## 2 Materials and Methods
### Finite Element Analysis for Transducer Design  
Figure 4 shows the procedure of the numerical simulation for the design optimization of the 1-3 piezoelectric composite transducer. First, the piezoelectric volume fraction exhibiting the maximum coupling coefficient was determined. Next, the thickness of the 1-3 composite was determined to obtain the target operation frequency at the piezoelectric volume fraction. The thickness and the volume fraction, estimated through the analytical simulation, were used as the reference information for the numerical simulation (i.e., FEA).  
Figure 5a shows the finite element model and the boundary conditions. For the simulation, we used the ANSYS Workbench (v. 19.2, Canonsburg, PA, USA), a commercially available software. The 1-3 composite transducer had a piezoelectric volume fraction of 64% so that it could exhibit the maximum electromechanical coupling. It is worth noting that the analytical model cannot explain the influence of the kerf width in the composite material, which must be investigated through FEA. A voltage of 100 V (AC) and ground conditions were applied at the upper and lower surfaces of the piezoelectric rods. The mechanical vibration of the 1-3 composite transducer was transferred to the acoustic model through the fluid-structure interaction condition at the outer surfaces of the transducer. The acoustic media was set as water. The lower surface of the transducer was regarded as having air backing after applying the impedance condition of 500 Ravls [11, 31]. For computation efficiency, a quarter model was utilized for the simulation.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Material** & & **Properties** & **Value** \\ \hline \multirow{8}{*}{PZT-4} & Density & \(\rho\) (kg/m\({}^{3}\)) & 7500 \\ \cline{2-4}  & & \(c_{11}^{\rm E}\) (\(\times 10^{9}\) Pa) & 139.0 \\ \cline{2-4}  & Stiffness under free electric field & \(c_{53}^{\rm E}\) (\(\times 10^{9}\) Pa) & 115.4 \\ \cline{2-4}  & & \(c_{13}^{\rm E}\) (\(\times 10^{9}\) Pa) & 74.3 \\ \cline{2-4}  & & \(c_{15}^{\rm E}\) (\(\times 10^{9}\) Pa) & 25.6 \\ \cline{2-4}  & Piezoelectric coefficient & \(e_{31}\) (C/m\({}^{2}\)) & \(-\)5.2 \\ \cline{2-4}  & & \(e_{33}\) (C/m\({}^{2}\)) & 15.1 \\ \cline{2-4}  & Dielectric permittivity & \(c_{11}^{\rm S}/\varepsilon_{0}\) & 762 \\ \cline{2-4}  & Density & \(e_{33}^{\rm S}/\varepsilon_{0}\) & 663 \\ \hline \multirow{2}{*}{Epoxy} & Density & \(\rho\) (kg/m\({}^{3}\)) & 1100 \\ \cline{2-4}  & Stiffness & \(c_{31}^{\rm p}\) (\(\times 10^{9}\) Pa) & 5.3 \\ \cline{1-1} \cline{2-4}  & & \(c_{33}^{\rm p}\) (\(\times 10^{9}\) Pa) & 3.1 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Material properties in the numerical simulation [14, 29, 37].  
Figure 3: Prediction of the resonance frequency in the thickness vibration mode of the 1-3 piezoelectric composite transducer.  
The pressure and the acoustic intensity outputs, produced by the single-phase and the 1-3 composite transducers, were computed through FEA and were compared with each other at the same frequency conditions for fair comparison (Figure 5b). Once the frequency effect was confirmed, the influence of the lateral size of piezoelectric rods was examined. It is noticeable that the analytical solution was estimated by the overall volumetric ratio of the piezoelectric material in the composite, ignoring the influence of the lateral vibration of the active elements.