Propagation Of Harmonic Waves Research Articles

Time harmonic wave propagation in one dimensional weakly randomly perturbed periodic media

In this work we consider the solution of the time harmonic wave equation in a one dimensional periodic medium with weak random perturbations. More precisely, we study two types of weak perturbations: (1) the case of stationary, ergodic and oscillating coefficients, the typical size of the oscillations being small compared to the wavelength and (2) the case of rare random perturbations of the medium, where each period has a small probability to have its coefficients modified, independently of the other periods. Our goal is to derive an asymptotic approximation of the solution with respect to the small parameter. This can be used in order to construct absorbing boundary conditions for such media.

On the characterization and stability of plane waves under hyperbolic two-temperature generalized thermoelasticity

The propagation and stability (Whitham’s criteria) of harmonic plane waves are described in the context of the hyperbolic two-temperature generalized thermoelasticity in which heat conduction in deformable bodies depends upon the difference between the double derivative of conductive and dynamic temperature. The exact dispersion relation solutions for the longitudinal plane wave are derived analytically. Several characterizations of the wave field, like phase velocity, specific loss, penetration depth, amplitude coefficient factor, and phase shift are examined for the low as well as high frequency asymptotic expansions. For the validity of analytical findings and to study the effect of varying hyperbolic two-temperature parameter on different characterizations, the numerical computation of a particular example is illustrated and displayed graphically. The results of some earlier works have been deduced and discussed from the present investigation as speciallimiting cases.

Sweeping preconditioners for stratified media in the presence of reflections

In this paper we consider sweeping preconditioners for time harmonic wave propagation in stratified media, especially in the presence of reflections. In the most famous class of sweeping preconditioners Dirichlet-to-Neumann operators for half-space problems are approximated through absorbing boundary conditions. In the presence of reflections absorbing boundary conditions are not accurate resulting in an unsatisfactory performance of these sweeping preconditioners. We explore the potential of using more accurate Dirichlet-to-Neumann operators within the sweep. To this end, we make use of the separability of the equation for the background model. While this improves the accuracy of the Dirichlet-to-Neumann operator, we find both from numerical tests and analytical arguments that it is very sensitive to perturbations in the presence of reflections. This implies that even if accurate approximations to Dirichlet-to-Neumann operators can be devised for a stratified medium, sweeping preconditioners are limited to very small perturbations.

Third harmonic shear horizontal waves for material degradation monitoring

Early detection of incipient damage in structures through material degradation monitoring is a challenging and important topic. Nonlinear guided waves, through their interaction with material micro-defects, allow possible detection of structural damage at its early stage of initiations. This issue is investigated using both the second harmonic Lamb waves and the third harmonic shear horizontal waves in this article. A brief analysis first highlights the selection of the primary–secondary S0 Lamb wave mode pair and primary–tertiary SH0 mode pair from the perspective of cumulative high-order harmonic wave generation. Through a tactic design, an experiment is then conducted to compare the sensitivity of the third harmonic shear horizontal waves and the second harmonic Lamb waves to microstructural changes on the same plate subjected to a dedicated thermal heating treatment. The third harmonic shear horizontal waves are finally applied to monitor the microstructural changes and material degradation in a plate subjected to a thermal aging sequence, cross-checked by Vickers hardness tests. The experiment results demonstrate that the third harmonic shear horizontal waves indeed exhibit higher sensitivity to microstructural changes than the commonly used second harmonic Lamb waves. In addition, results demonstrate that the designed third harmonic shear horizontal wave–based system entails effective characterization of thermal aging–induced microstructural changes in metallic plates.

WAVES IN A POROUS MEDIUM WITH A GAS HYDRATE CONTAINING LAYER

This study describes the propagation of harmonic and pulsed waves in a layered porous medium, with one of its layers containing gas hydrate. It is shown that the resulting wave pattern makes it possible to predict the presence of a hydrate in a layer and its thickness provided that thermobaric conditions for the existence of the gas hydrate are fulfilled.

