## 3 Results and Discussion
### Complex Band Structure  
There are usually two forms of solutions used to obtain the band structure of phononic crystals: \(\omega(k)\) and \(k(\omega)\). In the form of \(\omega(k)\), the wave vector \(k\) is known and the angular frequency \(\omega\) is derived by solving the eigenvalues of the established characteristic equation. The drawback of the \(\omega(k)\) form is that the wave vector \(k\) can only be a real number, so this form cannot accurately describe all the characteristics of the band structure of phononic crystals [38]. However, the \(k(\omega)\) form does not have the above shortcoming, and the band structure of phononic crystals can also be determined by obtaining the wave vector \(k\) with a given angular frequency \(\omega\). Additionally, the dispersion relationship of the complex wave vector can be derived with the \(k(\omega)\) form to analyze the attenuation characteristics of elastic waves in phononic crystals [39; 40]. Figure 5: Vibration modes of the third and fourth bandgap edges: (**a**) Point E; (**b**) Point F; (**c**) Point G; (**d**) Point H.  
The wave vector dispersion relationship of the piezoelectric phononic crystal in the \(\Gamma\)X direction is shown in Figure 7. In the figure, the ordinate is the frequency, the abscissa \(\Gamma\)X in Figure 7a represents the wave vector, and \(\delta\) in Figure 7b is described by the following equation:  
\[\delta=(\min(|\text{Im}(k_{\text{x}}l)|))/(2\pi). \tag{35}\]  
From the comparison between Figure 7a,b, one can observe that the range of the four broad bandgaps in Figure 7a coincides perfectly with the amplitude attenuation interval in Figure 7b. In addition, compared with Figure 7a, the calculated results of the complex  
Figure 6: Iso-frequency curves of the proposed piezoelectric phononic crystal. Figure 7: (**a**) Band structure in the \(\Gamma\)X direction; (**b**) Attenuation constant in the \(\Gamma\)X direction. wave vector dispersion relation in Figure 7b can also reflect the attenuation intensities of the elastic waves in the bandgaps. The attenuation levels of the elastic waves in the first bandgap and the second bandgap are basically the same. Furthermore, both their attenuation peaks appear near the 7 th band. The third bandgap and the fourth bandgap both exhibit a strong inhibition effect on the propagation of the elastic waves in the piezoelectric phononic crystal. In the \(\Gamma\)X direction, the fourth bandgap has the largest attenuation capacity for the elastic waves, while the lowest attenuation intensity occurs at the second bandgap. To investigate the effects of series and parallel circuits on the \(\Gamma\)X direction bandgaps of the designed piezoelectric phononic crystal, a finite length piezoelectric phononic crystal is established. It should be noted that the external circuit is coupled to the phononic crystal via the piezoelectric material PZT-5H, which is polarized along the \(z\)-axis. The established 4 \(\times\) 8 piezoelectric phononic crystal and employed external circuit are illustrated in Figure 8. In Figure 8a, with the displacement as the variable, a unit force in the negative direction along the \(z\)-axis is applied as the excitation, which is indicated by the green arrow on the left while the displacement picking point is represented by the red arrow on the right. The reason why the pick point is not selected at the boundary of the piezoelectric phononic crystal is mainly to avoid the influence of the finite boundary on the accuracy of the displacement measurement. Moreover, a periodic boundary condition is adopted in the \(y\)-direction of the finite length piezoelectric phononic crystal to observe the wave transmission loss from \(\Gamma\)X direction. For the employed external circuit in Figure 8b, a resistor, inductor and capacitor are related in series to form a series circuit, and the left end of the resistor is coupled to the piezoelectric material while the right end of the capacitor is connected to ground.