Wave Propagation In Periodic Structures Research Articles

Rayleigh wave manipulation based on phase shift between two periodic surface corrugations

The propagation of Rayleigh waves along solid surfaces has attracted great attention due to its ubiquity in seismic waves, non-destructive evaluation, acoustic surface wave devices, and so on. Moreover, Rayleigh waves in periodic structures and their related band gaps caused by Bragg resonances have also been investigated. In this paper, we theoretically and experimentally demonstrate the propagation mechanism of Rayleigh waves along the side wall of a periodically corrugated aluminum plate. As expected, the band gap structures are observed in the spectrum of Rayleigh waves. However, not all the band gaps are caused by the Bragg resonances. The additional band gaps created by the interference of different transverse modes are observed. Different from the well-known Bragg resonance, the different mode interference, producing stronger energy attenuation and wider band gap, is called non-Bragg resonances. The two resonances can be manipulated by varying the phase shift of two sidewall corrugations, achieving an effective prohibition of Rayleigh wave propagation. These findings on Rayleigh wave propagation in periodic structures provide a comprehensive understanding of wave-structure interactions, which can be used to regulate elastic waves in various applications, such as multifunctional elastic wave separators, filters, and modulators.

Adaptive Gaussian Process with PCA for prediction of complex dispersion relations for periodic structures

The complex dispersion relation is important for understanding the wave propagation in periodic structures. However, its calculation is much more expensive compared to calculation of the real dispersion relation, due to the need to solve the dispersion equation at large number of frequencies. For uncertainty quantification in the dispersion relations, the computational challenge is even higher. This paper proposes an adaptive Gaussian process (AGP) model to efficiently predict the complex dispersion relations for periodic structures with different properties. Instead of directly building GP for the dispersion relation, which is challenging due to discontinuity in the dispersion relation, we first reformulate the problem as predicting the coefficients of the dispersion equation at selected frequencies, where these coefficients are continuous and smooth functions of the properties of periodic structures. Then GP model is trained to predict these coefficients, based on which the dispersion equation is then analytically solved to establish the complex dispersion relation. Second, Principal component analysis (PCA) is used to reduce the dimension of these coefficients to facilitate efficient training of GP model. Third, an adaptive procedure is integrated to iteratively add training data that can most effectively improve the GP model accuracy. Results verify the great efficiency and accuracy of the proposed approach for both undamped and damped periodic structures. The proposed approach has great promise in improving efficiency for uncertainty quantification, sensitivity analysis, and design optimization of periodic structures.

All waves have a zero tunneling time

Abstract Zero tunneling time and thereby a faster than light traversal velocity was calculated nearly a hundred years ago and has been observed recently. We report about experimental results and estimations, which confirm the zero time tunneling for elastic as well as for electromagnetic and Schrödinger waves. Zero time tunneling was first observed with microwaves 1992 (H. Aichmann and G. Nimtz, Found. Phys., vol. 44, p. 678, 2014; A. Enders and G. Nimtz, J. Phys. I, vol. 2, p. 169, 1992). In 2008, zero time was also observed for tunneling electrons (P. Eckle, A. N. Pfeiffer, C. Cirelli, et al., Science, vol. 322, p. 1525, 2008). Presumably, this effect took place with atoms quite recently (R. Ramos, D. Spierings, I. Racicot, and A. M. Steinberg, Nature, vol. 583, p. 529, 2020). The Einstein relation E 2 = (ħk)2 c 2 is not satisfied in the tunneling process, since the wave number k is imaginary (E is the total energy, ħ the Planck constant, and c the vacuum velocity of light), Zero time tunneling is described by virtual photons (A. Stahlhofen and G. Nimtz, Europhys. Lett., vol. 76, p. 189, 2006). The tunneling process itself violates the Special Theory of Relativity. Remarkably, Brillouin conjectured that wave mechanics is valid for all waves independent of their field (L. Brillouin, Wave Propagation in Periodic Structures, Chap. VIII, New York, Dover Publications, 1953).

On a Possible Wave Mechanism for an Increase in the Yields of Low-Energy Nuclear Reactions in Crystal Structures

The possibility of using one of the properties of wave propagation in periodic structures is considered to explain the known effect of an increase in the yields of nuclear DD reactions in crystalline films. The effect is observed under the bombardment of deuterated metal foil targets with an oxide film by low-energy deuterons (less than 100 keV) in the majority of metals: the yield increases many times as compared with the expected yield when extrapolating the data obtained for high-energy deuterons. Proceeding from the analogy between equations describing the propagation of a light wave in a photonic crystal and that of a particle in a crystal, the conclusion that the properties of the wave functions of particles in a crystal and of the light field in a photonic crystal are analogous is drawn. Forbidden bands with transparency windows can exist in the spectrum of the particles; their properties are also analogous to those of the windows in a photonic crystal. The amplitude of the wave function of a massive particle incident on a crystal can increase in them, which must be taken into account when considering the increase in the yields of nuclear reactions with the participation of moving particles.

