iii) the coefficients of irregularly reflected and transmitted waves are found to increase and decrease with increase of corrugation and frequency parameters respectively. The results of Singh and Khurana [Singh, S.J., Khurana, S., 2001. Reflection and transmission of P- and SV-waves at the interface two between monoclinic elastic half-spaces. Proc. Natl. Acad. Sci. India 71(A) (IV), 305–319] have been reduced from the present problem.     

Parametric excitation of flexural-gravity edge waves in the fluid beneath an elastic ice sheet with a crack

The properties of elastic-gravity oscillations of deep water beneath a thin elastic plate with a crack are investigated in the paper. The dependence of the reflection and transition coefficients of the waves through the crack on wave frequency and incident angle are found. The shape of the fluid surface deformed by edge waves, propagating along the crack and decreasing exponentially away from the crack, is investigated in the vicinity of the crack. The asymptotic equations describing the parametric excitation of counterpropagating edge waves by flexural-gravity waves which hit the crack at normal incidence are derived.     

弹性波在准饱和土和弹性土界面的反射与透射

考虑土颗粒和孔隙流体的压缩性,根据准饱和土体的变形连续条件和平衡条件,推导了P1波(快压缩波)、P2波(慢压缩波)和S波的计算公式．根据Snell定理,研究了平面P1波从准饱和土体入射到弹性土层时在界面上的反射和透射,并绘制了反射系数、透射系数和界面应力、位移随入射角的变化曲线,结果表明:饱和度是影响界面效应的一个重要因素,当饱和度低于99％时,反射和透射系数以及应力和位移基本相等,变化不大,但当饱和度为100％时,虽然饱和度比99％提高了1％,但反射和透射系数以及应力和位移值比饱和度为99％时明显降低．     

Reflection and refraction of SH-waves at a corrugated interface between two laterally and vertically heterogeneous viscoelastic solid half-spaces

Reflection and transmission coefficients due to incident plane SH-waves at a corrugated interface between two isotropic, laterally and vertically heterogeneous visco-elastic solid half spaces are obtained. The density and complex rigidity of each medium are considered to vary along horizontal and vertical directions. Closed form expressions of reflection and transmission coefficients are derived using Rayleigh’s method of approximation. These coefficients are found to be the function of corrugation, heterogeneity, angle of incidence, angle between propagation and attenuation vectors and visco-elasticity of the media. Numerical computations are made for a specific model to study the nature of dependence of these coefficients. Variations of reflection and transmission coefficients for the first order of approximation of the corrugation versus angle of incidence, corrugation and angle between propagation and attenuation vectors are computed and depicted graphically. Comparison is made between these coefficients in viscoelastic media and in uniform elastic media. The problems investigated earlier by Asano [Bull. Earthq. Res. Inst. 38 (1960) 177], Singh et al. [Acta Geophys. Pol. XXVI (1978) 209], Kaushik and Chopra [Geophys. Res. Bull. 18 (1980) 111] and Gupta [Geophys. Trans. 33 (1987) 89] have been reduced as particular cases.     

Numerical estimation of the influence of joint stiffness on free vibrations of frame structures via the scattering of waves at elastic joints

In the structural design of mechanical products, natural frequencies must be controlled to reduce noise and vibration. In particular, the stiffness of the joints which assemble the structural components affects the natural frequencies. Therefore, it is important to predict the influence of joint stiffness on natural frequencies. Generally, these effects are determined by iterative finite element analyses of assembled structural models. Because this results in high computational costs, the sensitivity of natural frequencies to joint stiffness should be determined by a different approach to make the structural design process more efficient. Therefore, this paper proposes the use of reflection and transmission coefficients of elastic joints to predict the dependency of natural frequencies on joint stiffness. First, we formulate the reflection and transmission coefficients of joint stiffness, and then organize the coefficients using a ray tracing method. These formulations enable us to discuss the mechanisms which determine the natural frequency of a structure based on a wave approach using the phase-closure principle. Therefore, by applying the phase-closure principle to the frame structure, we investigate the formation of bending modes, which suggests that the effects of joint stiffness on natural frequencies correspond to the dependence of the reflection and transmission coefficients on joint stiffness. Therefore, these coefficients are useful indicators for estimating the influence of joint stiffness.     