Fundamental Solution and Study of Plane Waves in Bio-Thermoelastic Medium with DPL

The fundamental solution of the system of differential equations in bio-thermoelasticity with dual phase lag (DPL) in case of steady oscillations in terms of elementary function is constructed and basic property is established. The tissue is considered as an isotropic medium and the propagation of plane harmonic waves is studied. The Christoffel equations are obtained and modified with the thermal as well as bio thermoelastic coupling parameters. These equations explain the existence and propagation of three waves in the medium. Two of the waves are attenuating longitudinal waves and one is non-attenuating transverse wave. The thermal property has no effect on the transverse wave. The velocities and attenuating factors of longitudinal waves are computed for a numerical bioheat transfer model with phase lag. The variation with frequency, thermal parameters, blood perfusion parameter and phase lag parameter are presented graphically. Also the reflection of plane wave from a stress free isothermal boundary of isotropic bio-thermoelastic half space in the context of DPL theory of thermoelasticity is studied. The amplitude ratios of various reflected waves are obtained and these amplitude ratios are further used to obtain the energy ratios of various reflected waves. These energy ratios are function of the angle of incidence and bio-thermoelastic properties of the medium. The expressions of energy ratios have been computed numerically for a particular model to show the effect of Poisson ratio, blood perfusion rate and phase lag parameters.

Analysis of harmonic plane wave propagation predicted by strain and temperature-rate-dependent thermoelastic model

ABSTRACT The present work is concerned with a novel generalized thermoelasticity theory introduced by Yu et al. [A modified Green-Lindsay thermoelasticity with strain rate to eliminate the discontinuity. Meccanica. 2018;53:2543–2554] which considers the strain-rate along with temperature-rate term in the constitutive equations. This theory is an attempt to modify the Green-Lindsay (GL) thermoelasticity theory. For an unbounded isotropic homogeneous elastic medium, the present work analyzes the propagation of plane harmonic waves in the context of this new theory. In order to investigate the effects of strain-rate term, the unified governing equations related to three thermoelastic models namely, classical coupled thermoelastic model, GL model and new model have been considered. The dispersion relation for the propagation of longitudinal plane waves in the contexts of all three models has been derived in a unified way, which is further solved by the computational tool for a particular material. The behavior of different wave components for longitudinal waves is examined through graphical representation and the effects of temperature-rate and strain-rate terms on the plane wave propagation are investigated. Significant differences in predictions by this new model as compared to other models are highlighted.

Second Harmonic Generation by ultra-fast Lasers in Plasma.

The interaction of laser with plasma gives rise to several novel relativistic nonlinear optical effects. Higher harmonic generation is an important nonlinear effect. High power ultrafast lasers uses chirped pulse amplification technique to reduce duration of pulse and hence produces pulses of extremely high power densities which in turn oscillates electrons in the medium. The velocity of oscillating electron beats with applied frequency giving rise to higher order harmonics. This paper presents the idea of generation of second harmonic waves with the ultra-fast lasers in plasma, phase matching conditions of second harmonic generated by self modulation under paraxial approximations and its application in imaging microscopy.

Body waves propagation in a fluid-saturated transversely isotropic poroelastic solid with a potential method

This study is focused on the propagation of plane harmonic body waves in a transversely isotropic linear poroelastic fluid-saturated medium, where the material symmetry axes for both solid and fluid are coincide. Simplified formulation is considered as the framework. A set of two scalar potential functions is employed to decouple the coupled fluid continuity equation and equations of motion. The velocities and corresponding attenuation coefficients of both longitudinal and transverse waves are extracted from the presented acoustic equations for the body waves. To show the validity of the analytical solution given in this paper, degeneration to the case of a single phase transversely isotropic and consequently isotropic solid is presented to provide interesting comparisons with the solutions reported in the literatures. The incompressible solid and fluid are degenerated from the general slowness obtained in this study for some special cases. In addition, the effects of mechanical and hydraulic parameters of materials on the velocity of propagation and attenuation coefficient of the waves are investigated in more detail. To this end, various synthetic poroelastic transversely isotropic materials are defined and the dependency of wave motion to these parameters is illustrated by plotting the graphs.

Generation of circumferential harmonic travelling waves on thin circular plates

A paradigm to generate harmonic circumferential travelling waves on a thin circular plate is proposed. The plate is considered to be connected to three circumferentially placed point actuators that excite the plate transversely with appropriate phasing. Uniform angular distribution of the actuators is observed to lead to wave locking at certain frequencies. The idea of angular detuning in the placement of the actuators is introduced to prevent wave locking. It is found that a simple kinematic phasing, which can generate travelling waves with uniformly distributed actuators, cannot be used in the case of non-uniformly placed actuators. An algorithm to calculate the phase detuning is proposed for a perturbed angular configuration of actuators for a smooth generation of travelling waves near any modal frequency. The mathematical model and the wave generation paradigm is validated using a finite element model developed in ANSYS. The results presented herein is expected to be helpful in studying the mechanics of wave-based transport of granular materials interacting with such surface travelling waves.

