Mathematical analysis of the waves propagation through a rectangular material structure in space-time

We consider propagation of waves through a spatio-temporal doubly periodic material structure with rectangular microgeometry in one spatial dimension and time. Both spatial and temporal periods in this dynamic material are assumed to be the same order of magnitude. Mathematically the problem is governed by a standard wave equation t(ρut)−z(kuz)=0 with variable coefficients. We consider a checkerboard microgeometry where variables cannot be separated. The rectangles in a space-time checkerboard are assumed filled with materials differing in the values of phase velocities but having equal wave impedance . The difference between dynamic materials and classical static composites is that in the former case the design variables will also be time dependent. Within certain parameter ranges, the formation of distinct and stable limiting characteristic paths, i.e., limit cycles, was observed in [K.A. Lurie, S.L. Weekes, Wave propagation and energy exchange in a spatio-temporal material composite with rectangular microstructure, J. Math. Anal. Appl. 314 (2006) 286-310]; such paths attract neighboring characteristics after a few time periods. The average speed of propagation along the limit cycles remains the same throughout certain ranges of structural parameters, and this was called in [K.A. Lurie, S.L. Weekes, Wave propagation and energy exchange in a spatio-temporal material composite with rectangular microstructure, J. Math. Anal. Appl. 314 (2006) 286-310] a plateau effect. Based on numerical evidence, it was conjectured in [K.A. Lurie, S.L. Weekes, Wave propagation and energy exchange in a spatio-temporal material composite with rectangular microstructure, J. Math. Anal. Appl. 314 (2006) 286-310] that a checkerboard structure is on a plateau if and only if it yields stable limit cycles and that there may be energy concentrations over certain time intervals depending on material parameters. In the present work we give a more detailed analytic characterization of these phenomena and provide a set of sufficient conditions for the energy concentration that was predicted numerically in [K.A. Lurie, S.L. Weekes, Wave propagation and energy exchange in a spatio-temporal material composite with rectangular microstructure, J. Math. Anal. Appl. 314 (2006) 286-310].     

Analyzing of the effect of the dusty fluid,anisotropy and layered of the wall on the wave propagation

This study considers the propagation of time harmonic waves in, prestressed, anisotropic elastic tubes filled with viscous fluid containing dusty particles. The fluid is assumed to be incompressible and Newtonian. The tube material is considered to be incompressible, anisotropic, and elastic. The tube is subjected to a static inner pressure P_i and an axial stretch λ. Utilizing the theory of “Superposing small deformations on large initial static deformations”, differential equations governing wave propagation inside the tube are obtained in terms of cylindrical coordinates. Analytical solutions for the equations of motion for the dust and the fluid are obtained, and expressed numerically. The dispersion relation is obtained as a function of the stretch, the thickness ratio and the parameters for dusty particles.     

Wave propagation in advected acoustics within a non-uniform medium under the effect of gravity

We investigate linear wave propagation in non-uniform medium under the influence of gravity. Unlike the case of constant properties medium here the linearized Euler equations do not admit a plane-wave solution. Instead, we find a “pseudo-plane-wave”. Also, there is no dispersion relation in the usual sense. We derive explicit analytic solutions (both for acoustic and vorticity waves) which, in turn, provide some insights into wave propagation in the non-uniform case.     

Wave propagation in piezoelectric medium of hexagonal symmetry

The wave propagation at large distances from a source of disturbance (isolated harmonic electric charge or body force of fixed frequency) in an infinite piezoelectric medium belonging to classes (6), (6 m m) or ceramic (α m) and (6 2 2) is discussed by means of asymptotic evaluation (at large distances) of Fourier integrals. Numerical results are given for the (6 m m) crystal class using the constants of cadmium selenide crystal.     

Dispersion of elastic waves in the contact-impact problem of a long cylinder

Numerical dispersion of two-dimensional finite elements was studied. The outcome of the dispersion study was verified by the numerical and analytical solutions to the longitudinal impact of two long cylindrical bars. In accordance with the results of the dispersion analysis it was demonstrated that the quadratic elements showed better accuracy than the linear ones.     

High-resolution difference methods with exact evolution for multidimensional waves

We consider the generalization of high-order upwind Strang methods for simulating waves. In $1+1$ dimensions the methods can be defined via the exact evolution over a single time step of an odd-order piecewise polynomial interpolant of the grid data. We construct a true multidimensional version for acoustic waves by applying the solution operator in integral form to the interpolant. We also examine the replacement of polynomials by bandlimited interpolation functions (BLIFs). Numerical experiments with turbulent wave fields are presented to verify the accuracy and stability of the multidimensional methods and to assess the relative effectiveness of the two interpolation techniques.     

