## 1 Introduction
An acoustic band structure with periodic acoustic lattice was proposed in the last decades that has shown novel properties of sound propagation [63; 64; 65; 66; 67; 68; 69; 70; 71; 72]. Although the AMM is composed of periodic structures, the working mechanism of AMM is locally resonant rather than the Bragg scattering in the earlier acoustic band gap material, which is called a phononic crystal (PC) [73; 74; 75]. Properties of AMMs can be described with homogenized parameters, the effective mass density, and the effective modulus [76; 77]. For traditional composite materials, the effective mass density reflects the inertial characteristic of a material. For a compound containing two components, in general, the static mass density \(\rho_{\mathit{eff}}\) can be modeled as the volume average of the density of the two components [78]:  
\[\rho_{\mathit{eff}}=\rho_{1}f+\rho_{2}(1-f) \tag{1}\]  
where \(f\) means the volume percentage of component 1, and \(\rho_{1}\) and \(\rho_{2}\) are the mass density of component 1 and 2, respectively. However, for an artificial composite material with local resonators, there will be a relative motion between the matrix and the resonators if the displacement between them is not synchronous and even inverse [79]. The effective mass of this system can be acquired by [80; 81]:  
\[D_{\mathit{eff}}V=M_{0}+\frac{m\omega_{0}^{2}}{\omega_{0}^{2}-\omega^{2}} \tag{2}\]  
where \(D_{\mathit{eff}}\) is the effective density, \(V\) is the total volume, \(\omega_{0}\) is the resonant frequency, \(\omega\) is the angular frequency of the time harmonic excitation, and \(M_{0}\) and \(m\) are the mass of the matrix and the solid components, respectively. Therefore, the effective density of the whole structure could be negative at some frequencies The first locally resonant sonic material, which is the prototype of AMMs, was demonstrated by [PERSON] et al. in 2000 [82]. The unit of phononic crystal-like material is made of a solid material core with relatively high density and a soft coating, as shown in Figure 1a. This AMM exhibits dipolar resonant modes at certain frequency ranges where the wavelengths in the bulk medium are at least two orders of the lattice parameter. According to Equation (2), its effective mass density is calculated to be negative, so that the propagating wave becomes an evanescent wave, and low-frequency band gaps appear in the transmission. The gaps of the AMMs can be ascribed to Bragg scattering and local resonance. The Bragg scattering occurs around the frequency where the periodic constant of the acoustic material is of the order of the acoustic wavelength. The band-folding rising from the Bragg scattering can lead to band gaps in which the acoustic waves cannot propagate. Negative dynamic mass will occur when the frequency is close to the resonant frequency of the crystal. The negative mass density is analogous with a mass-spring model in which the matrix can be regarded as a spring, and the core is similar to the mass [81]. Negative modulus can be designed by a class of AMMs consisting of an array of subwavelength-sized Helmholtz resonators, as shown in Figure 1b [83]. The system did not adopt the combination of different materials, but changed the geometry to limit the propagation of the acoustic wave in fluid [84]. The periodic shunted Helmholtz resonators can also be regarded as a simple mass-spring model [85]. Since the volume of the neck is much smaller than the volume of the cavity, it can be approximated that fluid in the neck is incompressible. Therefore, the fluid in the neck can be modeled as mass, while the fluid in the cavity can be considered a spring. In the framework of effective medium theory (EMT), the effective bulk modulus of fluid and solid composite structures \(B_{\mathit{eff}}\) is [78]:  
\[\frac{1}{B_{\mathit{eff}}}=\frac{1-f}{B_{1}}+\frac{f}{B_{2}} \tag{3}\]  
where \(B_{1}\) and \(B_{2}\) refer to the modulus of the matrix and solid components, respectively.