Erratum to: A theory of porous media and harmonic wave propagation in poroelastic body

Erratum to: A theory of porous media and harmonic wave propagation in poroelastic body

Highly accurate space-time coupled least-squares finite element framework in studying wave propagation

Simulation of stress wave propagation through solid medium is commonly carried using Galerkin weak-form cast over decoupled space and time domains. In this paper, accuracy of this commonly utilized framework is compared to that of the variationally-consistent least-squares form of the wave equation cast over space-time domain. The two formulations are tested for numerical dispersion and numerical diffusion, through two test cases. The first case studies the dispersion in harmonic shear wave propagation through a soil column over a wide range of forcing frequencies. The second test case investigates numerical diffusion in an axial wave propagation generated by constant force; which is removed after a certain time to allow free vibration to take place. Low numerical dispersion and numerical diffusion as well as high rates of convergence are the main advantages of the coupled least-squares (CLS) computational framework; when compared to the decoupled Galerkin (DG) framework. Based on studies presented here, CLS has low dispersion; yielding errors with one to two orders of magnitude less than that of DG. Also, the numerical diffusion present in DG framework causes a %40 error in DG’s prediction of the stress-wave intensity. Furthermore, accumulative error during evolution is virtually nonexistent for CLS, whereas, the error steadily increases as the solution evolves in DG framework. It is also demonstrated that CLS feature of temporal meshing allows for faster computations.

Propagation of harmonic waves in a cylindrical rod via generalized Pochhammer-Chree dynamical wave equation

In this manuscript, three novel schemes (Generalized direct algebraic; Improved simple equation and Modified F-expansion methods) are successfully utilized to find the solitary solutions of generalized form of Pochhammer-Chree equation.The concerned wave model has been used in the scrutiny for the propagating of harmonic waves in a cylindrical rod and several problems in fluid mechanics and waves theory in physics. The obtained results have imperative role in the field of the nonlinear science.

Dual‐phase‐lag model on magneto‐thermoelastic rotating medium with voids and diffusion under the effect of initial stress and gravity

AbstractIn this paper, the propagation of harmonic plane waves is considered in a generalized thermoelastic medium with diffusion and voids in the presence of initial stress, magnetic field, rotation, and gravity in the context of thermoelastic models; classical, Lord Shulman, Green Lindsay as well as dual‐phase‐lag models. We applied the boundary conditions in the physical domain using the normal mode method technique on the surface to obtain the displacements, stresses, temperature, diffusion concentration, and the volume fraction field. Influence of initial stress, magnetic field, rotation, and gravity on temperature, stresses, concentration of diffusion, and the volume fraction is observed through a numerical example. The results obtained will be compared in the presence and absence of the new considered variables, also with the previous results obtained by the others and displayed graphically.

Propagation of Rayleigh waves at the boundary of an orthotropic elastic solid: Influence of initial stress and gravity

Propagation of harmonic plane waves is considered in an orthotropic elastic medium in the presence of initial stress and gravity. Roots of a quadratic equation define the propagation of one quasi-longitudinal wave and one quasi-transverse wave in a symmetry plane in this medium. These two waves are coupled in the identical phase to define the propagation of Rayleigh waves at the boundary of the medium. Two conditions at the stress-free boundary translate into a complex frequency equation, which explains the dispersive behavior of this Rayleigh wave. For the presence of radical terms, this complex equation is rationalized into a real algebraic equation. Only one root of this algebraic equation satisfies the mother frequency equation and hence represents the propagation of dispersive Rayleigh waves at the boundary of the orthotropic solid. The influence of initial stress and gravity on velocity and polarization of Rayleigh waves is observed through a numerical example.