Numerical computation of the spatial decaying wave characteristics for the design of locally resonant acoustic metamaterials

In structural dynamics, periodic resonant inclusions are widely used to improve the vibro-acoustic properties of lightweight structures. Depending on the design of these inclusions, the wave propagation in the structure is modified for specific frequency ranges and stop bands, where in theory no free wave propagation is possible, can be generated. The Wave Finite Element Method (WFEM) investigates the wave propagation in periodic structures based on a Finite Element (FE) model of a single unit cell. Using the direct approach of the WFEM, it is possible to depict the wave characteristics in the frequency range of the resonant frequency of the inclusions. Consequently, a spatial decay for the different wave types can be determined in the frequency range of the potential stop band. The frequency dependent decay characteristics can be used to evaluate the performance of the resonant inclusions and find optimal parameters for the desired application of the metamaterial. The procedure is applied to design beam-like resonators and find an optimal resonator spacing for a beam-like acoustic metamaterial. For a fixed percentage of added mass, the spacing of the resonators can be adjusted to optimize the maximum spatial decay or the stop band size.

Deaf band based engineered Dirac cone in a periodic acoustic metamaterial: A numerical and experimental study

A Dirac-cone-like dispersion at the center of the Brillouin zone where the wave number $\stackrel{P\vec}{k}=0$ [L. Brillouin, Wave Propagation in Periodic Structures (Dover, New York, 1953)], is rare and only happens due to accidental degeneracy. At certain frequencies, the Dirac cone breaks the time-reversal symmetry of acoustic waves, which has not yet been fully explored. In this paper, even the simplest geometric microarchitecture of phononic crystals (PnCs) in a periodic structure can be modulated to obtain the accidental triple degeneracies that make a Dirac-like cone at the \ensuremath{\Gamma} point ($\stackrel{P\vec}{k}=0$). While doing so, it was observed that the frequency of a nondispersive band obtained from any arbitrary periodic structure made of similar PnCs remains unaltered. Then, a deaf band based predictive modulation of the PnCs is realized, and multiple occurrences of the Dirac-like points are demonstrated. The claims are validated using a numerical and experimental study of a baseline periodic structure having a square array of cylindrical polyvinylchloride inclusions in an air matrix. Phenomena such as orthogonal wave transport, negative refraction, and wave vortex are verified to exist at the deaf band based engineered Dirac cone.

Band gap characteristics of periodic gyroscopic systems

The stop and pass bands (i.e. the band gaps) characteristics are determined for gyroscopic systems by developing an approach which is compatible with such class of systems which is based on the concept of Bloch wave propagation in periodic structures. In this approach, the dispersion curves of the periodic gyroscopic systems are determined for different rotational speeds. The obtained characteristics are compared with non-rotating systems in an attempt to quantify the effect of the gyroscopic forces on the “band gap” characteristics. The developed approach is illustrated by a new class of drill strings with passive periodic inserts. These inserts are utilized to filter out the vibration transmission along the drill string. Such mechanical filtering capabilities allow the vibrations to propagate along the periodic drill string only within specific frequency bands called the ‘pass bands’ and completely block it within other frequency bands called the ‘stop bands’. The inserts introduce impedance mismatch zones along the vibration transmission path to impede the propagation of vibration along the drill string. The design and the location of the inserts are selected to confine the dominant modes of vibration of the drill string within the stop bands generated by the periodic arrangement of the inserts in order to completely block the propagation of the vibrations.A finite element model (FEM) that simulates the operation of this new class of drill strings is developed to describe the complex nature of the vibration encountered during drilling operations. The FEM is used to extract the dispersion characteristics of the gyroscopic unit cell of the drill string in order to determine its stop and pass band characteristics. Experimental prototype of the passive periodic drill string is built and tested to demonstrate the feasibility and effectiveness of the concept of periodic drill string in mitigating undesirable vibrations. The experimental results are used to validate the developed theoretical model in order to develop a scalable design tool that can be used to predict the dynamical behavior of this new class of drill strings.

