Nonsymmetric Waves Research Articles

A Functionally Arranged Phononic Crystal Rod Enabling Ultra-Wideband Vibration Attenuation and Orderly Frequency Distillation

In this study, a functionally arranged phononic crystal rod, incorporating sub-periodic structures featuring distinct lattice constants, is designed. This rod seamlessly combines two distinct capabilities: highly efficient broadband vibration suppression and orderly frequency distillation. In the pursuit of this design, the dispersion characteristics of a periodic rod are first analytically calculated to determine the precise cut-on and cut-off frequencies of the resulting band gaps. Following this, the transmission of longitudinal waves traveling through a finite functionally arranged rod is elucidated. Based on these calculations, careful parameter selection yields a compact configuration for a functionally arranged phononic crystal rod, boasting an exceptionally wide band gap spanning frequencies from 100 Hz to 10,000 Hz. Finite element simulations vividly demonstrate its ultra-broadband vibration attenuation capability, nonsymmetrical wave propagation behavior, and orderly frequency distillation phenomenon. Our study underscores the advantages of employing constituent materials with a higher impedance ratio in obtaining a more compact topological structure while preserving the broadband required for vibration attenuation. This research provides essential insights for tackling the need for broadband vibration suppression in engineering and also sets the stage for achieving orderly frequency extraction and energy localization.

Classification of solutions for guided waves in fluid-loaded viscoelastic composites with large numbers of layers.

Guided ultrasonic waves are used for the inspection of multilayered composite aerospace structures. Calculating the corresponding dispersion diagrams is challenging for thick-walled composites with more than 100 layers, such as in modern rocket booster pressure vessels. The Dispersion Calculator (DC) is an open source software for calculating such dispersion diagrams and mode shapes of guided waves. Attenuation caused by viscoelasticity and fluid-loading makes the dispersion curve tracing much more difficult than in the nonattenuated case because the modal solutions are sought in the complex wavenumber plane. The tracing problem is mastered by a reliable algorithm. Whereas leaky Lamb and Scholte waves in coupled and decoupled cases are modeled using the stiffness matrix method, shear horizontal (SH) waves are traced using the transfer matrix method without facing the numerical instability. Through implementation of mode family specific dispersion equations in both matrix techniques for nonattenuated and attenuated cases, symmetric, antisymmetric, and nonsymmetric leaky Lamb, Scholte, and SH waves can be traced separately with better efficiency and robustness. The capabilities of DC are demonstrated by calculating dispersion diagrams and mode shapes for a viscoelastic composite with 400 layers immersed in water. Results are compared against DISPERSE (Imperial College London, London, UK) for selected cases.

Strongly non-linear Boussinesq-type model of the dynamics of internal solitary waves propagating in a multilayer stratified fluid

We propose a system of first-order balance laws that describe the propagation of internal solitary waves in a multilayer stratified shallow water with non-hydrostatic pressure in the upper and lower layers. The construction of this model is based on the use of additional variables, which make it possible to approximate the Green–Naghdi-type dispersive equations by a first-order system. In the Boussinesq approximation, the governing equations allow one to simulate the propagation of non-linear internal waves, taking into account fine density stratification, a weak velocity shear in the layers, and uneven topography. We obtain smooth steady-state soliton-like solutions of the proposed model in the form of symmetric and non-symmetric waves of mode-2 adjoining to a given multilayer constant flow. Numerical calculations of the generation and propagation of large-amplitude internal waves are carried out using both the proposed first-order system and Green–Naghdi-type equations. It is established that the solutions of these models practically coincide. The advantage of the first-order equations is the simplicity of numerical implementation and a significant reduction in the calculation time. We show that the results of numerical simulation are in good agreement with the experimental data on the evolution of mode-2 solitary waves in tanks of constant and variable height.