Wave reflection at a free interface in an anisotropic pyroelectric medium with nonclassical thermoelasticity

In this paper, the well-established two-dimensional mathematical model for linear pyroelectric materials is employed to investigate the reflection of waves at the boundary between a vacuum and an elastic, transversely isotropic, pyroelectric material. A comparative study between the solutions of (a) classical thermoelasticity, (b) Cattaneo–Lord–Shulman theory and (c) Green–Lindsay theory equations, characterised by none, one and two relaxation times, respectively, is presented. Suitable boundary conditions are considered in order to determine the reflection coefficients when incident elasto–electro–thermal waves impinge the free interface. It is established that, in the quasi-electrostatic approximation, three different classes of waves: (1) two principally elastic waves, namely a quasi-longitudinal Primary (qP) wave and a quasi-transverse Secondary (qS) wave; and (2) a mainly thermal (qT) wave. The observed electrical effects are, on the other hand, a direct consequence of mechanical and thermal phenomena due to pyroelectric coupling. The computed reflection coefficients of plane qP waves are found to depend upon the angle of incidence, the elastic, electric and thermal parameters of the medium, as well as the thermal relaxation times. The special cases of normal and grazing incidence are also derived and discussed. Finally, the reflection coefficients are computed for cadmium selenide observing the influence of (1) the anisotropy of the material, (2) the electrical potential and (3) temperature variations and (4) the thermal relaxation times on the reflection coefficients.     

The low frequency scatter of SH waves by a rough interface in an elastic plate

The study of the reflection and transmission of low frequency SH waves incident upon a rough interface in an elastic plate is undertaken by employing a theory of acoustic wave scattering from rough surfaces originally due to Biot and subsequently generalised to the case of elastic media. In this theory the interface is replaced by a distribution of voids/asperities whose individual size is small compared to the excitation wavelength. We plot the absolute values of the reflection and transmission coefficients versus frequency when a single symmetric SH plate mode is used as the input excitation. The different types of inclusions are used to simulate the rough surface are the hollow, fluid filled and aluminum spheres. Lastly, the loss of energy due to scattering is also estimated for the different inclusion distributions considered.     

Hydroelastic analysis of gravity wave interaction with submerged horizontal flexible porous plate

The present study deals with the surface gravity wave interaction with submerged horizontal flexible porous plate under the assumption of small amplitude water wave theory and structural response. The flexible porous plate is modeled using the thin plate theory and wave past porous structure is based on the generalized porous wavemaker theory. The wave characteristics due to the interaction of gravity waves with submerged flexible porous structure are studied by analyzing the complex dispersion relation using contour plots. Three different problems such as (i) wave scattering by a submerged flexible porous plate, (ii) wave trapping by submerged flexible porous plate placed at a finite distance from a rigid wall and (iii) wave reflection by a rigid wall in the presence of a submerged flexible porous plate are analyzed. The role of flexible porous plate in attenuating wave height and creating a tranquility zone is studied by analyzing the reflection, transmission and dissipation coefficients for various wave and structural parameters such as angle of incidence, depth of submergence, plate length, compression force and structural flexibility. In the case of wave trapping, the optimum distance between the porous plate and rigid wall for wave reflection is analyzed in different cases. In addition, effects of various physical parameters on free surface elevation, plate deflection, wave load on the plate and rigid wall are studied. The present approach can be extended to deal with acoustic wave interaction with flexible porous plates.     

含单排周期微裂纹平板结构中的弹性波传播

受损伤固体中含有的微裂纹或微孔洞往往具有周期性，对含周期性缺陷结构中的弹性波分析是力学研究中的重要课题，它直接关系到结构的强度和使用寿命。目前对损伤固体中弹性波散射与透射研究结果主要是弹性动力学平面问题。1995年。Scarpetta和Sumbatyan采用解析法研究了平面波在双周期裂纹弹性介质中的传播问题。并推出显式分析结果。本文基于弹性动力学理论，分析研究了含有单排横向周期裂纹的平板中弯曲波的反射与透射问题。给出了含单排裂纹时反射波与透射波系数的数值结果。对于多排裂纹情况，可采用具有退化核第一类Fredholm积分方程方法分析求解，在求解中给出相应的无量纲数，例如无量纲波数、裂纹尺寸比等。本文分析结果可望能在工程振动控制中应用。