Wave attenuation in nonlinear periodic structures using harmonic balance and multiple scales

We study the attenuation, caused by weak damping, of harmonic waves through a discrete, periodic structure with frequency nominally within the Propagation Zone (i.e., propagation occurs in the absence of the damping). The period of the structure consists of a linear stiffness and a weak linear/nonlinear damping. Adapting the transfer matrix method and using harmonic balance for the nonlinear terms, a four-dimensional linear/nonlinear map governing the dynamics is obtained. We analyze this map by applying the method of multiple scales upto first order. The resulting slow evolution equations give the amplitude decay rate in the structure. The approximations are validated by comparing with other analytical solutions for the linear case and full numerics for the nonlinear case. Good agreement is obtained. The method of analysis presented here can be extended to more complex structures.     

Wave dispersion in periodic post-buckled structures

Wave propagation in pinned-supported, post-buckled beams can be described with the Korteweg de Vries (KdV) equation. Finite-element simulations however show that the KdV is applicable only to post-buckled beams with strong pre-compression. For weak and moderate pre-stress, a dispersive front is present and it is the aim of the current paper to analyze sources of dispersion beyond periodicity given three support types: guided, pinned, and free. Bloch theorem and a transfer-matrix method are employed to obtain numerical dispersion relations and characteristic wave modes, which are used to analyze the effects of pre-stress, initial curvature, and the influence of support types. Additionally, a new method is proposed to obtain a semi-analytical dispersion equation for the acoustic branch. Powers of frequency and the propagation constant are explicitly expressed and their coefficients are based on stiffness and mass-matrix components obtained from finite elements. This allows a physical interpretation of the dispersion sources, based on which, equivalent mass–spring models of post-buckled beam are proposed. It is found that mass and stiffness coupling are significant dispersion sources. In the present paper, a reduced form of Bloch theorem is presented exploiting glide-reflection symmetries, reducing the size of the unit cell and allowing an easier representation and interpretation of results.     

Wave propagation along transversely periodic structures

The dispersion curves for guided waves have been of constant interest in the last decades, because they constitute the starting point for NDE ultrasonic applications. This paper presents an evolution of the semianalytical finite element method, and gives examples that illustrate new improvements and their importance for studying the propagation of waves along periodic structures of infinite width. Periodic boundary conditions are in fact used to model the infinite periodicity of the geometry in the direction normal to the direction of propagation. This method allows a complete investigation of the dispersion curves and of displacement/stress fields for guided modes in anisotropic and absorbing periodic structures. Among other examples, that of a grooved aluminum plate is theoretically and experimentally investigated, indicating the presence of specific and original guided modes.     

Wave propagation in two-dimensional periodic lattices

Plane wave propagation in infinite two-dimensional periodic lattices is investigated using Floquet-Bloch principles. Frequency bandgaps and spatial filtering phenomena are examined in four representative planar lattice topologies: hexagonal honeycomb, Kagomé lattice, triangular honeycomb, and the square honeycomb. These topologies exhibit dramatic differences in their long-wavelength deformation properties. Long-wavelength asymptotes to the dispersion curves based on homogenization theory are in good agreement with the numerical results for each of the four lattices. The slenderness ratio of the constituent beams of the lattice (or relative density) has a significant influence on the band structure. The techniques developed in this work can be used to design lattices with a desired band structure. The observed spatial filtering effects due to anisotropy at high frequencies (short wavelengths) of wave propagation are consistent with the lattice symmetries.     

Flow acoustics in periodic structures

A recent paper [A.A. Krokhin, J. Arriaga, L.N. Gumen, Speed of sound in periodic elastic composites, Phys. Rev. Lett. 91 (2004) 264302-1-4] addresses the speed of sound in periodic elastic composites (phononic crystals) with particular emphasis to the case where air bubbles are present in water and arranged periodically. In such periodically arranged mixtures, the well-known phenomena of the drop of the speed of sound may occur and applications related to, e.g., sound-beam focusing and acoustic surgery are possible [F. Cervera, L. Sanchez, J.V. Sanchez-Perez, R. Martinez-Sala, C. Rubio, F. Meseguer, C. Lopez, D. Caballero, J. Sanchez-Dehesa, Phys. Rev. Lett. 88 (2002) 023902]. In this paper, the analysis is extended theoretically to include cases where a background flow in a periodic structure is maintained. Calculations of dispersion relations and group velocities are presented in cases with one- and two-dimensional material periodicity for background flow values in the range: 0-1m/s. Materials considered in the calculations are periodic water-air mixtures. It is shown that acoustic waves couple to the group velocities only if the (acoustic) wave vector has a component along the background flow velocity direction.     

Acoustic attenuation in self-affine porous structures

As acoustic waves propagate through fluid-filled porous materials possessing heterogeneity in the elastic compressibility at scales less than wavelengths, the local wave-induced fluid-pressure response will also possess spatial heterogeneity that correlates with the compressibility structure. Such induced fluid-pressure gradients equilibrate via fluid-pressure diffusion causing wave energy to attenuate. This process is numerically simulated using finite-difference modeling. It is shown here, both numerically and analytically, that in the special case where the compressibility structure is a self-affine fractal characterized by a Hurst exponent H, the wave's quality factor Q (where Q(-1) is a measure of acoustic attenuation) is a power law in the wave's frequency omega given by Q proportional to omega(H) when /H/<1, and given by Q proportional to omega(tanhH) in general.     

