Propagation Of Harmonic Waves Research Articles

Tomographic analysis of high-order harmonic generation by integrating a phase-matching profile measurement with disruptive interaction-length control

Three-dimensional (3D) analysis on the growth process of high-order harmonic generation (HHG) is achieved by integrating in situ measurement of a 3D phase-matching profile with interaction length control. Incorporating the phase-matching profile, calculation of high-order harmonic wave generation, and propagation reproduces the measured harmonics growth curve as well as distinctive features of the far-field beam profiles. The method resolves contributions of HHG from different positions of the driving beam affected by different sources of phase mismatch, and thus provides illuminating insight into the HHG process.

Detection, Integration, and Optimization of Acoustic Field Simulation in the Closed Space

Noise pollution in the closed space such as railway carriage is an important problem because the noise pollution seriously affects comfort and health of people in the closed space. We propose the method to detection, integration, and optimization of acoustic field simulation in the closed space. First, we analyze the acoustic field distribution in the virtual 3D close space. We use harmonic sound wave propagation in the closed space and present the distribution according to geometric analysis. Second, we introduce Delaunay triangulation and k-means clustering into visualization to form the quiet zone and show it in 3D perspective. Our method used acoustic simulation to develop the sound barrier system. The simulation results show that our method can improve the analysis of the noise problem in the closed space.

On control of harmonic waves in an acoustic metamaterial

Generation of harmonic waves in a mass-in mass model of an acoustic metamaterial is considered. It is shown numerically that outside the band gap, the boundary harmonic excitation gives rise to a formation of periodic waves whose profile gradually coincides with the profile of the exact traveling wave harmonic solution. No waves are generated by the boundary excitation inside the band gap. A switch-on/off control is developed to achieve the formation of the harmonic waves inside the band gap.

Benchmarking preconditioned boundary integral formulations for acoustics

AbstractThe boundary element method (BEM) is an efficient numerical method for simulating harmonic wave propagation. It uses boundary integral formulations of the Helmholtz equation at the interfaces of piecewise homogeneous domains. The discretization of its weak formulation leads to a dense system of linear equations, which is typically solved with an iterative linear method such as GMRES. The application of BEM to simulating wave propagation through large‐scale geometries is only feasible when compression and preconditioning techniques reduce the computational footprint. Furthermore, many different boundary integral equations exist that solve the same boundary value problem. The choice of preconditioner and boundary integral formulation is often optimized for a specific configuration, depending on the geometry, material characteristics, and driving frequency. On the one hand, the design flexibility for the BEM can lead to fast and accurate schemes. On the other hand, efficient and robust algorithms are difficult to achieve without expert knowledge of the BEM intricacies. This study surveys the design of boundary integral formulations for acoustics and their acceleration with operator preconditioners. Extensive benchmarks provide valuable information on the computational characteristics of several hundred different models for multiple reflection and transmission of acoustic waves.

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.

Harmonic and shock wave propagation in bistable periodic structure: Regularity, randomness, and tunability

Nonlinear periodic structures can present abundant nonlinear wave physics. The model consisting of periodic bistable oscillators (i.e., the bistable periodic structure) is essentially different from those nonlinear periodic systems consisting of monostable oscillators due to multiple equilibria in bistable periodic structure. Despite the extensive attention received, properties of harmonic and shock wave propagation in bistable periodic structure, especially the randomness and tunability behind regularity, have not been fully understood. This article reports the answers based on numerical method. We consider the varying trends of the band gap, vibration center, wave amplitude, and transmission and show their effects on energy transport. We find that the snap-through behavior always presents local intrinsic randomness with the regularity in whole, that is, it does not happen in sequence. For both harmonic and shock wave, most energy is localized inside the snap-through oscillators that changes the regularity for energy transport and is meaningful for shock wave protection. Bistable periodic structure can present very low-frequency and broadband wave attenuation by shifting the initial frequency of the band gap to nearly zero through tuning the wave amplitude to a critical value, which offers dynamic tunability. The damping and intensity of the shock pulse have significant effects on the shock wave propagation. This work provides guidance for the design and application of bistable periodic structure for elastic wave attenuation and shock wave protection.

