## 2 Piezoelectric Control of Acoustic Metamaterials  
According to the converse piezoelectric effect, the mechanical parameters of piezos (PZT) change under voltage, so emphatic research on the local resonant gap and the negative material properties of the AMM is significant to design a new AAMM. Since an active acoustic cloak consisting of an array of fluid cavities separated by piezoelectric boundaries was designed by [PERSON] [107], lots of research combining PZT with AMM has been proposed [114; 115; 116; 117]. A theoretical analysis of one-dimensional AMMs with piezoelectric boundaries has been discussed by [PERSON] et al. [84]. In comparison to the equivalent transmission line theory, lumped parameter models can be employed to approach the acoustic response composite structures. For example, the electrical current, electric potential, the electrical capacitor, and electrical inductor are in analogy with the acoustic volume flow rate, acoustic pressure difference, the acoustic cavity compliance, and the acoustic mass. [PERSON] and [PERSON] demonstrated the experimental research on shunted piezos [118; 119; 120]. A beam is fixed with periodic shunted piezoelectric patches whose electrodes are connected to the [PERSON]'s circuit (shown in Figure 2b) because of its high value of inductance. The equivalent inductance of the circuit is \(L=Z_{1}Z_{3}Z_{4}C_{1}/Z_{2}\), where \(C_{1}\) is the capacitance of the piezoelectric patch. Then, the resonance frequency can be calculated with:  
\[f_{0}=1/2\pi\sqrt{LC_{p}} \tag{6}\]  
where \(C_{p}\) is the equivalent capacitance of this sample, and \(f_{0}\) could be tuned by changing the value of \(Z_{4}\)[119]. Since the effective parameters can be tuned in real time by introducing piezoelectric patches into compound structures, AAMMs with tunable bands and resonant frequencies can be acquired consequently. [PERSON] et al. proposed a series of schemes with acoustic fluid domains and piezoelectric subdomains [121; 122; 61; 123]. A straight acoustic cavity with a rigid wall is coupled from both ends to bimorph diaphrags, as shown in Figure 3a. The periodical structure consists of an array of identical composite units that are fabricated by depositing PZT layers on both surfaces of the Brass disk. The composite cell is sensitive to the effect of actuated voltage, and the stiffness along the propagation path will be tuned. Meanwhile, these two bimorph diaphrags play different roles as sensor and actuator in adjusting the transmission of sound waves (Figure 3b). When an incident acoustic pressure is exerted on the cell, the sensing bimorph will measure the sound signal and submit it to the actuating bimorph by a feedback control circuit with a controller and amplifier. Then, the effective mass density of the cell could be tuned actively by changing the stiffness of the bimorphs via the power amplifier. The subsequent analysis and experimental results were demonstrated by this research group as well [122]. The sample was measured by being mounted on an impedance tube, and it showed homogenized dynamic properties over a frequency bandwidth approaching 3 kHz. In Figure 3c, the PZT layers operating as sensor and actuator can also be used in the sonic acoustic domain by bonding on both sides of a Brass plate [124]. Another AAMM is composed of a square of ultra-thin foil fixed by a rigid boundary; a pair of piezoelectric patches are attached to the center of the foil [55]. The finite element method and an impedance tube were used to numerically and experimentally study sound transmission behaviors, respectively. The sound transmission loss (STL) [125] can be calculated by:  
\[\mathrm{STL}=10\log_{10}(W_{in}/W_{tr}) \tag{7}\]  
Figure 2: (**a**) A unit cell with shunting through an electrical impedance; (**b**) A circuit diagram of [PERSON]’s circuit. Reproduced under a Creative Commons CC BY 3.0 Licence [120] Copyright 2011, IOP Publishing. where \(W_{in}\) and \(W_{tr}\) are the energy of incidence and transmission acoustic waves, respectively.