Propagation Of Plane Harmonic Waves Research Articles

Complex slowness vector for generalised propagation of harmonic plane waves at the boundaries of real materials

Complex slowness vector for generalised propagation of harmonic plane waves at the boundaries of real materials

Dispersion Analysis of Three-Dimensional Elastic Wave Propagation in Transversely Isotropic Media Using Optimally Blended Spectral-Element Method

The accuracy of the optimally blended spectral-element method for wave propagation in a homogeneous transversely isotropic elastic medium was investigated. A nonstandard quadrature rule obtained by combining Gauss quadrature rule and Gauss–Lobatto–Legendre quadrature rule was used to compute elementary matrixes. The mass and stiffness matrixes were represented as the triple tensor-product of elementary matrixes as second-order tensors. The solution of the eigenvalue problem representing the semidiscretized version of elastic wave equation for plane harmonic wave propagation was obtained using the Rayleigh quotient approximation technique. The resulting eigenvalues subsequently were used to estimate the phase and group velocity of three bulk waves. The variations of the errors in these velocities of the bulk waves with the number of grid points per wavelength were depicted graphically. These variations were shown for different polynomial orders and various angles of deviation from the symmetry axis. The effect of a change in the order of time discretization on error variation also was shown. The solution obtained by the optimally blended spectral-element method was found to be more accurate than that from the classical spectral-element method for low-order polynomials. The improvement in solution accuracy was demonstrated through dispersion analysis.

Plane waves in Moore–Gibson–Thompson thermoelasticity considering nonlocal elasticity effect

The Moore–Gibson–Thompson (MGT) generalized thermoelasticity theory together with the Eringen’s nonlocal elasticity model is used to study the propagation of harmonic plane waves in an infinite elastic medium that conducts heat. Two sets of coupled longitudinal thermoelastic waves have been initiated. These waves are dispersive in nature and experience attenuation. In conjunction with the coupled waves, there also exists one uncoupled vertically shear–type (SV-type) wave. The SV-type wave is dispersive in nature but without any attenuation. The elastic nonlocality parameter influences all these waves. Furthermore, the SV-type wave faces critical frequency, while the coupled longitudinal waves may face critical frequencies conditionally. We also explore the problem of reflection of thermoelastic waves at a stress-free insulated and isothermal plane boundary of the medium considered here. The reflection coefficients and their respective energy ratios of the reflected waves to that of incident wave are determined analytically and numerically by considering an appropriate material. The numerical results are presented graphically to highlight the effects of various parameters of interest. Some important observations about the prediction of the MGT model in comparison to the other existing models (Lord–Shulman (LS) and the Green–Nagdhi theory of type III (GN-III) models) are highlighted.

Propagation of generalised Rayleigh wave at the surface of piezoelectric medium with arbitrary anisotropy

AbstractMathematical model for mechanical and electrical dynamics is solved for three‐dimensional propagation of harmonic plane waves in a piezoelectric medium with arbitrary anisotropy. A system of modified Christoffel equations is derived to explain the existence and propagation of bulk waves or decaying phases in the considered medium. At the free plane boundary, a superposition of decaying phases form a generalised Rayleigh wave, which is governed by a linear system of four homogeneous equations. A complex determinantal secular equation ensures a solution to this system. True surface wave at the boundary of the considered medium demands a real solution of this complex secular equation. The linear system of four equations is then transformed to replace the complex secular equation with a real one, which can be solved by standard numerical methods. A real solution of this real secular equation provides the phase velocity for generalised Rayleigh wave at the boundary of piezoelectric medium with arbitrary anisotropy. This phase velocity defines a complex slowness vector, which is used to calculate the motion of material particles and the wave‐induced electric field. A numerical example is considered to compute the phase velocity as well as group velocity for given (arbitrary) propagation directions of Rayleigh wave at the boundary. Variations in particle motion and electric field, induced by Rayleigh wave, are analysed at different depths for different propagation directions at the boundary.

Spatial Dispersion of Elastic Waves in Transversely Isotropic Media Using Lagrange Spectral Element Method

Dispersion is studied for the two-dimensional propagation of elastic waves in transversely isotropic media using the Lagrange spectral element method. Spectral element matrices are derived as the tensor product of elementary second-order tensors. Gauss-Lobatto-Legendre points are used for the interpolation of Lagrange-type shape functions as well as for the numerical integration to obtain elementary matrices. The Rayleigh quotient approximation technique is employed to find the solution of the eigenvalue problem, which is obtained from the semidiscretized form of the elastic wave equation for propagation of plane harmonic waves. Variations of errors in the phase/group velocities of bulk waves are depicted graphically with the order of interpolation polynomial, angle with the symmetry axis, and the time discretization. Error analysis clearly demonstrated the effectiveness of the Lagrange spectral element method for wave simulation in a transversely isotropic medium.

