## 1 Introduction
When the damping of the system is negligible, the effective bulk modulus of a Helmholtz resonator can be written as [65]:  
\[\frac{1}{B_{\mathit{eff}}}=\frac{1}{B}(1-\frac{\omega_{\mathit{sh}}^{2}}{ \omega^{2}}) \tag{4}\]  
where \(B\) is the bulk modulus of air, \(\omega_{\mathit{sh}}\) is the cutoff frequency of the side hole, and \(\omega\) is the angular of the harmonic excitation. For the monopolar resonant mode of this AMM, the effective bulk modulus is found to be negative. The monopolar and dipolar resonances of each scattering object are the key to realizing the negative effective modulus and effective mass density, respectively [50]. The negative effective properties indicate an out-of-phase wave response to the applied input; to be specific, it tends to move toward the left with the force to the right when the effective mass density is negative, while it expands with exerted pressure when the effective bulk modulus is negative [86]. When both situations occur simultaneously, a frequency range appeared in which propagation in the AMM is allowed;  
Figure 1: **(a)** Photograph of a local resonant acoustic structure. Adapted with permission from [82]. The American Association for the Advancement of Science, 2000. **(b)** Schematic of periodical daisy-chained Helmholtz resonators. Adapted with permission from [83], Springer Nature, 2006.  
then, the acoustic structure is classified as a double negative acoustic metamaterial. In these negative composite structures, the effective wave velocity \(V_{\textit{eff}}\) is given by [87]:  
\[V_{\textit{eff}}=\sqrt{\frac{B_{\textit{eff}}}{D_{\textit{eff}}}} \tag{5}\]  
In Equation (5), when both negativities occur simultaneously, the terms within the square root are positive and the velocity is real. The acoustic wave is a transmission wave rather than an evanescent wave in single negative structures. Then, a pass band appears in the band structure, which is called a double negative acoustic metamaterial. A one-dimensional structure was first designed and demonstrated experimentally for double negative acoustic metamaterials [88]. The dipole resonance was regulated by the elastic membrane arranged periodically, and the monopolar response was provided through the side opening tube. Researchers found a double negative band at the low frequency limitation; the same configuration was employed for demonstrating the reverse Doppler shift in the double negative zone [68]. Later various attempts were exploited for tuning both the density and the bulk modulus for developing composite acoustic structures with tunable properties. For example, a classical design with double negativities was designed by [PERSON] et al. [89], in which an array of glass bottles are attached to a plastic tube wall, and the membranes are also fixed among the resonators as well. These bottles operate as Helmholtz resonators with negative mass density, and the membranes convey collective acoustic oscillation that is similar to that in plasmonics to achieve negative mass density. The manipulation of acoustic waves has been investigated intensively in last decades [90; 91; 92; 93]. The development of AMMs provides a new platform for the manipulation of the propagation of acoustic waves, including the airborne acoustic wave [94; 95; 96], underwater acoustic wave [97; 98; 99; 100], elastic wave [76; 101; 102], and surface acoustic wave [58]. The theoretical and experimental research of an acoustic cloak [103], acoustic rectification [53], acoustic lens [104; 105], and near-zero index [106] have been carried out. Although a great deal of investigations have been reported, many studies mainly focused on passive metamaterials with constant material properties [107]. The acoustic properties of passive metamaterials are difficult to change once they are fabricated, resulting in a fixed and narrow working band.