Propagation Of Harmonic Waves Research Articles

Multiscale Modeling of Elastic Waves in Carbon-Nanotube-Based Composite Membranes

A multiscale model is developed for vertically aligned carbon nanotube (CNT)-based membranes that are made for water purification or gas separation. As a consequence of driving fluids through the membranes, they carry stress waves along the fiber direction. Hence, a continuum mixture theory is established for a representative volume element to characterize guided waves propagating in a periodically CNT-reinforced matrix material. The obtained coupled governing equations for the CNT-based composite are found to retain the integrity of the wave propagation phenomenon in each constituent, while allowing them to coexist under analytically derived multiscale interaction parameters. The influence of the mesoscale characteristics on the continuum behavior of the composite is demonstrated by dispersion curves of harmonic wave propagation. Analytically established continuum mixture theory for the CNT-based composite is strengthened by numerical simulations conducted in COMSOL for visualizing mode shapes and wave propagation patterns.

Strong enhancement of third harmonic generation from a Tamm plasmon multilayer structure with WS2.

Improving the conversion efficiency is particularly important for the generation and applications of harmonic waves in optical microstructures. Herein, we propose to enhance the efficiency of third harmonic generation by integrating a monolayer WS2 with the metal/dielectric/photonic crystal multilayer structure. The numerical simulations show that the multilayer structure enables to generate the Tamm plasmon mode between the metal film and photonic crystal around the telecommunication wavelength, which is consistent with the experimental result. By measuring with a self-built nonlinear optical micro-spectroscopy system, we find that the third harmonic signal can be reinforced by 16-fold through inserting the monolayer WS2 in the dielectric spacer. This work will provide a new way for improving nonlinear optical response, especially THG in multilayer photonic microstructures.

Minimal Conditioned Stiffness Matrices with Frequency-Dependent Path Following for Arbitrary Elastic Layers over Half-Spaces

This paper introduces an efficient computational procedure for analyzing the propagation of harmonic waves in layered elastic media. This offers several advantages, including the ability to handle arbitrary frequencies, depths, and the number of layers above an elastic half-space, and efforts to follow dispersion curves and flag up possible singularities are investigated. While there are inherent limitations in terms of computational accuracy and capacity, this methodology is straightforward to implement for studying free or forced vibrations and obtaining relevant response data. We present computations of wavenumber dispersion diagrams, phase velocity plots, and response data in both the frequency and time domains. These computational results are provided for two example cases: plane strain and axisymmetry. Our methodology is grounded in a well-conditioned dynamic stiffness approach specifically tailored for deep-layered strata analysis. We introduce an innovative method for efficiently computing wavenumber dispersion curves. By tracking the slope of these curves, users can effectively manage continuation parameters. We illustrate this technique through numerical evidence of a layer resonance in a real-life case study characterized by a fold in the dispersion curves. Furthermore, this framework is particularly advantageous for engineers addressing problems related to ground-borne vibrations. It enables the analysis of phenomena such as zero group velocity (ZGV), where a singularity occurs, both in the frequency and time domains, shedding light on the unique characteristics of such cases. Given the reduced dimension of the problem, this formulation can considerably aid geophysicists and engineers in areas such as MASW or SASW techniques.

Rotation impact on the radial vibrations of frequency equation of waves in a magnetized poroelastic medium