Active metamaterials for realizing odd mass density

Solids built out of active components have exhibited odd elastic stiffness tensors whose active moduli appear in the antisymmetric part and which give rise to non-Hermitian static and dynamic phenomena. Here, we present a class of active metamaterial featured with an odd mass density tensor whose asymmetric part arises from active and nonconservative forces. The odd mass density is realized using metamaterials with inner resonators connected by asymmetric and programmable feed-forward control on acceleration and active forces along the two perpendicular directions. The active forces produce unbalanced off-diagonal mass density coupling terms, leading to non-Hermiticity. The odd mass is then experimentally validated through a one-dimensional nonsymmetric wave coupling where propagating transverse waves are coupled with longitudinal ones whereas the reverse is forbidden. We reveal that the two-dimensional active metamaterials with the odd mass can perform in either energy-unbroken or energy-broken phases separated by exceptional points along principal directions of the mass density. The odd mass density contributes to the wave anisotropy in the energy-unbroken phase and directional wave energy gain in the energy-broken phase. We also numerically illustrate and experimentally demonstrate the two-dimensional wave propagation phenomena that arise from the odd mass in active solids. Finally, the existence of non-Hermitian skin effect is discussed in which boundaries host an extensive number of localized modes. It is our hope that the emergent concept of the odd mass can open up a new research platform for mechanical non-Hermitian system and pave the ways for developing next-generation wave steering devices.

Radiation of Nonsymmetric Whistler Waves from a Phased Antenna Array in a Magnetoplasma

Radiation of Nonsymmetric Whistler Waves from a Phased Antenna Array in a Magnetoplasma

Lamb waves in discrete homogeneous and heterogeneous systems: Dispersion properties, asymptotics and non-symmetric wave propagation

In this paper, we study Lamb waves propagating in a discrete strip, whose microstructure is represented by either a monatomic or a diatomic triangular lattice. In considering the in-plane vector problem, we derive an analytical solution for the dispersion relation of Lamb waves. Additionally, we investigate the main features of the eigenmodes of the system, which describe how the lattice strip vibrates at different frequencies. Further, we discuss how the dispersion properties depend on the number of the lattice’s rows and on the chosen boundary conditions. For heterogeneous systems, we focus the attention on the internal stop-band and on the flat bands appearing in the dispersion diagram. Different asymptotic models are employed to approximate the low-frequency behaviour of the lattice strip, starting from the classical Euler–Bernoulli beam. The effective behaviour of a lattice strip with dense microstructure is also investigated, and we present a comparative numerical analysis with the analogous continuum for which the classical Lamb wave problem is posed. The theory developed is exploited here to design a structured medium capable of manipulating wavemodes, and, through conversion and selection, generating uni-directional wave phenomena. We envisage that the present work can fill a gap in the research field related to the analytical study of dispersive waves in microstructured media, whose dynamic performance is influenced by the presence of multiple external boundaries.

Efficiency of semi-implicit alternating direction implicit methods for solving cardiac monodomain model

It is well known that numerical simulations of the cardiac monodomain model require fine mesh resolution, which increases the computational resources required. In this paper, we construct three operator-splitting alternating direction implicit (ADI) schemes to efficiently solve the nonlinear cardiac monodomain model. The main objective of the proposed methods is to reduce the computational time and memory consumed for solving electrocardiology models, compared to standard numerical methods. The proposed methods have second-order accuracy in both space and time while evaluating the ionic model only once per time-step. Several examples using regular wave, spiral wave reentry, and nonsymmetrical scroll wave are conducted, and the efficiency of the proposed ADI methods is compared to the standard semi-implicit Crank–Nicolson/Adams–Bashforth method. Large-scale two- and three-dimensional simulations are performed.

Hyperbolic model of internal solitary waves in a three-layer stratified fluid

We derive a new hyperbolic model describing the propagation of internal waves in a stratified shallow water with a non-hydrostatic pressure distribution. The construction of the hyperbolic model is based on the use of additional instantaneous variables. This allows one to reduce the dispersive multi-layer Green–Naghdi model to a first-order system of evolution equations. The main attention is paid to the study of three-layer flows over an uneven bottom in the Boussinesq approximation with the additional assumption of hydrostatic pressure in the intermediate layer. The hyperbolicity conditions of the obtained equations of three-layer flows are formulated, and solutions in the class of travelling waves are studied. Based on the proposed hyperbolic and dispersive models, numerical calculations of the generation and propagation of internal solitary waves are carried out and their comparison with experimental data is given. Within the framework of the proposed three-layer hyperbolic model, a numerical study of the propagation and interaction of symmetric and non-symmetric soliton-like waves is performed.

