# Combined Attenuation Zones of Combined Layered Periodic Foundations
## 1 Introduction  
In recent years, periodic structures and phononic crystals have generated significant interest among researchers for their physical properties and potential applications in acoustic and elastic vibration isolation and mitigation [1; 2]. Periodic structures possess a unique dynamic property of frequency attenuation zones (AZs) in which the elastic waves of certain frequencies cannot propagate. Inspired by the concept of AZs of periodic structures, PFs (also known as a type of seismic metamaterials) that consist of periodic unit cells have been proposed as a novel base isolation method to protect structures exposed to seismic excitations [3; 4]. Seismic waves with frequencies in the AZs of PFs will be significantly reduced after propagating though the PFs [5]. Based on the dimensions of the periodicity of unit cells, PFs can be divided into LPFs [6; 7], two-dimensional PFs [8; 9; 10; 11] and three-dimensional PFs [12; 13; 14]. Due to the simplicity of configuration, LPFs have been receiving increasing attention in the field of seismic isolation. [PERSON] et al. [6] studied the AZs of LPFs composed of alternating rubber and concrete layers by the transfer matrix method and assessed the effectiveness of the LPFs in seismic isolation through numerical simulations. By taking the thickness of unit cells as 0.4 m, the first AZ from 6.6 to 15.0 Hz was obtained in [6] by tuning the thickness ratio of rubber layers to concrete layers. [PERSON] et al. [15] extended the differential quadrature method to analyze the AZs of LPFs, and designed the first AZ from 2.6 to 4.3 Hz with the thickness of unit cells as 1 m. [PERSON] et al. [16] investigated the seismic responses of a steel structure with LPF by shake table tests. The results in [16] show that the LPF can significantly reduce the seismic responses of the upper structure when the main frequencies of seismic waves are in the AZs of the LPF. [PERSON] et al. [17] proposed using discrete rubber blocks instead of rubber layers in LPF to obtain AZs with a lower startingfrequency, where the first AZ from 2.2 to 15.0 Hz was obtained with the thickness of unit cells being 0.4 m. [PERSON] et al. [18] studied the effect of initial stress caused by the self-weight of superstructures on the AZs of LPFs, and found that the starting frequencies of the AZs of LPFs are lower but their widths of them are narrower due to the compressive initial stress. [PERSON] et al. [19] considered the effects of material damping on the dispersion properties of damped LPFs by modeling rubber material as linear viscoelastic hysteretic material. [PERSON] et al. [20] and [PERSON] et al. [21] conducted the global sensitivity analyses of the AZs of LPFs by the Monte Carlo simulation-based and the Gauss-Lobatto integration-based Sobol' decomposition method, respectively. The main frequency components of seismic waves span the range of 1-20 Hz [22], which requires AZs with both low starting frequencies and large bandwidths. The previous studies on LPFs mainly concentrated on those composed of identical unit cells. However, in such cases, the starting frequencies and bandwidths of AZs simultaneously decrease as the thickness of unit cells increases, and vice versa [15; 16; 17; 18]. Thus, it is difficult to design AZs with both low starting frequencies and large bandwidths for traditional LPFs with identical unit cells. In order to solve the above problems, in the present paper, CLPFs with two traditional LPFs with different unit cells in tandem are proposed and the generation mechanism of combined attenuation zones (CAZs) is examined. The remaining sections are organized as follows: the computational methods of dispersion curves for SH waves in traditional LPFs with identical unit cells and frequency response functions of CLPFs are given in Section 2. In Section 3, the effect of the filling ratio on AZs of traditional LPF with identical unit cells is discussed. In addition, the calculation method and generation mechanism of the CAZs of CLPFs are investigated.