The effect of magnetic field on the propagation of Rayleigh waves in fiber-reinforced viscoelastic media

The propagation of harmonic plane waves is considered in a linear viscoelastic solid reinforced by two families of fibers. These families are orthogonal, and a composite material is orthotropous to wave motion. The impact of a magnetic field and fiber parameters in a linear magneto-viscoelastic model considers the propagation of harmonic plane waves. . In this scattered medium, propagation of the Rayleigh wave is governed by a complex, irrational equation. The implicit radicals in this complex equation are rationalized by squaring. The resulting cubic equation is solved to define three complex velocities. Each complex velocity represents an inhomogeneous wave that decays while traveling across the medium. Thus, qualified, complex velocity is resolved to determine a Rayleigh wave’s phase velocity and attenuation coefficient. Consequently, there may be more than one Rayleigh wave in this fiber-reinforced viscoelastic medium. The corresponding particle motion at any given position in the medium is ccalculated using the Rayleigh wave’s complex velocity. The influence of fibers and magnetic fields on the velocity, attenuation, and polarization of Rayleigh waves is investigated using a numerical example.

Reflection of plane harmonic wave in rotating media with fractional order heat transfer and two temperature

The aim of the present investigation is to examine the propagation of plane harmonic waves in transversely isotropic homogeneous magneto thermoelastic rotating medium with fractional order heat transfer and two temperature. It is found that, for two dimensional assumed model, there exist three types of coupled longitudinal waves (quasi-longitudinal, quasi-transverse and quasi-thermal) in frequency domain. The amplitude ratios, energy ratios, phase velocities, specific loss, penetration depth, attenuation coefficients of various reflected waves are computed and depicted graphically. The conservation of energy at the free surface is verified. The effects of two temperature and fractional order parameter by varying different values are represented graphically

On propagation of harmonic plane waves under the Moore–Gibson–Thompson thermoelasticity theory

ABSTRACT The present work is aimed at a detailed investigation of harmonic plane wave propagation through an isotropic, unbounded and homogeneous thermoelastic medium under the recently proposed Moore–Gibson–Thompson (MGT) thermoelasticity theory. The dispersion relation is derived and its solution for the longitudinal plane wave propagation is obtained analytically. Two different modes of longitudinal wave are identified as elastic and thermal mode waves. The asymptotic expressions of various important characteristics of the wave fields such as phase velocity, specific loss and penetration depth for both the waves are obtained for limiting cases of very high and low frequency values. The Whitham stability of harmonic plane waves is established. Further, the computational tool is used to achieve numerical results for these wave characteristics and to illustrate the analytical findings graphically. Some important observations about the prediction of the MGT model in comparison to the other existing models, i.e. classical coupled thermoelastic model, LS model and GN-III model, are highlighted. Effects of material parameters on the propagation of waves are analyzed in a detailed manner. From the theoretical as well as numerical results, it is investigated that the MGT model accounts for finite speed of elastic wave as well as thermal wave.

The propagation of plane waves in nonlocal visco-thermoelastic porous medium based on nonlocal strain gradient theory

The present paper deals with propagation of plane harmonic waves in nonlocal visco-thermoelastic porous medium based on the nonlocal strain gradient theory. For the first time, the assumptions of Green-Naghdi model type III of hyperbolic thermoelasticity with voids in conjunction with nonlocal strain gradient theory are taken into account in formulation of the problem. The constitutive relations and field equations for nonlocal thermoelastic materials with voids are presented. In addition to transverse waves, three longitudinal waves (elastic, thermal, volume fraction) can be observed. The fundamental solution to equations for steady oscillations is derived. The effects of nonlocal length scale parameter, voids, viscosity on phase velocities, attenuation coefficients and penetration depths of waves with diverse frequencies are comprehensively studied and graphically presented.

Propagation of plane harmonic waves in flexoelectric microstructured solids

In this paper, we specialize the governing equations of microstructured flexoelectric solids in a plane and solve for possible plane wave propagation. It is shown that there exist two plane waves, namely quasi-P and quasi-SV waves, in a microstructured flexoelectric solid half-space. A numerical example is considered to plot the speeds of quasi-P and quasi-SV waves against the wave number for given propagation angle. The speeds of quasi-P and quasi-SV waves are also computed against the angle of propagation for given wave number. The micro-inertial, strain gradient elastic, and flexoelectric effects on plane wave speeds are illustrated graphically.