Two implementations of fractional-order relaxation oscillators

This work proposes general formulas for designing two different topologies of fractional-order relaxation oscillators. One topology contains an Operational Amplifier and the other one relies on an Operational Trans-Resistance Amplifier. The design procedure hinges on the general fractional-order natural and step responses of RC, which is proved in this work depending on Mittag Leffler function. The proposed topologies can be controlled to generate symmetrical and non-symmetrical square wave signals. They also benefit from the employment of fractional-order capacitors (FOCs), which makes it possible to obtain higher frequencies using simple components. Furthermore, these topologies are verified through numerical solutions, circuit simulations, and experimental implementations. This encourages the authors to build circuit emulators for the FOC, which is achieved by applying the Foster-I synthesizing technique to the Matsuda’s approximation of $$s^\alpha$$.

N-fold Darboux transformation of the two-component Kundu–Eckhaus equations and non-symmetric doubly localized rogue waves

The two-component Kundu–Eckhaus (KE) equations were introduced for the first time in 1999. Very recently, the two-component KE equations considered as a model of describing the effect of quintic nonlinearity on the ultra-short optical pulse propagation in non-Kerr media have been intensively studied. In this paper, we construct an analytical and explicit representation of the Darboux transformation (DT) for the two-component KE equations. Compared with the DT constructed by researchers before, the DT here is expressed by the initial eigenfunctions, spectral parameters, and ‘seed’ solution. As applications of DT, the explicit expressions of non-symmetric rogue wave of two-component KE equations and KE equation are displayed, and the differences between the non-symmetric and symmetric rogue wave for the KE equation are discussed in detail.

Axially moving materials with granular microstructure

The mechanics of axially moving media is significant because of their broad engineering applications. In many engineering applications, it is beneficial to understand the dynamical material response from a microstructural viewpoint. Here we focus upon wave propagation in axially moving materials with granular microstructure. To this end, the granular micromechanics approach is utilized since the resulting continuum model is known to predict wave dispersion. To consider axially moving materials, this approach is enhanced to account for the axial velocity using an Eulerian description of the system accounting for convective terms in the material derivatives and utilizing variational approach. The dispersive behavior of axially moving 1D materials are then derived and compared with the dispersive behavior of non-moving materials. In the absence of microstructure, the axially moving material model simplifies to published literature and shows non-dispersive non-symmetric forward and backward waves. In the case of axially moving materials with granular microstructure, the model predicts dispersive non-symmetric waves. In this case, there are two acoustic and two optical wave branches. Axial velocity leads to narrowing and widening in the frequency band gaps in the forward and backward waves, respectively. Negative group velocity is also observed in certain wavenumber ranges. Clearly, the stopbands created by the axial velocity and the non-symmetric dispersive behavior studied here should be considered in engineering designs for vibration control when the axially moving material possesses granular microstructure. The results presented here can also be used to help obtain parameters needed for axially moving granular metamaterials to be designed for particular applications.

Non-symmetric flexural wave scattering and one-way extreme absorption.

The possibility of asymmetric absorption and reflection for flexural waves is demonstrated through analytical and numerical examples. The emphasis is on the one-dimensional (1D) case of flexural motion of a beam for which combinations of point scatterers are considered, which together provide asymmetric scattering. The scatterers are attached damped oscillators characterized by effective impedances, analogous to effective configurations in 1D acoustic waveguides. By selecting the impedances of a pair of closely spaced scatterers it is shown that it is possible to obtain almost total absorption for incidence on one side, with almost total reflection if incident from the other side. The one-way absorption is illustrated through numerous examples of impedance pairs that satisfy the necessary conditions for zero reflectivity for incidence from one direction. Examples of almost total and zero reflection for different incidences are examined in detail, showing the distinct wave dynamics of flexural waves as compared with acoustics.

Small-amplitude steady water waves with critical layers: Non-symmetric waves

The problem for two-dimensional steady water waves with vorticity is considered. Using methods of spatial dynamics, we reduce the problem to a finite dimensional Hamiltonian system. The reduced system describes all small-amplitude solutions of the problem and, as an application, we give a proof of the existence of non-symmetric steady water waves whenever the number of roots of the dispersion equation is greater than one.

