## 2 Mathematical Formulation  
### Impedance Matrix Equation  
#### 2.1.1 Single Transducer  
Tonpilz transducers are piezoelectric transducers mainly used for relatively low frequencies and high-power sound emission. Figure 1 shows a free body diagram of a tonpilz transducer. The transducer consists of its four basic components: the piston head mass, piezoelectric stack, tail mass, and stress rod. \(u_{1}\), \(u_{2}\), and \(u_{3}\) are the velocities, and \(I\) is the current of a transducer. \(F_{1}\), \(F_{2}\), and \(F_{3}\) are the forces, and \(V\) is the driving voltage. Here, each force can be written as  
\[ \begin{array}{l}F_{1}=F_{r}+F_{1,\text{Head}}+F_{1,\text{PZT}}+F_{1,\text{st }}\\ F_{2}=F_{2,\text{PZT}}+F_{2,\text{Tail}}\\ F_{3}=F_{3,\text{Tail}}+F_{3,\text{st}}\end{array}  \tag{1}\]  
where \(F_{r}\) is the reaction force of the medium to the motion of the wet surface (the radiating surface) of the head mass, and \(F_{1,\text{Head}}\) is the inertia force of the head mass. \(F_{1,\text{PZT}}\) and \(F_{2,\text{PZT}}\) are the forces exerted by the piezoelectric stack on the top and bottom of the piezoelectric stack, respectively. \(F_{1,\text{st}}\) and \(F_{3,\text{st}}\) are the forces exerted by the stress rod on the top and bottom of the stress rod, respectively. \(F_{2,\text{Tail}}\) and \(F_{3,\text{Tail}}\) are the forces exerted by the tail mass on the top and bottom of the tail mass, respectively. The transducer's head mass is assumed to be rigid because it needs to provide uniform longitudinal motion, and its first flexural resonance should be significantly above the operating frequency band. Thus, the force \(F_{1,\text{Head}}\) and the mechanical impedance of the head mass \(Z_{1}\) are expressed using the Lumped Parameter Model (LPM):  
\[F_{1,\text{Head}}=Z_{1}\,u_{1}\ \ \ \ \ \ \ \ \textit{where}\ Z_{1}=\text{j}\omega\,M_{\text{Head}} \tag{2}\]  
where \(\text{j}=\sqrt{-1}\), \(\omega\) is the circular frequency, and \(M_{\text{Head}}\) is the mass of the head. \({}_{2}Z_{c^{\prime}}\,Z_{b^{\prime}}\,3_{c^{\prime}}\,3_{c^{\prime}}\,Z_{b^{ \prime}}\,4_{c}\) and \({}_{4}Z_{b}\) denote the mechanical impedances of the components of the transducer other than the head, obtained by the T-network. The right subscripts \(b\) and \(c\) indicate the shunt arm impedance and the sum of the shunt arm impedance and series arm impedance of the T-network, respectively. The left subscripts 2, 3, and 4 denote the components of the transducer: piezoelectric elements, tail, and stress rod, respectively. Figure 1: Free body diagram of a tonpilz transducer. The free body diagram and EC model of a piezoelectric stack are shown in Figure 2. From the EC model shown in Figure 2b, the following equations can be obtained. \[ \begin{array}{l}I\quad=N(u_{1}-u_{2})+(1/Z_{0})V\\ F_{1,\text{PZT}}=+_{2}Z_{c}u_{1}-_{2}Z_{b}u_{2}-NV\\ F_{2,\text{PZT}}=-_{2}Z_{b}u_{1}+_{2}Z_{c}u_{2}+NV\end{array} \qquad\text{ where }\quad_{2}Z_{c}={}_{2}Z_{a}+{}_{2}Z_{b} \tag{3}\]  
Using Equation (3), the following equations can be obtained. \[ \begin{array}{l}F_{1,\text{PZT}}=\left(+_{2}Z_{c}+N^{2}Z_{0}\right)u_{1}+ \left(-_{2}Z_{b}-N^{2}Z_{0}\right)u_{2}-NZ_{0}I\\ F_{2,\text{PZT}}=\left(-_{2}Z_{b}-N^{2}Z_{0}\right)u_{1}+\left(+_{2}Z_{c}+N^{2 }Z_{0}\right)u_{2}+NZ_{0}I\\ V=-NZ_{0}u_{1}+NZ_{0}u_{2}+Z_{0}I\end{array}  \tag{4}\]  
For a segmented 33 bar, the shunt electrical impedance \(Z_{0}\) can be obtained from the geometry and material properties of the PZT (lead zirconate titanate) elements using the LPM:  
\[ \begin{array}{l}Z_{0}=1/\text{j}\omega C_{0}\\ C_{0}=C_{f}(1-k_{33}^{2})\\ C_{f}=n_{PZT}\varepsilon_{33}^{T}A_{PZT}/t_{PZT}\\ k_{33}^{2}=d_{33}^{2}/s_{33}^{E}s_{33}^{T}\\ N=d_{33}A_{PZT}/ts_{33}^{E}\end{array}