This research delves into the propagation of radial free harmonic waves in a poroelastic cylinder, conceptualized as a magnetically and rotationaly influenced hollow structure. The primary aim is to elucidate the magnetic field's and rotation impact on the vibrational behavior of such systems. The investigative method encompasses the resolution of motion equations, formulated as partial differential equations, through the application of Lame's potential theory. This analytical process is augmented by the implementation of fitting boundary conditions, culminating in the derivation of a comprehensive expression for the complex dispersion equation, predicated on the premise that the wavenumber embodies a complex entity. The precision of the model is corroborated through a comparative analysis with established literature, underpinned by an exploration of diverse scenarios. The research employed MATLAB for both numerical and graphical assessments, focusing on the dispersion and displacement attributes. Dispersion relations within the poroelastic medium were computed, considering varied magnitudes of magnetic field intensity, rotation and angular velocities. The outcomes are articulated through complex-valued dispersion relations, transcendental formulations, and numerical resolutions employing MATLAB's bisection technique. These insights hold substantial significance for the theoretical advancement in orthopedic research, particularly concerning cylindrical poroelastic media. This study deduces that the radial vibrational patterns and the corresponding frequency equation within a poroelastic medium are profoundly modified by the magnetic field's interference and rotation. This study formulate a novel governing equation for a poroelastic medium, highlighting the significance of radial vibrations and investigating the impact of magnetic field and rotation.

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

Two-Layer Shallow Water Equations with Momentum Conservative Scheme for Wave Propagation Simulation

In this paper, we discuss the implementation of momentum conservative scheme to shallow water equations (SWE). In shallow water model, the hydrodynamic pressure of the water is neglected. Here, the numerical calculation of mass and momentum conservation was applied on a staggered grid domain. The vertical interval was divided into two parts which made the computation quite efficient and accurate. Our focus is on the performance of the numerical scheme in simulating wave propagation and run-up phenomena, where the main challenge is to calculate the wave speed accurately and to count the non-linear term of the model. Here we also considered the wet and dry conditions of the topography. Three benchmark tests were picked out to validate the numerical scheme. A simulation of standing wave was carried out; the results were compared to the linear analytical solution and show a good fit. In addition, a simulation of harmonic wave propagation on a sloping beach was conducted, and the results closely align with the expected values from exact solution. Finally, we carried out a simulation of solitary wave with a sloping topography; and the results were compared to laboratory data. A good agreement was observed between the simulation results and experimental measurements.

Reflection of coupled elastic waves in nonlocal rotating viscoelastic media

In this paper, propagation of homogeneous harmonic plane waves has been studied in an infinite homogeneous isotropic nonlocal viscoelastic rotating medium. Due to the rotational effects, two coupled plane waves, namely a quasi longitudinal wave and a quasi transverse wave are found in the medium under study. The dispersion equations is derived analytically and represented numerically. The problem of reflection of the quasi longitudinal wave at a stress-free solid half-space of the considered medium is also investigated. Various analytical results for the phase speed, attenuation coefficients, reflection coefficients and their corresponding energy ratios are depicted graphically for a particular model of interest to highlight the effect of nonlocal as well as rotational parameter.

Seismic Beacon-A New Instrument for Detection of Changes in Rock Massif.

The seismic beacon is a new instrument that allows for the measurement of changes in a rock massif with high sensitivity. It is based on effects, which affect the propagation of harmonic seismic waves generated continuously with stable and precise frequency and amplitude. These seismic waves are registered by a system of seismic stations. The amplitude of the seismic signal is very small, and it is normally hidden in a seismic noise. Special techniques are applied to increase the signal-to-noise ratio. In 2020, the first prototype of the seismic beacon was constructed in a laboratory, and field tests were performed in 2022 and 2023. During the tests, the changes in spectral amplitude and phase of seismic waves were detected, which is interpreted as the changes in material properties. These measurements testified the basic functionality of the device. The seismic beacon has been developed primarily for the detection of critical stress before an earthquake, which is manifested by non-linear effects such as higher harmonics generation. In addition, it could be used, for example, in the detection of magma movements, groundwater level changes, changes in hydrocarbon saturation in rocks during the extraction of oil and natural gas, or the penetration of gases and liquids into the earth's crust.