Analysis of plane wave propagation under the purview of three phase lag theory of thermoelasticity with non-local effect

Present study characterizes the propagation of harmonic plane waves in Eringen's non-local thermoelastic medium. Three phase lag (TPL) theory of thermoelasticity has been applied to account for the interactions between elastic and thermal fields. Two sets of coupled longitudinal waves (elastic and thermal) and one independent vertically shear wave have been achieved. Longitudinal waves are found to be dispersive and experience attenuation. All these waves are found to be influenced by the elastic non-local parameter. Further shear-type wave obtains a critical frequency while the coupled longitudinal waves may face critical frequencies conditionally. High and low frequency asymptotes of physical fields have been calculated. By considering an illustrative example, numerical results have been computed to explore the physics of the problem. The results for the TPL model have been compared with those for the dual phase lag theory. The study reveals that the distribution of phase velocities and attenuation coefficients for the TPL model are significantly different from those of the dual phase lag models with respect to the frequency as well as non-local elastic parameter. TPL theory is found to be capable in predicting better results as compare to the dual phase lag theory of thermoelasticity.

Reflection of plane harmonic wave in rotating media with fractional order heat transfer

The aim of the present investigation is to examine the propagation of plane harmonic waves in transversely isotropic homogeneous magneto visco thermoelastic rotating medium with fractional order heat transfer and two temperature. It is found that, for two dimensional assumed model, there exist three types of coupled longitudinal waves (quasi-longitudinal, quasi-transverse and quasi-thermal) in frequency domain. phase velocities, specific loss, penetration depth, attenuation coefficients of various reflected waves are computed and depicted graphically. The effects of viscosity and fractional order parameter by varying different values are represented graphically.

Reflection of inhomogeneous waves at the surface of a cracked porous solid with penny-shaped inclusions

A mathematical model developed by Zhang et al. [Modeling wave propagation in cracked porous media with penny-shaped inclusions. Geophys. 2019;84(4):1–11. doi:10.1190/GEO2018-0487.1] for wave motion in a cracked porous solid with penny-shaped inclusions is solved for the propagation of harmonic plane waves. The solution is obtained in the form of Christoffel equations. The solution of Christoffel equations demonstrates the existence of four (three longitudinal and one transverse) waves in the cracked porous solid. The behavior of the considered medium is dissipative as both host medium and inclusions are filled with the same viscous fluid. So, all the waves are inhomogeneous in nature. A finite non-dimensional parameter is used to represent the inhomogeneity character of an inhomogeneous wave. The phenomenon of reflection is investigated in two situations (i.e. sealed and opened surface pores) at the stress-free plane surface of a cracked porous solid. Further, the partition of incident energy is calculated in the form of an energy matrix. A numerical example is taken to discuss the effects of various properties on the partition of incident energy. The conservation of incident energy at each angle of incidence is prevailed in both situations.

Complete phenomenon of reflection at the plane boundary of a dissipative anisotropic elastic medium

SUMMARYA complex slowness vector governs the 3-D propagation of harmonic plane waves in a dissipative elastic medium with general anisotropy. In any sagittal plane, this dual vector is specified with phase direction, propagation velocity and coefficients for attenuation. A generalized reflection phenomenon is illustrated for incidence of inhomogeneous waves at the stress free boundary of the medium. Each reflected wave at the boundary is characterized by its propagation direction, propagation velocity, inhomogeneity, amplitude ratio, phase shift and energy flux. These propagation characteristics are exhibited graphically for a numerical example of anisotropic viscoelastic medium.

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.

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.

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.

Fundamental solution of steady oscillations equations in nonlocal thermoelastic medium with voids

A new nonlocal theory of generalized thermoelasticity with voids based on Eringen’s nonlocal elasticity is established. The propagation of plane harmonic waves in nonlocal thermoelastic medium with voids is investigated in the context of dual-phase-lag model of generalized thermoelasticity. There exist three longitudinal waves, namely elastic (E-mode), thermal (T-mode) and volume fraction (V-mode) in addition to transverse waves which get decoupled from the rest of motion and not affected by thermal and volume fraction fields. The fundamental solution of the system of differential equations in case of steady oscillations in terms of the elementary functions has been constructed. The effect of nonlocal parameter and the effect of voids on phase velocities, attenuation coefficients and penetration depths are presented graphically.