# Theoretical Optimization of Trapped-Bubble-Based Acoustic Metamaterial Performance
## 1 Introduction  
The concept of the metamaterials with negative properties was proposed about half a century ago by [PERSON] [1]. However, this area remained almost undeveloped [2] up to the early 2000s when the interest was inspired by [PERSON] [3], who developed a theory for left-handed electromagnetic metamaterials. The analogy between electromagnetic and acoustic waves potentially opened new horizons to create left-handed acoustic metamaterials possessing negative mass density and negative elastic modulus. Since that time the area was actively explored and attracted researchers' attention both theoretically and experimentally. Two strategies were employed by researchers to model acoustic metamaterials. One of them was based on mass-spring model with further addition of dashpots and it resulted in obtaining dispersion relation that allowed tuning metamaterials properties varying mechanical parameters (masses, spring constants, and dissipation coefficients). Another approach capitalized on so-called Multiscattering Theory (MST) and Self-Consistent Approximation (SCA) dealing with mean field created by the array of independent scatterers that allowed significant simplification of output signal analysis. One of the key features expected from the metamaterials is a controllable and tunable transmission of signal, namely acoustic waves [4; 5]. The simplest one-dimensional (1D) spring-mass model was proposed by [PERSON] and [PERSON] [6] and verified experimentally by [PERSON] et al. [7]. More specifically, the authors demonstrated the possibility to create the bandgaps and, thus, control the dynamictransmission of the material due to negative effective mass effect. The next step was done by [PERSON] et al. [8]. The authors proposed 1D mass-in-mass-spring model. They built 1D lattice and demonstrated that it could be replaced by monatomic lattice for which an effective mass was introduced and shown to be negative in certain frequency ranges. However, the application of this model was limited, since it only exhibited two complete bandgaps and, thus, was rather more suitable for gaining fundamental understanding of negative effective mass concept then for practical applications in working prototypes. To tackle this problem, [PERSON] and [PERSON] [9] developed multiresonator (dual-resonator) acoustic metamaterial model. The authors demonstrated the presence of several varying bandgaps dependent on driving frequency. Dual-resonator model was experimentally verified by [PERSON] et al. [10]. Moreover, the authors demonstrated that acoustic wave attenuation can be achieved for the wider frequency range as compared to a single-resonator system. However, even this model had a significant disadvantage, namely it did not take into account the dissipation that was inevitable in real materials. To address this issue, [PERSON] and [PERSON] [11] first considered single resonator system containing a dissipative element. The authors covered the most commonly used dissipative models, including Kelvin-Voigt, Maxwell, and Zener. In particular, Kelvin-Voigt oscillator-based metamaterial acts as a low-pass filter exhibiting broad spectrum performance. For the Maxwell oscillator model, the bandgap properties vary from the stopping one for low damping to almost passing for high damping. Finally, the Zener model allows tailoring damping and stiffness and, thus, to span the bandgaps of two previous cases. However, as mentioned above, the single-resonator systems have a limited number of bandgaps which might be not suitable for practical applications. Thus, the dissipative multiresonator (dual-resonator) system was developed by [PERSON] et al. [12]. More specifically, three dashpots were introduced in each unit cell that allowed enhancing the attenuation and absorption by the metamaterial. Moreover, this model also allowed the merging of multiple bandgaps observed in non-dissipative multiresonator system and, thus, come up with the broadband wave mitigation region.