Non-classical theory of electro-thermo-elasticity incorporating local mass displacement and nonlocal heat conduction

A non-classical local gradient theory of nonferromagnetic thermoelastic dielectrics is presented, incorporating both the local mass-displacement process and the heat-flux gradient effect. The process of local mass displacement is related to the changes in material microstructure. The nonlocal heat conduction law is also addressed in the model. Thus, the generalized relationship between the higher-grade heat and entropy fluxes is adopted. The gradient-type constitutive relations and governing equations are derived using the fundamental principles of continuum mechanics, non-equilibrium thermodynamics, and electrodynamics. Due to the contribution of higher-grade flux, the nonlocal law of heat conduction is obtained. The constitutive relations for isotropic materials with the corresponding additional material constants are derived. To illustrate the local gradient theory and to show the electro-thermo-mechanical coupling effect in isotropic materials, a straightforward problem is analytically solved for a layered non-piezoelectric structure under non-uniform temperature distribution. The analytical results reveal that the thermal polarization effect can also be pronounced in isotropic materials. To illustrate the model considering the effect of nonlocal heat conduction, the propagation of spherical thermoelastic harmonic waves in a homogeneous and isotropic elastic medium with non-classical heat conduction law is studied.

Systematic investigations on frequency mixing response of ultrasonic shear horizontal and Rayleigh Lamb waves

The study of self- and cross-interactions of shear horizontal (SH) waves has garnered significant attention in the field of nonlinear ultrasonic practice. This interest stems from the potential high sensitivity of SH waves to microdamage and their low out-of-plane attenuation characteristics. However, due to the dispersive nature and intricate mechanisms governing guided wave propagations and interactions, the nonlinear responses of SH waves, particularly the third-order harmonic generation and wave mixing with Rayleigh Lamb (RL) waves, remain largely unexplored. This study presents a comprehensive investigation into the ultrasonic harmonic wave generation of SH guided waves. The research approach combines theoretical considerations and finite element modeling to investigate and discuss guided wave mixing involving SH and RL waves. Six cases of phase-matched wave mode pairs are developed based on deduced synchronism and symmetry conditions. A nonlinear material constitutive model, encompassing both the third- and fourth-order elastic constants, is incorporated into the simulation. The results of the numerical analysis reveal the existence of newly generated higher and combined harmonic waves in an intuitive manner. Specifically, the interactions between ultrasonic SH and RL guided waves are observed to generate second- and third-order combined harmonic waves at the mixing frequencies under internal resonant conditions. The nonlinear waves have a cumulative effect during the mixing period, with their amplitude remaining constant after passing through the mixing zone. This study provides the first observation of mode conversion and third-order combined harmonic generation of SH waves. The insights gained from the numerical perspective offer understandings of the frequency mixing response of SH waves.

Data‐driven 1D wave propagation for site response analysis

AbstractPrediction of ground motion triggered by earthquakes is a prime concern for both the seismology community and geotechnical earthquake engineering one. The subfield occupied with such a problem is termed site response analysis (SRA), its one‐dimensional flavor (1D‐SRA) being particularly popular given its simplicity. Despite the simple geometrical setting, a paramount challenge remains when it comes to numerically consider intense shaking in the soft upper soil strata of the crust: how to mathematically model the high‐strain, dissipative, potentially rate‐dependent soil behavior. Both heuristics models and phenomenological constitutive laws have been developed to meet the challenge, neither of them being exempt of either numerical limitations or physical inadequacies or both. We propose herein to bring the novel data‐driven paradigm to bear, thus giving away with the need to construct constitutive behavior models altogether. Data‐driven computational mechanics (DDCM) is a novel paradigm in solid mechanics that is gaining popularity; in particular, the multiscale version of it relies on studying the response of the microstructure (in the case of soil, representative volumes containing grains) to populate a dataset that is later used to inform the response at the macroscale. This article presents the first application of multiscale DDCM to 1D‐SRA: first, we demonstrate its capacity to handle wave propagation problems using discrete datasets, obtained via sampling grain ensembles using the discrete element method (DEM), in lieu of a constitutive law and then we apply it specifically to analyze the propagation of harmonic waves in a soft soil deposit that overlies rigid bedrock. The soil in this application displays elastic yet non‐linear response that is depth‐dependent due to evolving overburden pressure. We validate the implementation via comparison to regular finite elements analyses (FEA), discuss the benefits of using DD, and demonstrate that traditional amplification functions are recovered when using the DDCM. Finally, we discuss a number of future work opportunities that lie ahead of this proof‐of‐concept, chiefly in terms of site‐specific studies and incorporating more complex realistic traits of soil behavior. This project was developed using open‐source software and the relevant code is freely made available to other researchers.

