Spectro-spatial wave features as detectors and classifiers of nonlinearity in periodic chains

Analysis of wave propagation in nonlinear periodic structures is often limited to the determination of the spectral characteristics of wave motion (dispersion relations). Unfortunately, the physical features of wave propagation are only partially highlighted by a spectral description. The objective of this work is to elucidate the relation between topological/physical (space-time domain) and spectral (wavenumber-frequency domain) features of wave motion in periodic chains with weak nonlinearities. The analysis is based on full-scale transient analysis of finite chains, from which dispersion curves are generated and verified against the results of unit-cell based perturbation methods. The evolution of spatial and spectral features is monitored using signal processing techniques such as spatial-spectrogram and wavenumber filtering, and the interplay of dispersive and nonlinear mechanisms in the process of waveform distortion is evaluated. The relation between topological features and spectral characteristics is determined using appropriate topological wave packet descriptors. It is found that certain packet descriptors are sensitive to the parameters of the nonlinear constitutive relation and can therefore be used to construct inverse problems to estimate nonlinearity in systems with unknown constitutive models.