Propagation Of Harmonic Waves Research Articles

Harmonic and superharmonic wave propagation in 2D mechanical metamaterials with inertia amplification

An original parametric lattice model is proposed to investigate harmonic and superharmonic planar waves propagating in a two-dimensional mechanical metamaterial, whose periodic microstructure is characterized by local linkage mechanisms for pantographic inertia amplification. The free undamped dynamics in the metamaterial plane is governed by differential difference equations of motion, featuring geometric nonlinearities of both elastic and inertial nature. Within the weakly nonlinear oscillation regime, multi-harmonic wave solutions are achieved analytically, although asymptotically, by means of a suited perturbation method. At the lowest perturbation order, the linear dispersion properties (wavefrequencies and waveforms) of freely propagating monoharmonic waves are determined analytically as functions of the mechanical parameters. At higher perturbation orders, the amplitudes of the superharmonic wave components generated by quadratic and cubic nonlinearities are determined analytically, in the absence of internal resonances. Furthermore, the nonlinear corrections of the linear wavefrequencies are obtained. Smooth transitions from hardening to softening behaviors (or viceversa) are found to occur along particular propagation directions, depending on the wavelength. Physically, a pair of unexplored and interesting dynamic phenomena are disclosed. First, the free propagation of transversal waves along particular directions is characterized – independently of the wavenumber – by essentially nonlinear waveforms (floppy modes), featuring evanescent amplitude-dependent wavefrequency. Second, the generation of superharmonic components oscillating with double and triple frequency multiples – caused by quadratic and cubic nonlinearities – can determine a loss of polarization (superharmonic depolarization) in waves propagating with perfectly polarized waveforms in the linear field.

Constitutive modelling and wave propagation through a class of anisotropic elastic metamaterials with local rotation

Elastic metamaterials are typically periodic materials possessing unit cells endowed with engineered architecture much smaller than the typical phenomenological length scale. The development of continuum models capable of accurately representing the effects of this aforementioned architecture is extremely challenging, and hence a sparsely developed area. This paper develops a novel 2-D continuum model capable of capturing the dynamic behaviour of a class of anisotropic elastic metamaterials with local rotational elements in the long wavelength limit. A constitutive relation incorporating these local rotational effects is proposed, and ratified using a representative discrete model using linear Hookean springs and identical rigid disks. The new continuum model is used to generate a dispersion relation for harmonic plane waves propagating in an arbitrary direction, which is subsequently compared to the dispersion behaviour of the original discrete model. The general behaviour of this continuum when subjected to 2-D planar harmonic wave propagation in the anisotropic medium is then analyzed, with specific attention given to the effect of material anisotropy and wave propagation direction. This work is the first of its kind to create a new continuum model of a class of anisotropic elastic metamaterials with local rotational effects.

Green’s Tensor of BIOT’s Equations by Stationary Oscillations

Various mathematical models of deformable solids mechanics are used in the study of seismic processes in the earth's crust. The processes of waves propagation are most studied in elastic media. But these models do not take into account many real properties of the ambient array. These are, for example, the presence of groundwater, which complicates the construction and operation of surface and underground structures, affect the magnitude and distribution of stresses. Models, which take into account the water saturation form the earth's crust structures, the presence of gas bubbles, etc., are multi-component medium. A variety of multicomponent media, the complexity of the processes associated with their deformation, lead to a large difference in the methods of analysis and modelling used in the solution of wave problems. In this paper the processes of wave propagation in a two-component Biot medium under the action of periodic forces of various forms are considered. Using the Fourier transform of generalized functions, fundamental solutions are constructed - the Green's tensor of the Biot equations and its properties are studied. 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 origin of coordinates, described by a singular delta function. On its basis, generalized solutions of these equations are constructed under the action of various sources of periodic disturbances, which are described by both regular and singular generalized functions. For regularly acting forces, integral representations of solutions are given, which can be used to calculate the stress-strain state of a porous water-saturated medium

Flexural wave propagation in canonical quasicrystalline-generated waveguides

We investigate the propagation of harmonic flexural waves in periodic two-phase phononic multi-supported continuous beams whose elementary cells are designed according to the quasicrystalline standard Fibonacci substitution rule. The resulting dynamic frequency spectra are studied with the aid of a trace-map formalism which provides a geometrical interpretation of the recursive rule governing traces of the relevant transmission matrices: the traces of three consecutive elementary cells can be represented as a point on the surface defined by an invariant function of the square root of the circular frequency, and the recursivity implies the description of a discrete orbit on the surface. In analogy with the companion axial problem, we show that, for specific layouts of the elementary cell (the canonical configurations), the orbits are almost periodic. Likewise, for the same layouts, the stop-/pass-band diagrams along the frequency domain are almost periodic. Several periodic orbits exist and each corresponds to a self-similar portion of the dynamic spectra whose scaling law can be investigated by linearising the trace map in the neighbourhood of the orbit. The obtained results provide a new piece of theory to better understand the dynamic behaviour of two-phase flexural periodic waveguides whose elementary cell is obtained from quasicrystalline generation rules.

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.