## 4 Parametric Analysis
### Effects of the Unit Cell Square Hollow Size  
The square hollow size is also one of the significant factors affecting the bandgap properties of the designed piezoelectric phononic crystal. Therefore, this subsection describes the analysis of these effects on the characteristics of multi-bandgaps caused by the change of the square hollow side length, and the results are exhibited in Table 3. It can be seen from Table 3 that four bandgaps are generated when the square hollow side length increases from 10 mm to 50 mm.  
Figure 12: Effects of thickness on the upper and lower boundaries of each bandgap.  
To better describe the relationship between the change of the square hollow side length and the bandgaps, the bandgap frequency and bandgap width of the piezoelectric phononic crystal are considered simultaneously. Therefore, the transformation is carried out with the following formula:  
\[\alpha_{r}=\frac{\beta_{r}}{\gamma_{r}}, \tag{37}\]  
where \(\alpha_{r}\) is the relative bandwidth of the bandgaps, \(\beta_{r}\) and \(\gamma_{r}\) are defined by the following equations:  
\[\beta_{r}=g_{h}-g_{l}, \tag{38}\]  
\[\gamma_{r}=\frac{g_{h}+g_{l}}{2}. \tag{39}\]  
In Equations (38) and (39), \(g_{h}\) and \(g_{l}\) represent the upper and lower boundaries of the multi-bandgaps, respectively.  
As indicated in Figure 13, the properties of each bandgap vary uniquely as the side length of the square hollow increases. The first bandgap gradually declines with the increase of \(x_{1}\) and the maximum value is 0.53, while the minimum value of the first bandgap is 0.08. The second bandgap shows an entirely opposite trend to the first bandgap. The minimum value for the second bandgap is 0.26, whereas the maximum value is 0.35.  
The variation trend of the third bandgap is different from those of the first two bandgaps. Specifically, with the gradually increase of \(x_{1}\), \(\alpha_{r}\) changes slightly around 0.1. For the fourth bandgap, when \(x_{1}\) is 10 mm, the maximum value is achieved, which is 0.3, and the minimum value of \(\alpha_{r}\) is 0.03 when \(x_{1}\) increases to 20 mm.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \(x_{1}\) (mm) & **1 st Gap** & **2 nd Gap** & **3 rd Gap** & **4 th Gap** \\ \hline
10 & [2260.10, 3886.33] & [4066.74, 5283.68] & [6775.12, 7554.94] & [11091.72, 15008.91] \\ \hline
20 & [2261.77, 3667.03] & [3850.89, 5077.89] & [6522.89, 7234.90] & [7686.61, 7947.03] \\ \hline
30 & [2268.89, 3356.36] & [3547.67, 4795.41] & [6204.55, 6894.88] & [7345.23, 7877.57] \\ \hline
40 & [2345.95, 2992.60] & [3196.86, 4456.71] & [5852.81, 6560.45] & [7046.26, 7718.97] \\ \hline
50 & [2457.48, 2649.43] & [2870.53, 4077.25] & [5465.80, 6213.06] & [8238.10, 8676.57] \\ \hline \hline \end{tabular}
\end{table}
Table 3: Multiple bandgaps with different values of the square hollow side length \(x_{1}\).  
Figure 13: Influences of square hollow side length on the relative bandwidth.  
In summary, more attention should be paid to the first and second bandgaps, because they are more valuable than the other bandgaps. Specifically, the frequency ranges of the first and second bandgaps should be lower and wider. To achieve the above results, it is necessary to increase the thickness of the piezoelectric phononic crystal, and the side length of the square hollow should be as small as possible.