Buckling design of large eccentrically filled steel silos

Large steel silos are typical kinds of thin-walled structure which are widely used for storing huge quantities of granular solids in industry and agriculture. In the present analyses, buckling design of circular steel silos subject to large eccentricity filling pressure is demonstrated in accordance with Eurocode: EN1990, 1991, 1993 . The finite element model is established by using the commercial general purpose computer package ANSYS. Five types of buckling analyses are carried out for the geometrically perfect and imperfect models, with and without the consideration of the material plasticity, which are designated as LBA, GNA, GMNA, GNIA, and GMNIA in EN 1993 Part 1–6.Buckling behavior of five example steel silos with capacity of 40,000 to 60,000m3 is investigated whose slenderness ranges from 1.89 to 0.46, comprising intermediate slender and squat silos widely applicable in practical engineering. The results show that the buckling deformations are nonsymmetrical and the GMNIA analysis gives out the least buckling load factor for all example silos from all proposed buckling analysis types, and the load displacement curves are highly nonlinear and predict a distinct maximum load followed by a descending path, in which the maximum load is taken as the critical buckling point λcr for the equilibrium path. The buckling mode in GMNIA analysis takes the form of the well-known elephant-foot deformation at the bottom part of the shell wall, combined with nonsymmetrical waves in meridional direction throughout the whole height of the silo wall due to the distribution of weld imperfection. The geometrical nonlinearity is beneficial while material nonlinearity is strong and detrimental to buckling behavior of example silos. The effect of weld imperfection is also harmful to buckling resistance of silo, which is more serious for relatively slender silos than squat silos. The buckling is mainly governed by the nonuniform distribution of the solid pressure other than other influential factors as the weld imperfection, geometrical and material nonlinearity, compared with the load case of symmetrical filling. The economical design of steel silos can be effectively measured by the economic index called the ratio of capacity to steel consumption (RCS). It is validated that the index RCS increases rapidly with the decrease of silo slenderness, and the storage efficiency of example silos is increased by about 2.1 times with the slenderness varying from 1.89 to 0.46. It also suggests that the eccentricity in filling of steel silo should be reduced as far as possible for improvement of buckling strength of structure under eccentric filling.

Effect of the sheath thickness of thin inhomogeneous plasma layer on the propagation constant for the surface waves

Based on the dispersion equation obtained from the Helmholtz equation in the plasma sheath region and main plasma column for surface waves, the propagation constant properties on the interface of the plasma sheath region and main plasma column region are studied. The analytic formula of propagation constant for the azimuthally symmetric surface wave (m = 0 mode) and non-symmetric surface waves (m = 1, 2, 3, and 4 modes) on the interface is presented, which indicates the effect of the thickness of the plasma sheath region on the propagation constant characteristics. The analytical result shows that the thickness has a significant influence on the propagation constant when the thickness is 0.01 times smaller than the thickness of the main plasma column region for both the symmetric and non-symmetric surface waves, while there is a significant difference in the propagation constant amplitude between the m = 1 mode and other four modes.

Investigation of symmetry breaking in periodic gravity–capillary waves

In this paper, fully nonlinear non-symmetric periodic gravity–capillary waves propagating at the surface of an inviscid and incompressible fluid are investigated. This problem was pioneered analytically by Zufiria (J. Fluid Mech., vol. 184, 1987c, pp. 183–206) and numerically by Shimizu & Shōji (Japan J. Ind. Appl. Maths, vol. 29 (2), 2012, pp. 331–353). We use a numerical method based on conformal mapping and series truncation to search for new solutions other than those shown in Zufiria (1987c) and Shimizu & Shōji (2012). It is found that, in the case of infinite-depth, non-symmetric waves with two to seven peaks within one wavelength exist and they all appear via symmetry-breaking bifurcations. Fully exploring these waves by changing the parameters yields the discovery of new types of non-symmetric solutions which form isolated branches without symmetry-breaking points. The existence of non-symmetric waves in water of finite depth is also confirmed, by using the value of the streamfunction at the bottom as the continuation parameter.