Focusing-to-defocusing crossover in nonlinear periodic structures

We demonstrate experimentally a transition from nonlinear beam trapping to defocusing in a two-dimensional periodic photonic structures by varying the modulation depth of the lattice. The observed effect illustrates the fundamental crossover from discrete to cw transport mechanisms. At the threshold modulation, the output beam is highly sensitive to refractive index and power variations, which can be potentially applied for high-sensitivity refractive index or temperature sensing.     

Epitaxial periodic p-n structures in PbS

The electrical and photoelectrical properites of periodic p-n structures in an epitaxial film of PbS are studied. The plane of the junctions is perpendicular to the plane of the film. The possibility of using such structures in the photodiode mode as low inertia detectors of IR radiation is shown. Qualitative agreement of the dark characteristics (volt-ampere curve, temperature dependence of resistance) of the p-n structure and photosensitive polycrystalline films was noted. This confirms the hypothesis that in sensitization of polycrystalline films, barriers of the p-n junction type are formed.     

Deterministic weak localization in periodic structures

In some perfect periodic structures classical motion exhibits deterministic diffusion. For such systems we present the weak localization theory. As a manifestation for the velocity autocorrelation function a universal power law decay is predicted to appear at four Ehrenfest times. This deterministic weak localization is robust against weak quenched disorders, which may be confirmed by coherent backscattering measurements of periodic photonic crystals.     

Quantum electrodynamics in three-dimensionally periodic structures

Abstract

Recently, the possibility of obtaining ISE in spectral bands of forbidden light propagation (photonic bandgaps) has been suggested by Yablonovitch.¹ These bandgaps are spectral regions admitting only complex wavevectors k (evanescent waves) in any direction, and boundecl by frequencies ω_max((min) at which the dispersion curve ω(k) becomes discontinuous. Because they are associated with definite k (band edges), such hndgaps are delocalized in space, i.e., they inhibit spontaneous emission independently of t.he spatial distribution of emitters in the system. The systems that have been proposed¹ for the demonstration of bandgaps are dielectric superlattices that exhibit strong three-dimensional (3D) periodic moclulations of the dielectric index with a period comparable to half the emission wavelength.     

Photon-phonon interaction on periodic structures

Features of the interaction between optical and acoustical waves with the same wavelength and multiple of the structural layer period, propagating simultaneously through periodic structures, are analyzed. The current state of and prospects for photon-phonon crystals being used for amplification, the generation of gigahertz acoustic waves, and other nonlinear optical and acoustic effects are examined.     

Dynamic analysis of periodic structures

The natural vibration analysis of a periodic structure with repeated identical substructures may be simplified by using some symmetrical properties of the substructure dynamic matrices, resulting in a set of linear difference equations in the displacements. These equations are readily solved for cyclic symmetric systems, simply supported systems and infinite systems. The order of the overall frequency equations is at most equal to one half of the total number of degrees of freedom retained for a single substructure regardless of the number of substructures in the system. With these natural modes, the system with general boundary conditions at end stations is analyzed by a fast converging method.     

Linear momentum quantization in periodic structures II

Fraunhofer interference of a single particle by a periodic array of scatterers, usually treated with a wave picture, can be fully explained on the basis of linear momentum quantization, as pointed out in a previous study by Van Vliet (1967) [4]. This analysis is now extended to scattering (or passing through slits) involving a finite number N of equidistantly spaced entities comprising the interferometer. The usual intensity probability distribution for W(sinθ) is obtained, noting that total momentum is conserved (as in the Compton effect), while the interferometer is treated as a quantum object—rather than a classical measuring apparatus, as perceived in the Copenhagen interpretation. Various aspects of the ‘orthodox view’ are examined and renounced.     

Collective transport in locally asymmetric periodic structures

In this paper we give an overview of the cooperative effects in fluctuation driven transport arising from the interaction of a large number of particles. (i) First, we study a model with finite-sized, overdamped Brownian particles interacting via hard-core repulsion. Computer simulations and theoretical calculations reveal a number of novel cooperative transport phenomena in this system, including the reversal of direction of the net current as the particle density is increased, and a very strong and complex dependence of the average velocity on both the size and the average distance of the particles. (ii) Next, we consider the cooperation of a collection of motors rigidly attached to a backbone. This system possesses dynamical phase transition allowing spontaneous directed motion even if the system is spatially symmetric. (iii) Finally, we report on an experimental investigation exploring the horizontal transport of granular particles in a vertically vibrated system whose base has a sawtooth-shaped profile. The resulting material flow exhibits complex collective behavior, both as a function of the number of layers of particles and the driving frequency; in particular, under certain conditions, increasing the layer thickness leads to a reversal of the current, while the onset of transport as a function of frequency occurs gradually in a manner reminiscent of a phase transition. (c) 1998 American Institute of Physics.