## 2 Mathematical Formulation
### Impedance Matrix Equation
#### 2.1.1 Single Transducer
 \tag{5}\]  
where \(C_{0}\) is the clamped capacitance, \(C_{f}\) is the free capacitance, \(k_{33}^{2}\) is the coupling coefficient, \(n_{\text{PZT}}\) is the number of rings in the drive stacks, \(t_{\text{PZT}}\) is the thickness of each ring, \(d_{33}\) is the piezoelectric coefficient, \(A_{\text{PZT}}\) is the cross-sectional area, \(s_{33}^{E}\) is the elastic compliance coefficient, \(\varepsilon_{33}^{T}\) is the permittivity coefficient, and \(N\) is the electromechanical turn ratio. As shown in Figure 2b, the mechanical part of the EC model of a piezoelectric stack is represented using a T-network [12] based on a simple longitudinal bar model. The T-network is one of the ways to express a distributed parameter system as an EC model. The mechanical impedances of piezoelectric element \({}_{2}Z_{c^{\prime}}\)\({}_{2}Z_{b}\) can be written as  
\[ \begin{array}{l}{}_{2}Z_{c}=-\text{j}\rho_{\text{PZT}}c_{\text{PZT}}A_{ \text{PZT}}\cot(k_{\text{PZT}}L_{\text{PZT}})\\ {}_{2}Z_{b}=-\text{j}\rho_{\text{PZT}}c_{\text{PZT}}A_{\text{PZT}}/sin(k_{ \text{PZT}}L_{\text{PZT}})\end{array}  \tag{6}\]  
Figure 2: Piezoelectric stack in the 33 mode: (**a**) Free body diagram. (**b**) Equivalent circuit (EC) model. where \(\rho,\ c,\ A,\) and \(L\) are the density, wave velocity, cross-sectional area, and longitudinal length of the part, respectively. \(k\) is the wave number. The subscripts PZT denote the piezoelectric stack. The tail mass and stress rod have the same form of T-network EC model as the PZT stack, except that there is no electrical part. Thus, the following equations can be obtained in a similar way:  
\[ \begin{array}{l}F_{2,\text{Tail}}=+_{3}Z_{c}u_{2}-_{3}Z_{b}u_{3}\\ F_{3,\text{Tail}}=-_{3}Z_{b}u_{2}+_{3}Z_{c}u_{3}\\ F_{1,\text{st}}=+_{4}Z_{c}u_{1}-_{4}Z_{b}u_{3}\\ F_{3,\text{st}}=-_{4}Z_{b}u_{1}+_{4}Z_{c}u_{3}\end{array}  \tag{7}\]  
The mechanical impedances of tail mass \({}_{3}Z_{c},\ {}_{3}Z_{b}\) and the mechanical impedances of stress rod \({}_{4}Z_{c},\ {}_{4}Z_{b}\) can be written as  
\[ \begin{array}{l}{}_{3}Z_{c}=-\text{i}\rho_{\text{Tail}}A_{\text{Tail}}\cot( k_{\text{Tail}}L_{\text{Tail}})\\ {}_{3}Z_{b}=-\text{i}\rho_{\text{Tail}}A_{\text{Tail}}/\sin(k_{\text{Tail}}L_{ \text{Tail}})\\ {}_{4}Z_{c}=-\text{j}\rho_{\text{st}}\varsigma_{\text{st}}A_{\text{st}}\cot( k_{\text{st}}L_{\text{st}})\\ {}_{4}Z_{b}=-\text{j}\rho_{\text{st}}\varsigma_{\text{st}}A_{\text{st}}/\sin (k_{\text{st}}L_{\text{st}})\end{array}  \tag{8}\]  
Here, the subscripts Tail, and st denote the parts of the transducer. The reaction force of the medium to the motion of the head surface \(F_{r}\) is  
\[F_{r}=\iint\limits_{S}p(\overrightarrow{r})dS \tag{9}\]  
where \(S\) is the wet surface (the radiating surface) of the transducer head, \(\overrightarrow{r}\) is the position vector on the radiating surface, and \(p(\overrightarrow{r})\) is the pressure produced by the transducer head. For a single transducer, \(F_{r}\) can be expressed in terms of the self-radiation impedance \(Z_{r}\) by dividing by the velocity of the head \(u_{1}\):  
\[Z_{r}=\frac{F_{r}}{u_{1}}\qquad\textit{where}\ \ Z_{r}=\frac{1}{u_{1}}\iint \limits_{S}p(\overrightarrow{r})dS \tag{10}\]  
By using the Equations (1), (2), (4), (7) and (10), the impedance matrix equation for a single tonpilz transducer can be expressed as  
\[\textbf{Zu=F} \tag{11}\]  
where \(\textbf{Z}\) is the impedance matrix, \(\textbf{u}\) is a vector of velocities and current of the transducer, and **F** is a vector of the external forces and voltage of the transducer. They can be written as  
\[\textbf{Z}=\left[ \begin{array}{cccc}Z_{r}+Z_{1}+_{2}Z_{c}+_{4}Z_{c}+N^{2}Z_ {0}&-_{2}Z_{b}-N^{2}Z_{0}&-_{4}Z_{b}&-NZ_{0}\\ -_{2}Z_{b}-N^{2}Z_{0}&{}_{2}Z_{c}+_{3}Z_{c}+N^{2}Z_{0}&-_{3}Z_{b}&+NZ_{0}\\ -_{4}Z_{b}&-_{3}Z_{b}&{}_{3}Z_{c}+_{4}Z_{c}&0\\ -N\,Z_{0}&&+NZ_{0}&0&Z_{0}\end{array} \right] \tag{12}\]  
\[\textbf{u}=\left[ \begin{array}{cccc}u_{1}&u_{2}&u_{3}&I\end{array} \right]^{ \text{T}},\qquad\textbf{F}=\left[ \begin{array}{cccc}F_{1}&F_{2}&F_{3}&V\end{array}