Traveling waves in trimer granular lattice II: Asymptotic prediction of weakly attenuated pulses

In the present study we consider the impulsive response of perfectly aligned, uncompressed, tri-atomic (trimer) granular lattice. In this study, we demonstrate that under particular choice of the system parameters - impulsively loaded, trimer granular lattice can support formation of highly localized, weakly attenuated pulses. These pulses are manifested by the completely non-symmetric wave profiles and can be attributed to the special family of solitary like waves forming in the non-homogenous, periodic trimer granular lattice in the state of acoustic vacuum. Using the recently developed analytical procedure based on the singular, multi-scale perturbation analysis, we derive a simplified reduced order model predicting the special regions in the space of the system parameters corresponding to the formation of the weakly attenuated pulses. Predictions of the asymptotical model are found to be in very good agreement with the results of numerical simulations of the full trimer granular lattice. From a practical point of view, these results can have important implications in complex, structural optimization problems of wave manipulation in the repetitive granular metamaterials.

A numerical and experimental study of oblique impact of ultra-high pressure abrasive water jet

An investigation of the abrasive water jet with an emphasis on the oblique impact of abrasive particles on the target plate is performed. Ultra-high jet pressure necessitates a close examination of the phenomena featured by small spatial and temporal scales. The effect of oblique impact is assessed from both numerical and practical aspects. Numerical simulation, implemented using the commercial code LS-DYNA, allows a detailed inspection of transient stress wave propagation inside the target plate. And impact experiments facilitate a qualitative description of resultant footprints of oblique water jet. Different incident angles of abrasive particles are adopted and a comparison is thereby unfolded. The results indicate that rebound, embedding, and penetration of single abrasive particle are three representative final operation states. Adjacent to the abrasive particle, the response of the target plate to oblique impact is reflected by von Mises stress distribution and plate deformation as well. Oblique impact arouses non-symmetrical stress wave distributions and distinct unbalanced node displacements at the two sides of the abrasive particle. As for the target plate, global surface morphology is in accordance with predicted effects. The most favorable surface roughness is not associated with vertical impact, and it hinges upon the selection of standoff distance. Furthermore, variation of surface roughness with incident angle is not monotonous.

Guided torsional wave generation of a linear in-plane shear piezoelectric array in metallic pipes

Cylindrical guided waves based techniques are effective and promising tools for damage detection in long pipes. The essential operations are generation and reception of guided waves in the structures utilizing transducers. A novel in-plane shear (d36 type) PMNT wafer is proposed to generate and receive the guided wave, especially the torsional waves, in metallic pipes. In contrast to the traditional wafer, this wafer will directly introduce in-plane shear deformation when electrical field is conveniently applied through its thickness direction. A single square d36 PMNT wafer is bonded on the surface of the pipe positioned collinearly with its axis, when actuated can predominantly generate torsional (T) waves along the axial direction, circumferential shear horizontal (C-SH) waves along circumferential direction, and other complex cylindrical Lamb-like wave modes along other helical directions simultaneously. While a linear array of finite square size d36 PMNT wafers was equally spaced circumferentially, when actuated simultaneously can nearly uniform axisymmetric torsional waves generate in pipes and non-symmetric wave modes can be suppressed greatly if the number of the d36 PMNT wafer is sufficiently large. This paper first presents the working mechanism of the linear d36 PMNT array from finite element analysis (FEA) by examining the constructive and destructive displacement wavefield phenomena in metallic pipes. Furthermore, since the amplitude of the received fundamental torsional wave signal strongly depends on frequency, a series of experiments are conducted to determine the frequency tuning curve for the torsional wave mode. All results indicate the linear d36 PMNT array has potential for efficiently generating uniform torsional wavefield of the fundamental torsional wave mode, which is more effective in monitoring structural health in metallic pipes.

Parametric excitation of azimuthally non-symmetric surface waves propagating in metal waveguides filled with isotropic plasma

The paper is devoted to developing the theory of parametric excitation of electromagnetic waves propagating across the axis of symmetry in cylindrical waveguides partially filled with isotropic plasma. The problem is studied theoretically in the fluid approximation and expressions for the wave fields are derived from Maxwell’s equations. The azimuthally non-symmetric electromagnetic waves propagate in the form of wave packets which are approximately described by the main azimuthal harmonic and two nearest satellite temporal harmonics. The boundary condition, which is cast in a nonlinear form, describes the flowing of a surface current on the plasma interface. This condition allows one to derive an infinite set of equations for harmonics of the tangential electric field of azimuthally non-symmetric surface waves. The dependence of the growth rate of the parametric instability of these waves on parameters of the plasma-filled waveguide and alternating electric field is studied both analytically and numerically.