Dynamic crack propagation in a 2D elastic body: The out-of-plane case

Already in 1920 Griffith has formulated an energy balance criterion for quasistatic crack propagation in brittle elastic materials. Nowadays, a generalized energy balance law is used in mechanics [F. Erdogan, Crack propagation theories, in: H. Liebowitz (Ed.), Fracture, vol. 2, Academic Press, New York, 1968, pp. 498-586; L.B. Freund, Dynamic Fracture Mechanics, Cambridge Univ. Press, Cambridge, 1990; D. Gross, Bruchmechanik, Springer-Verlag, Berlin, 1996] in order to predict how a running crack will grow. We discuss this situation in a rigorous mathematical way for the out-of-plane state. This model is described by two coupled equations in the reference configuration: a two-dimensional scalar wave equation for the displacement fields in a cracked bounded domain and an ordinary differential equation for the crack position derived from the energy balance law. We handle both equations separately, assuming at first that the crack position is known. Then the weak and strong solvability of the wave equation will be studied and the crack tip singularities will be derived under the assumption that the crack is straight and moves tangentially. Using the energy balance law and the crack tip behavior of the displacement fields we finally arrive at an ordinary differential equation for the motion of the crack tip.     

Wave propagation analysis in buried pipe conveying fluid

A study of wave propagation in buried pipe conveying fluid is presented in the paper. The Flüggle shell model is adopted for pipe and surrounding solid is modeled as elastic matrix by using Winkle model. Wave dispersion curves of a buried vacant pipe and a pipe conveying fluid are obtained numerically by considering coupling conditions. Results show that wave velocity exhibits sharp drop points in dispersion curves, and remains to an identical values before and after the points for both of vacant pipe and pipe conveying fluid. Effects of wall thickness, elastic matrix properties and fluid velocity are also discussed.     

One-mode wave propagation through a periodic array of interface cracks: Explicit analytical results

In the context of wave propagation in damaged composite elastic media, an analytical approach is developed to study the normal penetration of a longitudinal wave into a periodic array of interface thin defects (cracks) between two different materials. The problem is reduced to some integral equations which hold over the opening between adjacent cracks and are independent on frequency. By means of an original procedure, such equations are solved and some related integrals are calculated, so that an explicit analytical representation can be provided for the relevant scattering parameters. Finally, several graphs are set up which reflect the peculiarities of the structure; an excellent agreement is observed—in the concerned (one-mode) regime of propagation—between the obtained formulas and results from a full-numerical treatment of the problem.     

Influences of longitudinal magnetic field on wave propagation in carbon nanotubes embedded in elastic matrix

This paper reported the result of an investigation into the effect of magnetic field on wave propagation in carbon nanotubes (CNTs) embedded in elastic matrix. Dynamic equations of CNTs under a longitudinal magnetic field are derived by considering the Lorentz magnetic forces. The results obtained show that wave propagation in CNTs embedded in elastic matrix under longitudinal magnetic field appears in critical frequencies at which the velocity of wave propagation drops dramatically. The velocity of wave propagation in CNTs increases with the increase of longitudinal magnetic field exerted on the CNTs in some frequency regions. The critical/cut-off frequency increases with the increase of matrix stiffness, and the influence of matrix on wave velocity is little in some frequency regions. This investigation may give a useful help in applications of nano-oscillators, micro-wave absorbing and nano-electron technology.     

Characteristics of wave propagation in piezoelectric bent rods with arbitrary curvature

The wave propagation in the piezoelectric bend rods with arbitrary curvature is studied in this paper. Basic three-dimensional equations in an orthogonal curvilinear coordinate system (r, θ, s) are established. The Bessel functions in radial co-ordinate and triangle series in the angular co-ordinate are used to describe the displacements and electrical potential. Characteristics of dispersion, distributions of displacements and electrical potential over the cross section are calculated, respectively. In the numerical examples, the effects of the ratio of the two ellipse axes on the dispersion relations of the first three modes are observed. The characteristics of the distribution of displacements and electric potential in the cross section, along the radial and s direction are investigated.     

The influence of shelf zone topography and coastline geometry on coastal trapped waves

The results of numerical experiments with a model of coastal trapped waves are presented to identify two important features for regional modeling of the interaction of a shelf zone with open ocean. First, a wave train of this type can be formed by wind action at a considerable distance from the place of impact. The waves propagate along a coastline without significant loss of energy, provided that the coastline and shelf zone topography have no features comparable to the Rossby radius. However, the waves lose energy while passing over capes and submarine canyons and when the shelf width decreases. For regional modeling, remote generation of waves must be thoroughly investigated and taken into account. The other feature is that the propagating waves can use part of energy to form density anomalies on the shelf by raising intermediate waters from the adjacent offshore areas of the open ocean. Thus, coastal trapped waves can carry wind energy from wind action areas to other coastal areas to form density anomalies and other types of motion.     

