Wave equation system for dilatation and torsion for piecewise continuous materials

Summary The wave equations governing the propagation and scatter of dilatation and torsion are developed from first principles for visco-elastic, piecewise continuous materials. The wave equations describe the conversion of dilatational to torsion waves in materials of spatially varying properties. The conditions for maximum excitation transfer at interfaces are identified. The physical insights provided by the mathematical formulation of the coupled dilatation and torsion wave equations are used to interpret results obtained from finite element numerical model calculations of vibrations in and around inclusions in composite materials excited by a sinusoidal pressure wave or pulse generated by noisy machinery.     

Simulation of vibration of inhomogeneous,loaded plates

Summary The Timoshenko-Mindlin type wave equation is derived for a plate which is inhomogeneous in material and geometric properties. A numerical analogue to this wave equation for a plate is presented and results of propagation studies of Gaussian and sinusoidal wavepackets along a plate reinforced by ribs of up to five times the thickness of the plate are discussed. Results are presented for an infinite plate supported by two ribs. The plate equation includes consideration of shear and rotary inertia, external loading and plastic-visco-elastic loss. A spectral analysis of the inhomogeneous plate equation is presented.     

Symmetry Transformations and Exact Solutions of a Generalized Hyperelastic Rod Equation

In this paper, a nonlinear wave equation with variable coefficients is studied, interestingly, this equation can be used to describe the travelling waves propagating along the circular rod composed of a general compressible hyperelastic material with variable cross-sections and variable material densities. With the aid of Lou’s direct method1, the nonlinear wave equation with variable coefficients is reduced and two sets of symmetry transformations and exact solutions of the nonlinear wave equation are obtained. The corresponding numerical examples of exact solutions are presented by using different coefficients. Particularly, while the variable coefficients are taken as some special constants, the nonlinear wave equation with variable coefficients reduces to the one with constant coefficients, which can be used to describe the propagation of the travelling waves in general cylindrical rods composed of generally hyperelastic materials. Using the same method to solve the nonlinear wave equation, the validity and rationality of this method are verified.     

浅海沉积层中界面波的传播以及掩埋目标的回波

根据地声界面波的相关理论,采用时域有限差分法建立浅海沉积层中界面波传播模型。以沉积层中掩埋小目标为例,计算界面波对空腔掩埋物和实心掩埋物的不同作用结果。根据运动方程和应力应变关系采用中心差分网格建立声场的有限差分模型,计算了声场瞬态情况和质点振动的时间序列结果,从不同角度比较分析了地声界面波对掩埋目标的作用。根据浅海沉积层的性质计算了界面波的传播损失和对掩埋物体的目标强度,由于界面波传播衰减主要由沉积层的能量耗散引起,其规律接近横波的传播衰减,在高频情况下界面波在沉积层中传播的距离不远。文章试图建立一种对浅海沉积层中掩埋小尺度目标的探测新方法。通过一些典型模型的分析,从理论角度证明地声界面波对目标探测的可行性。     

Propagation of inhomogeneous plane waves in anisotropic viscoelastic media

A new technique is explained to study the propagation of inhomogeneous waves in a general anisotropic medium. The harmonic plane waves are considered in a viscoelastic anisotropic medium. The complex slowness vector is decomposed into propagation vector and attenuation vector for the given directions of propagation and attenuation of waves in an unbounded medium. The attenuation is further separated into the contributions from homogeneous and inhomogeneous waves. A non-dimensional inhomogeneity parameter is defined to represent the deviation of an inhomogeneous wave from its homogeneous version. Such a partition of slowness vector of a plane wave is obtained with the help of an algebraic method for solving a cubic equation and a numerical method for solving a real transcendental equation. Derived specifications enable to study the 3D propagation of inhomogeneous plane waves in a viscoelastic medium of arbitrary anisotropy. The whole procedure is wave-specific and obtains the propagation characteristics for each of the three inhomogeneous waves in the anisotropic medium. Numerical examples analyze the variations in propagation characteristics of each of the three waves with propagation direction and inhomogeneity strength.     

