# Theoretical Optimization of Trapped-Bubble-Based Acoustic Metamaterial Performance
## 1 Introduction
Soundproofing becomes a challenge at low frequencies and it requires the use of heavy or thick materials. To address this issue, metamaterials have recently been employed that included subwavelength resonant microstructures, namely acoustic bubbles. Free bubbles and bubbles trapped in a bulk of the soft media are known to exhibit low-frequency Minnaert resonances. Thus, bubbly media can be considered as an acoustic metamaterial with the bandgaps associated with bubbles' resonant frequency and small phase velocities at relatively low frequencies (audible range). An array of bubbles exhibits strong scattering properties. When modeling its interaction with an incident acoustic wave, the strong coupling between individual scatterers should be taken into account. This issue was resolved by [PERSON] et al. [13], who proposed to consider all of the bubbles to be affected by the same average field according to \(Fp_{e}=fp_{tot}\), where \(p_{tot}\) is the field acting on all the scatterers, \(p_{e}\) is an incident wave amplitude and \(f\) and \(F\) stand for the scattering functions of an individual scatterer and the bubble array, respectively. Interaction between the bubbles resulted in a shift of the resonant frequency of the system to the higher values according to:  
\[\omega_{0}=\frac{\omega_{M}}{\sqrt{1-\frac{2\sqrt{\pi}a}{d}}}, \tag{1}\]  
where \(a\) and \(d\) stand for the bubbles' radius and distance between the centers of adjacent bubbles, respectively, \(\omega_{M}\) is a Minnaert frequency of a single bubble. From this relation, it can be deduced that \(d>2\sqrt{\pi}a\) for the resonant frequency to be real, which is the essence of SCA. When it comes to the transmission and reflection of acoustic waves by bubble array, one more parameter becomes important, [PERSON]kappa=2\pi/(kd^{2})\), where \(k\) and \(d\) are acoustic wave number and distance between centers of adjacent bubbles. This parameter defines the mechanism of coupling for the bubble array and incident acoustic wave. Moreover, \(\kappa a\) term, which will appear in further considerations is responsible for radiative damping. The next step was done by [PERSON] et al. [14], who considered the case of the bubble metascreen being close to the interface where acoustic impedance of the media changed abruptly. It led to changes in transmission behavior. More specifically, the transmission maximum arrived in addition to preserved transmission minimum. This effect was observed both for the cases with one-side and two-side interfaces. The corresponding resonant frequencies are given by:  
\[\omega_{0}=\frac{\omega_{M}}{\sqrt{1-A+B}},\quad\omega_{0}=\frac{\omega_{M}}{ \sqrt{1-A+B/2}}, \tag{2}\]  
where \(A\) and \(B\) stand for \(A=\frac{2\sqrt{\pi}a}{d}\), \(B=\frac{4\pi a}{d^{2}}\frac{h}{d^{2}}\), respectively, \(h\) is the distance between the bubble's center and adjacent interface. As can be seen from these relations, there are two parameters that allow controlling the amplitude and position of the resonances, namely \(d\) and \(h\). In particular, a decrease in \(d\) caused deepening of the transmission minimum and shifted it to the higher frequencies. In contrast to it, an increase in \(h\) resulted in deeper transmission minimum and shift of the transmission maximum to lower frequencies. Nowadays, the field continues to attract more researchers. Most recent studies considered the absorption of acoustic waves by bubble metascreen [15] and locally resonant metamaterial architectures capitalizing on emerging fabrication techniques [16]. Following the trend, this paper proposes a novel design of the acoustic metamaterial. In contrast to what has been done previously and declared impossibility to generate stable ordered bubbles of the equal size [17], we develop a model to predict oscillatory and dissipative properties for the array of bubbles trapped in cylindrical cavities. We employ an approach proposed and developed in our previous study [18], where the trapped bubble (gas-liquid interface) was treated as a thin clamped circular plate.