Split fields and direction of propagation for the solution to first-order systems of equations

The standard wave-splitting approach for the wave equation in inhomogeneous media is first reexamined. Next, by analogy with the theory of wave propagation through singular surfaces, a characterization is given for a function in space-time to represent a wave propagating in a direction. The condition is applied in connection with a simple example and found to be quite restrictive. The same problem is then considered in the Fourier-transform domain where the unknown function is an n-tuple satisfying a system of ordinary differential equations. The condition for propagation in a direction is established for the Fourier components. Next, some physical problems are considered which are expressed by partial differential equations or by integro-differential equations. The associated first-order system of equations is examined in terms of the eigenvalues of a matrix. This shows that, for any eigenvalue, the direction of propagation may change with the frequency and that arguments about the dominance of the principal part of the operator may cease to hold.     

Propagation of waves in a micropolar elastic layer with stretch immersed in an infinite liquid

The effect of liquid on the propagation of waves in a micropolar elastic layer with stretch has been investigated. The frequency and wave velocity equations for symmetric and antisymmetric vibrations are derived. Propagation of monochromatic waves in a micropolar elastic layer with stretch is discussed. Results of this analysis reduce to those without stretch.     

Nonlinear transient thermal stress and elastic wave propagation analyses of thick temperature-dependent FGM cylinders,using a second-order point-collocation method

Nonlinear transient thermal stress and elastic wave propagation analyses are developed for hollow thick temperature-dependent FGM cylinders subjected to dynamic thermomechanical loads. Stress wave propagation, wave shape distortion, and speed variation under impulsive mechanical loads in thermal environments are also investigated. In contrast to researches accomplished so far, a second-order formulation rather than a first-order one is employed to improve the accuracy. The FDM method (as a point-collocation FEM method) is used. It is known that other FEM methods cannot show the actual trend jumps due to distributing the abrupt changes in the quantities as the numerical errors and the residuals of the governing equations among the nodal results. Furthermore, the required computational time and allocated computer memory are much reduced by the present solution algorithm. The cylinder is not divided into isotropic sub-cylinders. Therefore, artificial wave reflections from the hard interfaces are avoided. Time variations of the temperatures, displacements, and stresses due to the dynamic or impulsive loads are determined by solving the resulted highly nonlinear governing equations using an iterative updating solution scheme. A sensitivity analysis includes effects of the volume fraction indices, dimensions, and temperature-dependency of the material properties is performed. Results reveal the significant effect of the temperature-dependency of the material properties on the thermoelastic stresses and present some interesting characteristics of the thermoelastic and wave propagation behaviors.     

A new scheme for the solution of reaction diffusion and wave propagation problems

In this paper, a robust numerical scheme is presented for the reaction diffusion and wave propagation problems. The present method is rather simple and straightforward. The Houbolt method is applied so as to convert both types of partial differential equations into an equivalent system of modified Helmholtz equations. The method of fundamental solutions is then combined with the method of particular solution to solve these new systems of equations. Next, based on the exponential decay of the fundamental solution to the modified Helmholtz equation, the dense matrix is converted into an equivalent sparse matrix. Finally, verification studies on the sensitivity of the method’s accuracy on the degree of sparseness and on the time step magnitude of the Houbolt method are carried out for four benchmark problems.     

关于激光等离子体声波的数学模型

根据激光等离子体冲击波的传播规律,利用数学方法和物理知识解释了激光等离子体冲击波转化为激光等离子体声波的原理.     

On the propagation of elastic waves in two-phase media

At the present time a number of papers has been already devoted to the dynamics of two-phase media. One may mention the papers by Frenkel' [1], Rakhmatulin [2], Biot [3,4], Zwikker and Kosten [5], and others. However, the basic problem of the setting up of the equations of motion in two-phase media still cannot be considered solved and requires additional study and experimental verification.

This paper is concerned with the study of the simplest case of motion, which is the propagation of elastic waves in a homogeneous isotropic medium consisting of a solid and a fluid phase. The problems of the reflection of plane waves and surface waves at the free boundary of the half-space are solved. It is shown that the stress-strain relations established by Frenkel' are equivalent to the analogous relations proposed by Biot and that the equations of motion of the latter are more general.