Periodic structures can be designed to exhibit elastic wave propagation band gap behaviour by varying material or geometrical properties, i.e. phononic crystals, or by periodically distributed resonators or boundary conditions, i.e. acoustic metamaterials, with various applications in passive noise and vibration control. However, variability in the manufacturing process causes material and geometry uncertainties that affect their band gap robustness and consequently their dynamic attenuation performance. In this work, the effects of slowly varying spatial properties on the vibration suppression performance of metamaterials and phononic crystals are investigated. The spectral element and the wave and finite element approaches are used for modelling the unit cells such that a wave-like interpretation can be derived for nearly-periodic structures. A beam with evenly spaced attached resonators and an undulating beam are analysed. In both cases, the band gap formation is investigated considering both non-uniform deterministic and spatially stochastic material and geometric variability. The proposed approach provides a framework to represent variability and randomness with spatial correlation of the periodic unit cell and then to assess their effects on the vibration suppression performance. It is shown that even a slowly varying spatial profile, or the correlation length in the case of random fields, plays a role on the band gap performance and that the presence of a critical section, i.e. a transition region between propagating and non-propagating waves, can significantly affect the band gap width and the amplitude of vibration attenuation. Moreover, it is shown the slowly varying approach is suitable to represent the ensemble statistics of band gaps, even considering the occurrence of such critical sections.

周期性结构可以设计为通过改变材料或几何特性（即声子晶体）或通过周期性分布的谐振器或边界条件（即声学超材料）来表现弹性波传播的带隙行为，并在无源噪声和振动控制中有多种应用。但是，制造过程中的可变性会导致材料和几何形状的不确定性，从而影响其带隙的稳健性，进而影响其动态衰减性能。在这项工作中，研究了缓慢变化的空间特性对超材料和声子晶体的振动抑制性​​能的影响。频谱元素和波及有限元方法用于对单位单元进行建模，以便可以为近似周期的结构得出类似波的解释。分析了具有均匀间隔的谐振器的光束和起伏的光束。在这两种情况下，都考虑了非均匀确定性和空间随机材料以及几何变异性，研究了带隙的形成。所提出的方法提供了一个框架来表示具有周期性单位单元的空间相关性的可变性和随机性，然后评估它们对振动抑制性​​能的影响。结果表明，即使是缓慢变化的空间轮廓，或者在随机场的情况下，相关长度也会对带隙性能产生影响，并且还会出现临界截面，即传播波与非传播波之间的过渡区域，会严重影响带隙宽度和振动衰减幅度。而且，