Numerical simulation of a coupled system of Maxwell equations and a gas dynamic model

It is known that both linear and nonlinear optical phenomena can be produced when the plasmon in metallic nanostructures are excited by the external electromagnetic waves. In this work, a coupled system of Maxwell equations and a gas dynamic model including a quantum pressure term is employed to simulate the plasmon dynamics of free electron fluid in different metallic nanostructures using a discontinuous Galerkin method. Numerical benchmarks demonstrate that the proposed numerical method can simulate both the high order harmonic generation and the nonlocal effect from metallic nanostructures. Based on the switch-on-and-off investigation, we can conclude that the quantum pressure term in gas dynamics is responsible for the bulk plasmon resonance. In addition, for the dielectric-filled nano-cavity, a coupled effective polarization model is further adopted to investigate the optical behavior of bound electrons. Concerning the numerical setting in this work, a strengthened influence of bound electrons on the generation of high order harmonic waves has been observed.

GREEN TENSOR OF MOTION EQUATIONS OF TWO COMPONENTS BIOT’S MEDIUM BY STATIONARY VIBRATIONS

Here processes of wave propagation in a two-component Biot’s medium are considered which are generated by periodic forces actions. By use Fourier transformation of generalized functions, the Green tensor - a fundamental solutions of oscillation equations of this medium has been constructed. This tensor describes the process of propagation of harmonic waves of a fixed frequency in spaces of dimension N = 1,2,3 under the action of power sources concentrated at the coordinates origin, described by a singular delta -function. Based on it, generalized solutions of these equations are constructed under the action of various sources of periodic perturbations, which are described by both regular and singular generalized functions. For regular acting forces, integral representations of solutions are given that can be used to calculate the stress-strain state of a porous water-saturated medium.

A theory of porous media and harmonic wave propagation in poroelastic body

Abstract The present work is based on a mixture theory of poroelastic media which is consistent with the classical Darcy’s law and uplift force in soil mechanics. In addition, it also results in having an inertial effect on the motion of solid constituent as commonly expected, in contrast to Biot’s theory, where relative acceleration is introduced as an interactive force between solid and fluid constituents to account for the apparent inertial effect. The propagation of plane harmonic waves in homogeneously deformed region is considered. For different poroelastic models with either incompressible solid or incompressible fluid constituent, phase speeds and attenuation coefficients are analysed and numerically determined with convenient data from a nonlinear material model for comparison with some available results in the literature.

Shape optimization by conventional and extended isogeometric boundary element method with PSO for two-dimensional Helmholtz acoustic problems

In this paper, a new approach is developed for applications of shape optimization on the two-dimensional time harmonic wave propagation (Helmholtz equation) in acoustic problems. The particle swarm optimization (PSO) algorithm - a gradient-free optimization method avoiding the sensitivity analysis - is coupled with two boundary element methods (BEM) and isogeometric analysis (IGA). The first method is the conventional isogeometric boundary element method (IGABEM). The second method is the eXtended IGABEM (XIBEM) enriched with the partition-of-unity expansion using a set of plane waves. In both methods, the computational domain is parameterized and the unknown solution is approximated using non-uniform rational B-splines basis functions (NURBS).In the optimization models, the advantage of IGA is the feature of representing the three models; i.e. shape design/analysis/optimization, using a set of control points, which also represent control variables and optimization parameters, making communication between the three models easy and straightforward.A numerical example is considered for the duct problem to validate the presented techniques against the analytical solution. Furthermore, two different applications for various frequencies are studied; the vertical noise barrier and the horn problems, and the obtained results are compared against previously published numerical methods using sensitivity analysis and genetic algorithms to verify the efficiency of the proposed approaches.

Plane harmonic waves in a thermoelastic solid with double porosity

Propagation of time harmonic plane waves in an infinite thermoelastic solid medium with double porosity is studied in this paper. It is found that there may exist five basic waves consisting of four sets of coupled longitudinal waves and an independent transverse wave traveling at different speeds. Each set of coupled longitudinal waves is found to be dispersive, attenuating and depends upon the presence of both types of voids and thermal properties in the medium. The lone transverse wave is found to be non-dispersive and non-attenuating, and does not depend on the presence of voids and thermal properties, and travels with the speed of the shear wave of classical elasticity. Two sets of the coupled longitudinal waves face critical frequencies, below which these waves do not propagate in the medium. These critical frequencies depend upon the presence of voids and thermal parameters of the medium. The reflection phenomenon of a set of incident coupled longitudinal wave from the stress-free and thermally insulated boundary surface of a half-space has been studied. For a particular model, the phase speeds, attenuation coefficients, amplitude ratios and energy ratios have been computed numerically, presented graphically and discussed.