Thesis title: Wave effects in long-range connectivity: homogeneous and non-homogeneous periodicity
Metamaterials are artificial composites, developed to exhibit specific properties. Their characteristics majorly stem from the tailored structure itself rather than its composing materials. What drew attention to this family is the exotic behaviours they inherently present, which is unusual with respect to those observed in nature. Acoustic metamaterials, in particular, could provide attractive features for practical purposes, including cloaking and noise cancelling. A second original viewpoint on metamaterials suggests the possibility of enjoying the tantamount effects by introducing a domain within which distant points interact mutually. Thus, developing distinct configurations through embedding long-range connectors within host structures could provide the potential to manipulate wave paths across the domain.
Bearing in mind the need for a wide variety of propagation phenomena in different applications, systems with long-range inclusions, which challenges the common notion of exclusive first neighbour interaction, could be a potential candidate. By initiating the graph-periodicity concept, two major categories of long-range configurations are introduced, namely long-range homogenous and non-homogenous systems. With reference to the graph-periodicity concept, genuine periodic configuration can be built, where the system captures the tree-periodicity property. In this case, due to the contribution of select groups of points in constructing the configuration, a non-homogenous periodic configuration is met. On the other hand, long-range homogenous configurations are those in which all points adhere to an identical connectivity template. Although such arrangements are not genuinely periodic (no tree-periodicity), they can be leaf-periodic depending on the template.
This thesis presents the analysis of wave propagation in conventional structures, namely waveguide, membrane and shell, integrated with long-range connector. Regarding long-range homogenous configurations, the long-range forces are modelled by summation and integral terms. The long-range operators are majorly modelled by simple spring-like connections, and in some instances by resonator units. Wave propagation behaviour of the system is comprehensively realized by examining the dispersion relation, obtained analytically. Several phenomena, including wave-stopping, negative group velocity, and redistribution of eigenstates, are observed under different circumstances depending on the comparative stiffness of long-range connections.
Next, the problem of wave propagation in long-range non-homogenous periodic configurations is investigated. Although extracting the dispersion curves for such systems is not a trivial task due to non-homogeneous distribution of long-range connectors, the adopted connectivity template provides the opportunity to take the advantage of the well-known Floquet theorem to identify the band structure. The emergence of gaps over the frequency band is confirmed. The model implies the possibility of pulling the gaps towards ultra-low frequencies and even generating stopbands, which could virtually open from zero, when connections assume negative stiffness. Insertion loss analysis is performed to roughly validate the arrangements of stopbands over the frequency band, yielded by the Floquet theorem.
Given the fact that human beings are social creatures, this creates several layers of human interactions yielding a giant network of intercommunications in societies. Thus, mathematical modelling, analogous to the above, could be a big asset to shed light on behaviour of crowds in large scale. Correspondingly, dynamic memory, being the temporal counterpart of spatial long-range interactions, can be included in some long-established problems such as natural economic growth to achieve a more effective and accurate description of the system. Current study exclusively considers the effect of nonlocal interactions in the realm of physics and engineering though the underlying message conveyed by this concept can be extremely relevant in other fields as well.