一种由横波激发的特殊非线性声现象

目前对非线性超声的研究多集中在纵波激发的谐波性质以及对材料微观结构变化的实验检测上,横波激发的非线性声波性质少有研究。对横波激发的一维非线性声波方程入手,利用摄动法求解该方程,并改写为一阶偏微分方程,然后利用交错网格的有限差分形式进行数值求解。结果表明:采用横波激发,能产生线性横波和非线性纵波,且纵波的高次谐波内有两个信号,分别以纵波和横波两种速度传播。若采用较长的激发信号,纵波谐波能形成"拍"现象,成为一种奇特的声传播现象。     

用改进的Boussinesq方程模拟潜堤上的波浪变形

研究了一种改进的Boussinesq方程,采用预报校正格式对该方程进行了数值离散,并对淹没潜堤上的波浪变形进行了数值模拟,从数值模拟结果和实测值的比较结果来看,该方程能较好地模拟波浪在潜堤上传播时波面的变形过程,可以用于实际中的波浪问题计算。这种改进的Boussinesq方程本身及其求解方法需做进一步的完善。研究结果为实际应用Boussinesq方程来研究复杂地形上的波浪传播提供一定的理论指导。     

A finite element Laplace transform solution technique for the wave equation

A technique is described for the solution of the wave equation with time dependent boundary conditions. The finite element solution accompanied by the numerical Laplace inversion process seems to be an efficient procedure to treat such problems. The programming involved is straightforward in the sense that numerical Laplace inversion routines can be directly used as a time integration procedure after obtaining standard finite element differential equation solutions in the transformed domain. Some results are presented for one- and two- dimensional applications, such as wave propagation in longitudinal bars and wave propagation in harbours.     

Electrodynamics of layered media with boundary conditions corresponding to the total-current continuum

A consistent physical and mathematical model of propagation of electromagnetic waves in layered media, in which the induced surface charge is not explicitly taken into account, has been constructed on the basis of Maxwell equations, the total-current continuum equation, and the Dirichlet theorem. A numerical method of through counting for solving the system of equations defining layered media has been developed and substantiated. Examples of numerical simulation of the propagation of an electromagnetic wave in a two-dimensional cellular structure and the propagation of a modulated signal in a one-dimensional layered medium on condition that the current flows at the interfaces between the adjacent media are dependent are presented.     

Propagation of inhomogeneous waves in anisotropic piezo-thermoelastic media

A mathematical model for the propagation of harmonic plane waves in an anisotropic piezo-thermoelastic medium is explained through three relations. Two of them relate the stress-induced harmonic variations in temperature and electric potential to mechanical displacement of material particles. The third is a system that defines modified Christoffel equations for wave propagation in the medium. The solution of this system is ensured by a quartic equation whose complex roots explain the existence and propagation of four attenuating waves in the medium. The effects of piezoelectricity and thermoelasticity on the wave propagation are analyzed in the discussion of special cases. An angle between propagation direction and direction of maximum attenuation defines the attenuated wave as inhomogeneous wave. The complex slowness vector for each of the four attenuated waves in the medium is resolved to calculate the phase velocity and the attenuation factor for its propagation as an inhomogeneous wave along a general direction in three-dimensional space. The variations in phase velocities and attenuation factors with propagation direction are computed, for a realistic numerical model.     

Corner problems and global accuracy in the boundary element solution of nonlinear wave flows

The numerical model for nonlinear wave propagation in the physical space, developed by Grilli, et al.^12,13, uses a higher-order BEM for solving Laplace's equation, and a higher-order Taylor expansion for integrating in time the two nonlinear free surface boundary conditions. The corners of the fluid domain were modelled by double-nodes with imposition of potential continuity. Nonlinear wave generation, propagation and runup on slopes were successfully studied with this model. In some applications, however, the solution was found to be somewhat inaccurate in the corners and this sometimes led to wave instability after propagation in time.