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.

Generation of second harmonic terahertz surface plasmon wave over a rippled graphene surface

Abstract We propose a mechanism for the generation of second harmonic terahertz surface plasmon waves by incident terahertz electromagnetic radiation (ω, k 0) over a graphene surface deposited on the rippled dielectric substrate (SiO2). A p-polarized THz radiation incident obliquely on the graphene surface exerts a nonlinear ponderomotive force on free electrons in the rippled regime. This nonlinear ponderomotive force imparts oscillatory velocity to the electrons at frequency 2ω. Second harmonic oscillatory velocity couples with the modulated electron density and generates a nonlinear current density that drives second harmonic terahertz surface plasmon waves. Rippled surface provides an extra wave number for the phase matching condition to produce resonantly second harmonic at frequency 2ω and wavenumber (2k 0z + q). We examine the tunable response of second harmonic terahertz surface plasmon waves with respect to change in Fermi energy of graphene and laser incident angle. Second harmonic amplitude gets higher values by lowering the Fermi energy (E F) and increasing incident angle.

Electron distribution-controlled electron cyclotron harmonic wave generation in the magnetosphere

Electron cyclotron harmonic (ECH) waves, contributing significantly to magnetospheric dynamics, usually appear as a series of harmonics between the multiples of electron gyrofrequency. Previous studies have demonstrated that ECH waves are the electron Bernstein mode excited by the electron loss cone distribution. To investigate how the electron distribution controls the frequency and wave normal angle of excited ECH waves, we derive a concise analytic formula for ECH growth rates using the bi-Maxwellian distribution. Parametric studies show that the expansion of the loss cone size could effectively enlarge ECH growth rates and move the peak growth rates to a smaller wave number k (higher frequency) region, while the deepening of the loss cone can only increase the growth rates. It is also found that the growth rate pattern in the k space exhibits different trends on the two sides of 1.5 fce because the dominant resonance orders on the two sides are different. This study further develops our understanding of the generation mechanism of ECH waves.

Propagation of plane waves in a nonlocal transversely isotropic thermoelastic medium with voids and rotation

AbstractThe objective of present research work is to investigate the propagation of time harmonic plane waves in a rotating, nonlocal, transversely isotropic thermoelastic half‐space having void pores in the context of Lord–Shulman theory. It has been observed that four coupled plane waves travel in such type of medium with distinct speeds. Using appropriate boundary conditions, the amplitude and energy ratios for the reflected waves are presented. Numerical computations for a magnesium crystal‐like material are performed with the help of MATLAB software to study the effects of material anisotropy, nonlocal, void and rotation parameters on the amplitude ratios. The computational results are displayed and explained through graphs. The phase velocities of plane waves in transversely isotropic medium are found to depend upon the incident angle, whereas these attain constant values in an isotropic medium. The energy conservation law is also justified during the reflection phenomena.

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.