Electrodynamics of Reactive Power in the Space of Inter-Substation Zones of AC Electrified Railway Line

In railway traction, the definition of “electromagnetic field” is functionally connected to the concept of the reactive power consumed by the electric rolling stock, and characterized by the running and standing electromagnetic waves in the space of the inter-substation zones from the site of the AC traction system. Such a definition is established and theoretically justified by the theory of electromagnetic fields. This article uses the methodology of this theory, in particular, a method for power balance estimation in electromagnetic fields based on Maxwell’s equations, as well as methods for the analysis of running and standing electromagnetic waves based on the theory of reflection, propagation and transmission of plane harmonic waves. The research considers the regularities of standing electromagnetic waves in the space of inter-substation zones of electric traction systems, which occur due to the incomplete reflection of incident waves from the contact wire and metal parts of the roof surface and the frontal part of the body of the electric rolling stock. The flow of electricity to the roof surface and the frontal part of the body of an electric locomotive is considered. The possibility of using existing methods to reduce wave reflections and thereby to effectively compensate for reactive power in the space of inter-substation zones is discussed.

Nonlinear dispersion properties of one-dimensional mechanical metamaterials with inertia amplification

Architected metamaterials offering superior dynamic performances can be conceived by inducing local mechanisms of inertia amplification in the periodic microstructure. A one-dimensional cellular lattice characterized by a pantograph mechanism in the tetra-atomic cell is proposed as minimal physical realization of inertially amplified metamaterial. A discrete model is formulated to describe the undamped free dynamics of the cell microstructure. The ordinary differential equations of motion feature quadratic and cubic inertial nonlinearities, induced by the indeformability of the pantograph arms connecting the principal atoms with the secondary atoms, serving as inertial amplifiers. An asymptotic approach is employed to analytically determine the dispersion properties governing the free propagation of harmonic waves in the pantographic metamaterial. First, the linear wavefrequencies and waveforms are obtained by solving the eigenproblem governing the lowest asymptotic order. An invariant parametric form is achieved for the pass and stop band structure, corresponding to propagation and attenuation branches of the dispersion spectrum in the plane of complex wavenumbers. The major effects due to the mass ratio of the inertial amplifiers are discussed. Particularly, the existence conditions, amplitude and centerfrequency of the band gap separating the acoustic and optical pass bands are determined analytically. Second, the nonlinear wavefrequencies and waveforms are obtained by solving the hierarchical linear problems governing the higher asymptotic orders. Analytical, although asymptotically approximate, functions are achieved for the nonlinear wavefrequencies and waveforms, which show quadratic dependence on the oscillation amplitudes. The mechanical conditions for the softening/hardening behaviour of the nonlinear wavefrequencies and the different topological properties of the invariant manifolds associated to the nonlinear waveforms are discussed. Finally, numerical simulations are provided to validate the analytical results.

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.

Internal inertia gradient effect on two mode magneto‐elastic wave propagation

AbstractIn this study, two‐mode harmonic wave propagation for thick rods subjected to a transverse magnetic field is addressed. The analysis is based on the Mindlin‐Herrmann rod theory and the unified strain/inertia gradient model. The effects of the strain and inertial gradient parameters on lower and upper modes under different magnetic field intensities are presented in detail and discussed. This study also includes the single‐mode wave propagation analysis based on the Rayleigh‐Bishop rod theory, as a special case. An overview of the analysis results shows that the wave propagations in the lower and the upper modes are affected by the strain and inertial gradient parameters, as expected. However, some significant differences are observed in the effects of the internal inertial gradient parameter on these modes. In addition, this analysis reveals the relationship between the inertial gradient and cut‐off frequency.

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.

Longitudinal Pochhammer - Chree Waves. Secular Equations and Spectral Analysis

The paper presents the spectral analysis of the Pochhammer - Cree matrix dispersion equation for longitudinal axisymmetric modes. Analytical solutions of the matrix dispersion equation describing the propagation of longitudinal harmonic waves in an infinite cylindrical rod are analyzed. Expressions for displacement fields are obtained. The dependence of displacements on the surface on the angular frequency and Poisson’s ratio is analyzed. When the phase velocity is equal to the transverse wave velocity, the displacements on the surface completely disappear, which means that no axisymmetric Pochhammer - Cree waves propagate with such a phase velocity.

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.

Topology optimization of an acoustic diode?