In this paper, global and local accuracy of the model are improved by using a more stable free surface representation based on quasi-spline elements and an improved corner solution combining the enforcement of compatibility relationships in the double-nodes with an adaptive integration which provides almost arbitrary accuracy in the BEM numerical integrations. These improvements of the model are systematically checked on simple examples with analytical solutions. Effects of accuracy of the numerical integrations, convergence with refined discretization, domain aspect ratio in relation with horizontal and vertical grid steps, are separately assessed. Global accuracy of the computations with the new corner solution is studied by solving nonlinear water wave flows in a two-dimensional numerical wavetank. The optimum relationship between space and time discretization in the model is derived from these computations and expressed as an optimum Courant number of 0.5. Applications with both exact constant shape waves (solitary waves) and overturning waves generated by a piston wavemaker are presented in detail.     

Rayleigh surface waves problem in linear thermoviscoelasticity with voids

In this paper, we study the propagation of the Rayleigh surface waves in a half-space filled by an exponentially functionally graded thermoviscoelastic material with voids. We take into consideration the dissipative character of the porous thermoviscoelastic models upon the propagation waves and study the damped in time wave solutions. The propagation condition is established in the form of an algebraic equation of tenth degree whose coefficients are complex numbers. The eigensolutions of the dynamical system are explicitly obtained in terms of the characteristic solutions. The concerned solution of the Rayleigh surface wave problem is expressed as a linear combination of the five analytical solutions, while the secular equation is established in an implicit form. The explicit secular equation is obtained for an isotropic and homogeneous thermoviscoelastic porous half-space, and some numerical simulations are given for a specific material.     

Band gaps of elastic waves in 1-D phononic crystal with dipolar gradient elasticity

The dispersive relations of Bloch waves in the periodic laminated structure formed by periodically repeating of two different gradient elastic solids are studied in this paper. First, the various wave modes in the gradient elastic solid, which are different from those in the classical elastic solid, are formulated. Apart from the dispersive P wave and SV wave, there are two evanescent waves, which become the P type and S type surface waves at the interface of two different gradient elastic solids. Next, the continuity conditions of displacement vector, the normal derivative of the displacement vector and the monopolar and dipolar tractions across the interface between two different gradient elastic solids are used to derive the transfer matrix of the state vector in a typical single cell. At last, the Bloch theorem of Bloch waves in the periodical structure is used to give the dispersive equation. The in-plane Bloch waves and the anti-plane Bloch waves are both considered in the present work. The oblique propagation situation and the normal propagation situation are also considered, respectively. The numerical results are obtained by solving the dispersive equation. The influences of two microstructure parameters of the gradient elastic solid and the microstructure parameter ratio of two different gradient elastic solids on the dispersive relation are discussed based on the numerical results.     

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。     

Hydroelastic Behavior of a Floating Ring-Shaped Plate

The interaction between incident surface water waves and floating elastic plate is studied. This paper considers the diffraction of plane incident waves on a floating flexible ring-shaped plate and its response to the incident waves. An analytic and numerical study of the hydroelastic behavior of the plate is presented. An integro-differential equation is derived for the problem and an algorithm of its numerical solution is proposed. The representation of the solution as a series of Hankel functions is the key ingredient of the approach. The problem is first formulated. The main integro-differential equation is derived on the basis of the Laplace equation and thin-plate theory. The free-surface elevation, plate deflection and Green’s function are expressed in polar coordinates as superpositions of Hankel and Bessel functions, respectively. These expressions are used in a further analysis of the integro-differential equation. The problem is solved for two cases of water depth infinite and finite. For the coefficients in the case of infinite depth a set of algebraic equations is obtained, yielding an approximate solution. Then a solution is obtained for the general and most interesting case of finite water depth analogously in the seventh section. The exact solution might be approximated by taking into account a finite number of the roots of the plate dispersion relation. Also, the influence of the plate’s motion on wave propagation in the open water field and within the gap of the ring is studied. Numerical results are presented for illustrative purposes.     

