Abstract
The nonlinear propagation of intense acoustic waves in a periodic (phononic) medium is presented. Two cases are considered: a lattice of nonlinearly coupled oscillators, and a superlattice formed by layers with alternating linear acoustic properties and quadratic elastic nonlinearity (1D phononic crystal). The spacing between elements (masses or layers) is of the order of the wavelength; therefore Bragg effects such as band-gaps appear. We show that the interplay between strong dispersion and nonlinearity leads to new scenarios of wave propagation. The classical waveform distortion process typical of intense acoustic waves in homogeneous media can be strongly altered when nonlinearly generated harmonics lie inside or close to band gaps. This allows the possibility of engineer a medium in order to get a particular waveform. Examples of this include the design of media with effective (e.g. cubic) nonlinearities, or extremely linear media (where distortion can be cancelled). We first consider a discrete lattice where its elements interact with its the nearest-neighbors, via an inverse power law, which is typical of Coulomb-type interactions. The physical system is realized experimentally, in the form of an array of cylindrical magnets, forming a nearly one dimensional chain of masses coupled by its magnetic fields. The magnetic moments of neighbor magnets are oriented in the same directions, so they experience repulsive forces dependent on the separation between poles. Each magnet is the mass of a pendulum, which is forced to oscillate around an equilibrium position. We report theoretical and experimental evidence of nonlinear waves in the magnetic chain. In particular, we calculated the linear dispersion relation, and its dependence with the forcing amplitude. The generation of harmonics (second and higher order) is also demonstrated. Notably, is shown that harmonics, being forced oscillations, exist even when the corresponding frequency is higher than the linear cutoff frequency. In the case of diatomic chains of alternating masses, also subharmonics of the driving frequency are observed. In the case of superlattices, similar phenomena is reported. Here however the medium presents higher order propagation bands, separated by bansgaps, which allows to manage the dispersion of the harmonics in a unique way. We find that, as in the lattices, the efficiency of harmonic generation is weakened by dispersion, where instead of an unbounded growth, beatings in the amplitude of harmonics are observed. We show that the beating period is dependent on the amplitude (defining a coherence length). In the case of a periodic system, we show that it is also possible to get phase matching of particular sets of harmonics (for example, by fulfilling the condition for only a particular value of n), by allowing them to propagate in different dispersion bands. Using this fact we are able to engineer the structure in order to propagate efficiently only particular set of harmonics. The wave propagates as if the multilayered medium had an effective nonlinearity (e.g. cubic) different from that of the constituent layers (quadratic). We have developed a couple mode theory, that allows to treat the problem in a simpler way, and obtain analytical solutions for the different harmonics. Another interesting features of this problem, as KdV soliton propagation, are discussed. The presented ideas open a way towards the control of acoustic wave propagation in nonlinear regime.
BibTeX citation
@inproceedings{Sanchez-Morcillo2015b,
author = {Víctor Sánchez-Morcillo and Noé Jiménez and Ahmed Mehrem and Rubén Picó and Lluis M. García Raffi},
booktitle = {IMECE 2015: International Mechanical Engineering Congress \& Exposition. Huston, Texas. 13-19th November},
editor = {Book of abstracts},
month = {11},
organization = {American Society of Mechanical Engineers},
pages = {1},
publisher = {ASME},
series = {Oral},
title = {Nonlinear waves in phononic lattices and superlattices},
url = {www.asme.org/events/imece},
year = {2015}
}