By using topology optimization, we consider the problem of designing a passive acoustic device that allows for one-way flow of sound waves; such a device is often colloquially referred to as an acoustic diode. The Helmholtz equation is used to model the time harmonic linear wave propagation together with a Dirichlet-to-Neumann (DtN) type boundary condition, and the finite element method is used for discretization. The objective of this study is to maximize the wave propagation in one direction (from left to right) and minimize the wave propagation in the reverse direction (from right to left) for planar incoming waves. The method of moving asymptotes (MMA) solves the optimization problem, and a continuation approach is used for the penalizing intermediate design variables. The results for the optimized waveguide show that more than 99.8% of the power of planar incoming waves get transmitted from left to right while less than 0.3% gets transmitted in the reverse direction for planar incoming waves in the specified frequency range. Since a true diode is a non-reciprocal device and here we used a linear acoustic wave model, which is basically reciprocal, we discuss details about how it appears to be possible to obtain a one-way waveguiding effect using this linear model.

Generation, reflection and transmission of nonlinear harmonic waves by direct superposition of anharmonic dipoles

It is well known that the linear and nonlinear light reflection and transmission formula can be obtained from the standard macroscopic Maxwell equation. In the past, a microscopic approach based on the Ewald–Oseen extinction theorem to derive the reflection and transmission formula by direct dipole summation has been performed by Fearn et al where they successfully derived the Fresnel formula in linear optics. In this work, we extend their work for the first time to derive the nonlinear reflection and transmission formula by direct summation of anharmonically oscillating dipoles for the case of a normal incidence wave, yielding similar results with coupled-mode-theory (CMT). We demonstrate for a decaying incoming field that dipolar radiation is no longer forbidden inside the bulk of a centrosymmetric material. Using the simplified bond hyperpolarizability model (SBHM), we find that these additional bulk dipole contribution must be accounted along with the surface dipoles and bulk quadrupolar contribution to explain recent rotational anisotropy SHG intensity experimental data of Si(111) with high precision.

The Effect of Ion-to-Electron Mass Ratio on the Electron Beam-Plasma Interaction

Two-dimensional electromagnetic particle-in-cell simulations are carried out to investigate the effect of ion-to-electron mass ratio on the evolution of warm electron beam-plasma instability. Four cases are considered: A: mi/me = 0 (two-electron stream instability); B: mi/me = 1 (pair plasma); C: mi/me = 100; and D: mi/me = 1000. It is shown that the generation of Langmuir waves in the fundamental mode of electron plasma frequency and the subsequent dynamics of large-amplitude solitons are not affected by the ion species. However, it determines the decay process of solitons and the excitation of electromagnetic waves in the second harmonic. In the first two cases, mi/me = 0 and 1, there is no sign of emission in the second harmonic, while the strongest emission in the second harmonic is found for the case of largest mass ratio, mi/me = 1000. This confirms the two-step wave-wave coupling mechanism for the generation of second harmonic electromagnetic waves, which requires the excitation of ion-acoustic waves in the first step. Moreover, the dispersion diagrams of all excited waves are presented.

Three dimensional waves propagation in thermo-viscoelastic medium with two temperature and void

A dynamic model comprising anisotropic viscoelastic solid with voids along with two temperature and three phase lag is used to study the propagation of harmonic waves. The system of differential equation is made dimensionless by incorporating suitable quantities. Harmonic type plane wave solution is assumed to determine the Christoffel equation. From this equation five quasi-waves are noticed which vibrate with distinct speed in infinite medium and are under the impact of void, phase lag and viscosity. The velocity and attenuating factors of waves are computed numerically in three dimensional space. The influence of viscosity and void have been seen on the characteristic of waves. Different unique cases are also determined.

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.

On the Dynamics of Harmonic Filtration Waves in a Hydraulic Fracture Perpendicular to the Wellbore

The paper studies the propagation of low-frequency harmonic pressure waves in a fracture formed by hydraulic fracturing and perpendicular to the wellbore in a porous and permeable medium. The influence of reservoir characteristics and fracture (e.g., permeability, fracture width) has been analyzed. It was found that when waves propagate from a wellbore along radial fractures, the characteristic penetration distances of pressure perturbations can be significantly higher than the penetration depth of perturbations from an open well in bedding with no fracture.