Nonreciprocal wave scattering on nonlinear string-coupled oscillators

Abstract

We study scattering of a periodic wave in a string on two lumped oscillators attached to it. The equations can be represented as a driven (by the incident wave) dissipative (due to radiation losses) system of delay differential equations of neutral type. Nonlinearity of oscillators makes the scattering non-reciprocal: the same wave is transmitted differently in two directions. Periodic regimes of scattering are analyzed approximately, using amplitude equation approach. We show that this setup can act as a nonreciprocal modulator via Hopf bifurcations of the steady solutions. Numerical simulations of the full system reveal nontrivial regimes of quasiperiodic and chaotic scattering. Moreover, a regime of a "chaotic diode", where transmission is periodic in one direction and chaotic in the opposite one, is reported.PACS numbers: 05.60.-k 05.70.Ln 44.10.+iOne of the mostly general results of the linear wave theory is the reciprocity theorem, established in works of Rayleigh, Helmholtz and Lorentz. For the one-dimensional wave scattering it means the symmetry of the scattering matrix, so that transmission in both direction is the same. While in linear systems violations of reciprocity require violations of time-reversal symmetry, in nonlinear wave propagation reciprocity does not hold. In particular, scattering of linear waves on nonlinear objects may operate as a "wave diode", with different transmission properties in both directions. Here we consider a simple model of scattering of linear waves on two lumped nonlinear oscillators. If one neglects dispersion and dissipation in the medium and in the oscillators, the equations can be reduced to a system of delay-differential equations. We demonstrate in this paper different regimes of reciprocity violations. In the simplest case transmissions in both directions are different, while the waves remain periodic. We observe also more complex regimes, where reflected and transmitted waves are chaotic and different. Probably, mostly nontrivial regime reported is that of "chaotic diode": a periodic wave sent to the scatterer in one direction remains periodic, while when the same wave is sent in another direction, transmitted and reflected waves are chaotic.

Asymmetric acoustic energy transport in non-Hermitian metamaterials

“…nonlinear imaginary part of the resonance) present in the PC fork (the contrast ratio of transmission asymmetry (TMP/TPM), which reaches values above 5 within the range of amplitudes tested, can be seen in Fig.S7). Because we utilize dissipative nonlinearity, the outgoing signal has high frequency purity unlike in the use of conservative nonlinearities (Boechler et al, 2011;Lepri and Casati, 2011;Lepri and Pikovsky, 2014)-i.e. there are no significant higher harmonics observed, for example, as seen in the frequency response of the nLnH-MetaMater for the incidence at 1903 Hz and 1919 Hz (Fig.4f,g).…”

Modeling of vibration transmission through impact elements

“…J can be found from the impact conditions (3). At the same time, it is necessary to take into account the periodicity of the process and combine the impact with the beginning of the counting of time.…”

“…In any case, the ability to consider the peculiarities caused by heterogeneity and the formation of impact pairs is an interesting and multifaceted task. Many studies have been devoted to different approaches to its study, among which we note, for example, [1][2][3][4][5][6][7][8][9].…”

Wave propagation in a nonlinear acoustic metamaterial beam considering third harmonic generation

“…For ω F >ω r /3, the entire branch-1 in the FC solution corresponds to Im(k 3 )>0, leading to attenuation of the third harmonic with wave number k 3 . In band [35][36][37][38][39][40][41][42][43][44][45][46][47][48][49][50][51][52][53] Hz for highly efficient THG, both modes 3k 1 and k 3 for the third harmonics can propagate. Moreover, there are multiple branches in the FC solutions, in which two large-amplitude branches originate from the bifurcation near ω r /3, but the low-amplitude branch has k 3 >600, indicating slow waves or attenuated waves.…”

“…In NEMs, the second-and third harmonic generation (SHG and THG) are adopted to realize phase matching [25,34], which is of paramount importance for lasers; and enhanced nonlinear responses are desired [35,36]. High-order harmonics in nonlinear crystals/metamaterials can break the reciprocity of the wave propagation and phonon transport [37][38][39]. The broken reciprocity may leads to potential applications in realizing phononic computing and information processing [40,41].…”