Matrix Method for Determining Propagation Characteristics of Absorbing Waveguides

A numerical procedure is presented for calculating the complex propagation constants of absorbing waveguides by transforming the scalar wave equation into a matrix eigenvalue equation. The method is simple and can be used for arbitrary index profiles, and provides a more general treatment than perturbation techniques.     

Non-principal surface waves in deformed incompressible materials

The Stroh formalism is applied to the analysis of infinitesimal surface wave propagation in a statically, finitely and homogeneously deformed isotropic half-space. The free surface is assumed to coincide with one of the principal planes of the primary strain, but a propagating surface wave is not restricted to a principal direction. A variant of Taziev’s technique [R.M. Taziev, Dispersion relation for acoustic waves in an anisotropic elastic half-space, Sov. Phys. Acoust. 35 (1989) 535–538] is used to obtain an explicit expression of the secular equation for the surface wave speed, which possesses no restrictions on the form of the strain energy function. Albeit powerful, this method does not produce a unique solution and additional checks are necessary. However, a class of materials is presented for which an exact secular equation for the surface wave speed can be formulated. This class includes the well-known Mooney–Rivlin model. The main results are illustrated with several numerical examples.     

Spinor–electron waveguided optics based on the Dirac equation

A preliminary quantum analysis, based on the Dirac equation, of the propagation of spinor–electron waves in electron waveguides is presented. The wave equations for spin-up (SU) and spin-down (SD) electron waves in electron guides are derived and their analogy with TE and TM light modes in dielectric guides is stressed. The spinor–electron waveguided propagation in a electron wave slab waveguide is solved exactly using the Dirac equation, that is, the exact amplitudes and dispersion equations of the spinor–electron waveguided modes are calculated. The main consequences related to the spinor modal structure are discussed: phase retardations (spin polarization), modal cutoff conditions and the non-relativistic limit.     

椭球面镜声波反射聚焦数值模拟研究

据Hamilton线性理论解,给出沿椭球面镜轴线的电弧放电等离子声源(AD-PSS)反射波计算结果,分析由镜面引起的相位变化对中心波、边缘波及尾波传播影响。在聚焦前区中心波压力为正,边缘波、尾波压力为负;聚焦后区则相反。在线性条件下,反射波压力峰值出现于椭球面镜几何焦点,越过焦点后反射波压力幅值迅速衰减。利用有限元软件COMSOL对椭球面镜声反射过程进行数值模拟,揭示反射波传播演化过程及声场分布规律。据KZK方程及等效声源法,分析非线性效应对声传播过程影响。研究表明,非线性效应将使椭球面镜实际焦点位置偏离几何焦点,即正压实际焦点出现在几何焦点后,负压实际焦点出现在几何焦点前;随非线性系数增加正压实际焦点后移,负压实际焦点前移。     

Guided wave propagation in an elastic hollow cylinder coated with a viscoelastic material

The propagation of ultrasonic guided waves in an elastic hollow cylinder with a viscoelastic coating is studied. The principle motivation is to provide tools for performing a guided wave, nondestructive inspection of piping and tubing with viscoelastic coatings. The theoretical boundary value problem is solved that describes the guided wave propagation in these structures for the purpose of finding the guided wave modes that propagate with little or no attenuation. The model uses the global matrix technique to generate the dispersion equation for the longitudinal modes of a system of an arbitrary number of perfectly bonded hollow cylinders with traction-free outer surfaces. A numerical solution of the dispersion equation produces the phase velocity and attenuation dispersion curves that describe the nature of the guided wave propagation. The attenuation dispersion curves show some guided wave modes that propagate with little or no attenuation in the coated structures of interest. The wave structure is examined for two of the modes to verify that the boundary conditions are satisfied and to explain their attenuation behavior. Experimental results are produced using an array of transducers positioned circumferentially around the pipe to evaluate the accuracy of the numerical solution.