## 3 Results and Discussion
### Complex Band Structure
Similarly, the resistor, inductor, and capacitor are organized in parallel to constitute a parallel circuit, and one end of the parallel circuit is coupled to the piezoelectric material as the other end is connected to ground. It is worth noting that the circuits are applied on each unit cell, respectively. In addition, the value of the adopted resistor, inductor, and capacitor is 10 \(\Omega\), 1 H, and 1 C, respectively. The transfer loss calculation formula is as follows:  
\[TL=20\times\log_{10}(\frac{x_{out}}{x_{in}}), \tag{36}\]  
where \(x_{in}\) and \(x_{out}\) represent the displacement of the phononic crystal at the excitation point and the displacement at the picking point, respectively. Figure 8: (**a**) Finite length piezoelectric phononic crystal; (**b**) The employed external circuit. As can be seen from the transmission loss results in Figure 9, whether there is an external circuit or not, within the range of frequency from 0 to 8.5 kHz, there are four attenuation regions of transmission loss, and the frequency range of these attenuation regions is consistent with the frequency range of the bandgaps in Figure 7. Coupling an external series circuit with the finite length piezoelectric phononic crystal reduces the attenuation level of the first three-bandgaps, resulting in their transmission loss curves shifting upward, as shown by the blue dotted line in the figure. Although the decay strength of the transmission losses varies with the structure of the circuit, the attenuation regions in which the bandgaps are located hardly change. Furthermore, whether there is an external circuit or the structure of the external circuit is changed, the width and the attenuation intensity of the fourth bandgap are not affected. Figure 10a,b show the band structure and attenuation constant in the MT direction. The properties of the first bandgap and the second bandgap in Figure 10b are the same as those in Figure 7b. Compared with the first bandgap and the second bandgap, the third bandgap has a stronger attenuation capacity, especially near the lower boundary. Among the four bandgaps, the fourth bandgap has the strongest attenuation ability for the elastic waves, whereas the second bandgap has the weakest decay level. To understand the propagation characteristics of the elastic waves in different directions for the proposed piezoelectric phononic crystal, the attenuation amplitude at different frequencies is calculated. There are three curves in Figure 11 that represent the attenuation curves of 2.3 kHz, 4.5 kHz, and 6.6 kHz, and the sweep range is 0-360\({}^{\circ}\). Since 360\({}^{\circ}\) coincides with 0\({}^{\circ}\), it is not specifically marked in the figure and the values from the inner layer to the outer layer (0.0-0.4) represent the attenuation amplitude of the curves. The attenuation curve at 2.3 kHz is a circle and the attenuation amplitude is approximately 0.08, indicating that the attenuation level in any direction is similar at this frequency. The decaying curve at 4.5 kHz is an ellipse with the value of 0.2. The attenuation curve at 6.6 kHz is a square, with the maximum value of 0.39 at 0\({}^{\circ}\), 90\({}^{\circ}\), 180\({}^{\circ}\), and 270\({}^{\circ}\) while the minimum value of 0.33 at 45\({}^{\circ}\), 135\({}^{\circ}\), 225\({}^{\circ}\), and 315\({}^{\circ}\), showing the obvious directivity at the four angles and four edges. Figure 9: Transmission loss of the finite length piezoelectric phononic crystal. It can be found that the attenuation intensity varies with the angle, which indicates the anisotropy of the attenuation. Additionally, as the frequency increases, the level of anisotropy exhibited by the piezoelectric phononic crystal becomes more apparent.