Wave propagation in the polymer-filled star-shaped honeycomb periodic structure

The wave propagations in the periodic structure composed of auxetic star-shaped honeycombs are investigated. The matrix of the periodic structure is filled with polymer. The effective material properties of the unit cell of the periodic structure such as the Poisson’s ratio, the Young’s modulus, and the shear modulus are varied with different filling materials. The finite element method is used to solve the wave propagation within the auxetic star-shaped honeycombs. The dispersion analysis of the band structures and iso-frequency contour are presented. The auxeticity is found to have a significant effect on the wave propagation in the honeycomb structure, and the collimation of the auxetic star-shaped honeycomb structure is also studied. Effects of different types of filling on the wave propagation are also investigated. Among three filling types, the outer-filled honeycomb has the best self-collimation effect. The self-collimation of the polymer-filled auxetic honeycomb structure presented in this study can be applied to various acoustic devices.

Analysis of in-plane wave propagation in periodic structures with Sierpinski-carpet unit cells

In this study, the dispersion relationships and characteristics of directional periodic structure waveguides with Sierpinski carpet unit cells were investigated. Finite element method was used to analyze the Sierpinski-carpet periodic structures with square Pb cylinders inserted in the rubber background. The dispersion curves of unit cells with different stages were investigated first, then group velocities of periodic structures at given frequencies were calculated and used to analyze the directional waveguide characteristics in the Sierpinski-carpet periodic structures. The dynamic responses of finite periodic structures were investigated to observe the directional features, and the results were used to support the fact that Sierpinski carpet unit cells with different stages can control the propagation behavior of elastic waves in specific ways; the dispersion curves grew compressed as the stage increased. Multiple band characteristics were observed in the case of the second-stage fractal configuration. The directional propagation characteristics in Sierpinski carpet periodic structures can be utilized to further extend the range of vibration reduction zones. The results of this study have significant potential in terms of the application of Sierpinski-carpet periodic structures for vibration isolation.

Wave propagation in periodic track structures: band-gap behaviours and formation mechanisms

This paper presents the results of a study concerning the band-gap behaviours and formation mechanisms of periodic track structures. Based on the band-gap theories introduced from phononic crystal which concentrates on the elastic wave propagation in periodic structures, the railway track can be regarded as a novel locally resonant phononic crystal. The band-gaps are found by using the transfer matrix method combined with Bloch theorem, and the attenuation factors in band-gaps are also obtained firstly. Then, band-gap behaviours of periodic track structures are investigated with various parameters such as stiffness of rail pad, fastening spacing and thermal force in rail. Bounding frequencies and width of band-gaps are closely related to the parameters of track structures, resulting from the various wave motion modes at the bounding frequencies. Moreover, it has been found that Bragg band-gaps and locally resonance band-gaps coexist in periodic track structures. And formation mechanisms of band-gaps in periodic track structures can be explained by the Bragg scattering mechanism and locally resonance mechanism. The theoretical analysis is verified by the frequency response functions calculated through the finite element models at last.

Nonlinear waves in lattice materials: Adaptively augmented directivity and functionality enhancement by modal mixing

The motive of this work is to understand the complex spatial characteristics of finite-amplitude elastic wave propagation in periodic structures and leverage the unique opportunities offered by nonlinearity to activate complementary functionalities and design adaptive spatial wave manipulators. The underlying assumption is that the magnitude of wave propagation is small with respect to the length scale of the structure under consideration, albeit large enough to elicit the effects of finite deformation. We demonstrate that the interplay of dispersion, nonlinearity and modal complexity involved in the generation and propagation of higher harmonics gives rise to secondary wave packets that feature multiple characteristics, one of which conforms to the dispersion relation of the corresponding linear structure. This provides an opportunity to engineer desired wave characteristics through a geometric and topological design of the unit cell, and results in the ability to activate complementary functionalities, typical of high frequency regimes, while operating at low frequencies of excitation - an effect seldom observed in linear periodic structures. The ability to design adaptive switches is demonstrated here using lattice configurations whose response is characterized by geometric and/or material nonlinearities.

