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.

当与周期性非局部性结合时，本文关注一维波导中波传播中的新颖效应的出现。通过与常规波导叠加的连通性引入非局部性，并以树木和树叶作为图形来描绘，每个树木和树叶都有自己的周期性。合并非局部性和周期性概念会导致在同构和非同构的周期性配置之间进行区分。具体地，与非局部性的存在相关的各种非常规现象导致这种系统中能量传输的中断，从而揭示了用于振动隔离应用的新机会。为了证明这些效果，使用了简单的平面延伸/压缩波在配备有同轴弹簧状元件的均匀无限杆中传播的简单模型。通过直接的双重，时空，傅立叶变换来分析同质情况，从而导致讨论不寻常的色散效应，包括消失和负群速度。在非均匀情况下，规范的Floquet理论用于识别阻带并控制其在频域中的位置。将结果与单位周期性单元和有限结构的本征频率分析进行比较。接下来，考虑强迫问题，并计算具有非局部弹簧效应的半无限杆中的插入损耗，以证实Floquet理论的预测，从而为获得的结果提供物理解释。最后，强调了采用主动控制格式的非局部相互作用力来生成任意低频下的阻带的可能性。