Monitoring the behaviour of seven storeyed RC frame subjected to reversed cyclic loading by nonlinear NDT

Large-scale structures are more prone to risk than small-scale structures due to their increased weight, stiffness and ductility parameters. The main focus of this investigative study is on the structural monitoring of a building using the non-destructive method of non-linear ultrasonic pulse velocity. A single-bay seven-storeyed frame model subjected to reverse cyclic loading of varying amplitudes is monitored by non-linear ultrasonic technique based on third harmonic wave generation. The critical nodes are identified in two different ways: the points of possible plastic hinge formation (PPPHF) through the theoretical background of plastic hinge analysis and the highly sensitive regions in structures (HSR) through numerical analysis. The common nodes of PPPHF and HSR are considered to monitor the structure. The invisible primary structural damage in the initial stage and the visible structural damage up to ultimate load are observed using non-linear ultrasonic waves. The fundamental frequency value and the third harmonic frequency value are observed and processed in MATLAB in the form of a Fast Fourier Transform (FFT) algorithm. The relationship between the condition of the structure and the ratio of third harmonic parameter, load and displacement is compared. In the initial stage of loading, only slight change or no variation in third harmonic ratio is observed at the observation points indicating material disintegration. This varies consistently up to the development of visible cracks and the same varies inconsistently on further loading after the appearance of visual cracks.

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.

Elastic and thermal effects upon the propagation of waves with assigned wavelength

It is well known that harmonic longitudinal elastic waves propagate without damping in time, while the heat equation leads to standing modes that decrease exponentially over time. In this article, it is shown that the elastic deformation when coupled with thermal deformation leads to the following effects on the propagation of harmonic longitudinal waves and on standing modes: (a) the harmonic longitudinal waves become damped in time; (b) the standing modes suffer a slower decrease in time in relation to the pure thermal modes; and (c) the propagation speed of harmonic longitudinal waves increases in relation to that predicted by the purely elastic theory. The dependence of the dimensionless parameters that characterize the effects described at these previous points in relation to the material and coupling characteristics is explicitly presented and then is numerically simulated and graphically illustrated. As regards the Rayleigh waves class, although a rigorous mathematical proof is not offered, the numerical results presented for a lot of common thermoelastic materials show the same effects of thermoelastic coupling as in the case of longitudinal harmonic plane waves. The present article systematically synthesizes the thermal effects on wave propagation, and it provides a reference work with regard to the thermoelastic coupling.

Analytical spectral design of mechanical metamaterials with inertia amplification

Functional metamaterials offering superior dynamic performances can be conceived by introducing local mechanisms of inertia amplification in the periodic microstructure of cellular composite media. The minimal physical realization of an inertially amplified metamaterial is represented by a one-dimensional crystal lattice, characterized by an intracellular pantograph mechanism. The microstructural dynamics of the periodic tetra-atomic cell is synthetically described by a low-dimension lagrangian model. The lagrangian coordinates account for the longitudinal motion of a pair of elastically coupled principal atoms, rigidly connected by the pantograph arms to a pair of transversely-oscillating secondary atoms, serving as inertial amplifiers. Within the range of small-oscillations, the free propagation of undamped harmonic waves is described by a linear difference equation, governed by a symplectic transfer matrix. First, the band structure of the complex-valued dispersion spectrum is determined analytically, by properly exploiting the formal (time-to-space) analogy with the Floquet Theory for the stability of non-autonomous dynamic systems (direct problem). Second, a spectral design problem is stated and solved by inverting analytically the functions expressing parametrically the boundaries separating attenuation (stop) and propagation (pass) bands in the frequency spectrum (inverse problem). Specifically, the inverse problem solution has the merit of providing simple formulas for parametric design. Such formulas explicitly and exactly identify the mechanical parameters of the inertially amplified metamaterial possessing a desired pass-stop-pass band structure, assigned according to functional design requirements. The existence and uniqueness of the solution in the admissible range of mechanical parameters is discussed. The discussion provides (i) mathematical demonstration for the existence of feasible design solutions or multi-solutions, (ii) complete definition of the physically realizable band structures in the frequency domain, and (iii) alternative criteria to design iso-band structured metamaterials within the admissible domain of mechanical parameters. The extra-customization possibilities offered by iso-band structured metamaterials are analyzed in terms of static and dynamic performances. As feasible limit cases of the analytical spectral design, extreme mechanical metamaterials working as phononic superfilters and/or superpropagators are realizable.