Bloch theorem for isogeometric analysis of periodic problems governed by high-order partial differential equations

With the emergence of isogeometric analysis (IGA), numerical rotation-free solutions to high-order partial differential equations (PDEs) are facilitated. Indeed, IGA allows the use of high-order-continuous basis functions through the domain. For periodic systems, the Bloch theorem is useful for analyzing wave propagation, restricting the analysis to a single unit cell and applying periodic boundary conditions. However, in the literature, boundary conditions are only accounting for C0-continuity, and we propose in the present paper to enforce the G1-continuity as well or to maintain the angle in case of unit cells meeting at kinks. While G1-continuity is enforced by maintaining the collinearity between adjacent control points, in the presence of kinks, variations of the field derived with respect to opposite in-plane edge normals are linked by the Rodrigues rotation matrix. This method applied at collocation points belonging to the unit cell edges can be viewed as a novel approach of coupling multi-patch geometries with kinks, avoiding the use of nonlinear multipliers or additional bending strip elements. Both transfer-matrix and indirect methods of the Bloch theorem are investigated, for respectively quasi-one and quasi-two-dimensional wave propagation in periodic structures discretized by NURBS-based IGA. Dispersion curves of a set of numerical problems treating isogeometric rotation-free extensible elastica (beams) and Kirchhoff–Love shells are then analyzed.

A method for the analytical description of coupled-field waves in various anisotropic media

The formalism of the matricant method based on constructing the optimal structure of the matrix of fundamental solutions to the governing differential equations is presented. The structures of the coefficient matrices of the governing equations and the structure of the matricant are determined for the acoustic coupled-field waves in elastic, thermoelastic, piezoelectric, and piezomagnetic anisotropic media. The analytical expressions of matricants are obtained for various homogeneous anisotropic media. The efficiency of the method is demonstrated in the case for an analysis of wave propagation in periodic structures. As examples, the secular equations for dispersive waves in layered and periodic structures are derived.

Computationally efficient finite element method for predicting wave propagation in periodic structures

A generic numerical method for predicting the wave propagation in structures with two-dimensional periodicity is presented. The method is based on a combination of Finite Elements and periodic structure theory. A unit cell of the periodic structure is described with finite elements, and a Craig-Bampton reduction is applied to reduce the number of degrees of freedom. Periodic boundary conditions are then applied and the waves propagating in the structure are obtained by solving an algebraic eigenvalue problem. A number of analytical expressions are then used to derive the vibro-acoustic properties of the finite or infinite periodic structure. The method was recently extended to account for heavy fluid loading and material with frequency-dependent properties (typical of acoustic treatments). A number of examples are presented to validate the formulation and demonstrate the possible use of the method for design.

Analysis of Bloch’s Method and the Propagation Technique in Periodic Structures

Bloch analysis was originally developed by Bloch to study the electron behavior in crystalline solids. His method has been adapted to study the elastic wave propagation in periodic structures. The absence of a rigorous mathematical analysis of the approach, as applied to periodic structures, has resulted in mistreatment of internal forces and misapplication to nonlinear media. In a previous article (Farzbod and Leamy, 2009, “The Treatment of Forces in Bloch Analysis,” J. Sound Vib., 325(3), pp. 545–551), we clarified the treatment of internal forces. In this article, we borrow the insight from the previous work to detail a mathematical basis for Bloch analysis and thereby shed important light on the proper application of the technique. For example, we conclusively show that translational invariance is not a proper justification for invoking the existence of a “propagation constant,” and that in nonlinear media, this results in a flawed analysis. We also provide a simple, two-dimensional example, illustrating what the role stiffness symmetry has on the search for a band gap behavior along the edges of the irreducible Brillouin zone. This complements other treatments that have recently appeared addressing the same issue.

Wave characterization of cylindrical and curved panels using a finite element method

This paper describes a wave finite element method for the numerical prediction of wave characteristics of cylindrical and curved panels. The method combines conventional finite elements and the theory of wave propagation in periodic structures. The mass and stiffness matrices of a small segment of the structure, which is typically modeled using either a single shell element or, especially for laminated structures, a stack of solid elements meshed through the cross-section, are postprocessed using periodicity conditions. The matrices are typically found using a commercial FE package. The solutions of the resulting eigenproblem provide the frequency evolution of the wavenumber and the wave modes. For cylindrical geometries, the circumferential order of the wave can be specified in order to define the phase change that a wave experiences as it propagates across the element in the circumferential direction. The method is described and illustrated by application to cylinders and curved panels of different constructions. These include isotropic, orthotropic, and laminated sandwich constructions. The application of the method is seen to be straightforward even in the complicated case of laminated sandwich panels. Accurate predictions of the dispersion curves are found at negligible computational cost.

