Efficiency of nearly periodic structures for mitigation of ground vibration

Periodic structures are known to produce passbands and stopbands for propagation of vibration energy within the frequency domain. Sources vibrating harmonically at a frequency within a passband can lead to propagation of energy through propagating modes over long distances. However, sources vibrating at a frequency within a stopband excite only nearfields in the form of attenuating and evanescent modes, and the energy decays with distance. The decay phenomena are due to destructive interference of waves reflected and scattered by interfaces or obstacles placed periodically within or between the repeated cells of the structure. For a truly periodic structure, the vibration level within a stopband goes toward zero after infinitely many repetitions of the cell. For example, employing a two-dimensional model, Andersen [1] found that stopbands for ground vibration in the low-frequency range can be introduced by periodic inclusions or changes to the ground surface geometry. However, for vibration mitigation in the context of real civil-engineering problems related to ground-borne noise from railways, for example, the excitation is not strictly harmonic and a steady state of the response is usually not achieved. Further, only a limited number of repetitions of wave impedance blocks or barriers can be made in practice, and in three dimensions, the inclusions have finite extent in the direction orthogonal to the array. Similarly to the work by Andersen et al. [2], this paper addresses the question whether repeated structures of nearly periodic nature can be used to mitigate vibrations caused by non-stationary sources. For this purpose, wave impedance blocks with finite numbers of repetitions are compared to their truly periodic counterparts. Firstly, a two-dimensional study is conducted with focus on studying the nature of wave modes in a periodic array of wave impeding blocks. Secondly, three-dimensional analysis is performed in the frequency domain, focusing on the insertion loss provided by increasing numbers of repetitions of blocks with different height and embedment. Finally, the insertion loss provided by nearly periodic structures is examined, and the mitigation efficiency of wave-impeding-block arrays is quantified in the case of transient loads.