Propagation of waves in nonlocal-periodic systems

This paper is concerned with emergence of novel effects in wave propagation in one-dimensional waveguides, when integrated with periodic nonlocalities. The nonlocalities are introduced by a connectivity superimposed to a conventional waveguide and depicted as a graph with trees and leaves, each with its own periodicity. Merging nonlocality and periodicity notions induces a distinction between homogenous and non-homogenous periodic configurations. Specifically, various unconventional phenomena linked to the presence of nonlocalities result in disruption of the energy transmission in such systems, disclosing new opportunities for vibration isolation applications. To demonstrate these effects, simple models of propagation of plane extension/compression waves in a uniform infinite rod equipped with co-axial spring-like elements is used. The homogenous case is analysed by a direct double, space and time, Fourier transform, leading to discussion of unusual dispersion effects, including vanishing and negative group velocity. In the non-homogeneous case, the canonical Floquet theory is used to identify stopbands and control their positions in the frequency domain. The results are compared with eigenfrequency analysis of unit periodicity cells and finite structures. Next, the forcing problem is considered and the insertion losses in a semi-infinite rod with nonlocal spring effects are computed to corroborate predictions of Floquet theory, providing physical explanations of the obtained results. Finally, possibilities to employ the non-local interaction forces in an active control format to generate stopbands at arbitrarily low frequencies are highlighted.