Study of full band gaps and propagation of acoustic waves in two-dimensional piezoelectric phononic plates

Acoustic wave propagation in periodic structures composed of multi-constituents, namely the phononic crystals (PCs), has received much attention in the past decade. Recently, PCs constructed in the form of plate structures with two-dimensional lattices have been investigated, and some important phenomena, such as the full band-gap and waveguiding effects, induced by the periodicities of the phononic plates were reported. One of the virtues using Lamb waves is to perfectly confine the acoustic energy within the plate thickness and guided in structures with only 2D lattices rather than 3D lattices; however, the additional free surfaces derive much more complicated characteristics which are still awaiting study further. In this study, we investigate the band-gap and propagation properties of waves in two-dimensional phononic plates that consist of either elastic or piezoelectric materials by employing a revised full 3D plane wave expansion (PWE) method and so on. To apply the PWE method efficiently, Fourier expansions are performed in the thickness direction for an imaginary 3D periodic structure by stacking the phononic plates and vacuum layers alternatively. Moreover, effects of piezoelectricity, lattices, and filling ratios on band gaps are discussed.

An experimental investigation of guided wave propagation in corrugated plates showing stop bands and pass bands

Nonplanar surfaces are often encountered in engineering structures. In aerospace structures, periodically corrugated boundaries are formed by friction-stir-welding. In civil engineering structures, rebars used in reinforced concrete beams and slabs have periodic surface. Periodic structures are also being used to create desired acoustic band gaps. For health monitoring of these structures, a good understanding of the elastic wave propagation through such periodic structures is necessary. Although a number of research papers on the wave propagation in periodic structures are available in the literature, no one experimentally investigated the guided wave propagation through plates with periodic boundaries and compared the experimental results with theoretical predictions as done in this paper. The experimental results clearly show that elastic waves can propagate through the corrugated plate (waveguide) for certain frequencies called “pass bands,” and find it difficult to propagate for some other frequencies called “stop bands.” Stop bands are found to increase with the degree of corrugation. Experimental results are compared with the theoretical predictions, and good matching is observed for plates with small degree of corrugation. Only two parameters—the depth of corrugation and the wavelength of the periodicity—are sufficient for modeling the elastic wave propagation in slightly corrugated plates.

Minimally separating sets, mediatrices, and Brillouin spaces

Brillouin zones and their boundaries were studied in [J.J.P. Veerman et al., Comm. Math. Phys. 212 (3) (2000) 725] because they play an important role in focal decomposition as first defined by Peixoto in [J. Differential Equations 44 (1982) 271] and in physics [N.W. Ashcroft, N.D. Mermin, Solid State Physics, Holt, Rhinehart, and Winston, 1976; L. Brillouin, Wave Propagation in Periodic Structures, Dover, 1953]. In so-called Brillouin spaces, the boundaries of the Brillouin zones have certain regularity properties which imply that they consist of pieces of mediatrices (or equidistant sets). The purpose of this note is two-fold. First, we give some simple conditions on a metric space which are sufficient for it to be a Brillouin space. These conditions show, for example, that all compact, connected Riemannian manifolds with their usual distance functions are Brillouin spaces. Second, we exhibit a restriction on the Z 2 -homology of mediatrices in such manifolds in terms of the Z 2 -homology of the manifolds themselves, based on the fact that they are Brillouin spaces. (This will used to obtain a classification up to homeomorphism of surface mediatrices in forthcoming paper [J. Bernhard, J.J.P. Veerman, The topology of surface mediatrices, Portland State University].)

Vibration analysis of multi-supported curved panel using the periodic structure approach

This papers deals with the radial vibration of a row of cylindrical panels of finite length using the concept of wave propagation in periodic structures. For this study, the structure is considered as an assemblage of a number of identical cylindrically curved panels each of which will be referred to as a periodic element. For a given geometry dispersion curves of the propagation constant versus (non-dimensional) natural frequency have been drawn corresponding to the circumferential wave propagation. New conclusions that have emerged from this study are as follows. It is shown that by a proper choice of the periodic element the bounding frequencies and the corresponding modes in all the propagation bands can be determined. These have been shown to correspond to a single curved panel with all its edges simply supported. It is noted that there are no attenuation gaps in the entire frequency spectrum beyond the lowest bounding frequency. This is a unique feature of circumferential wave propagation around circular cylindrical shells and panels, as opposed to the wave propagation of periodically supported beams and rectangular panels without curvature. The natural frequency corresponding to every circumferential mode of the complete shell has been identified on the propagation constant curve. It has been observed that the natural frequencies of a cylindrically curved panel of a given curvature and length but of different circumferential arc length (corresponding to different angles subtended at the centre of any circular cross-section) may also be identified on the same propagation constant curve. Finally, it is shown that the same propagation constant curve may also be used to determine all the natural frequencies of a finite row of curved panels with the extreme edges simply supported. Wherever possible the numerical results have been compared with those obtained independently from finite element analysis and/or results available in the literature. Flutter analysis of multi-span curved panels using a wave approach is the ultimate objective of this work.