## 2 Mathematical Formulation
### Impedance Matrix Equation
#### 2.1.2 Array of Transducers  
For an array of transducers, depending on where the transducer is in the array, the reaction force of the medium, \(F_{r}\), has different characteristics. The force exerted on the \(i\)th transducer, \(F_{r,i}\), by the pressures from all the transducers is  
\[F_{r,i}=\sum\limits_{k=1}^{n}\iint\limits_{S}p_{k}(\overrightarrow{r}_{ik})dS _{i} \tag{14}\]where \(n\) is the number of transducers that make up the array, \(S_{i}\) is the wet surface of the \(i\)th transducer head, \(p_{k}(\overset{\rightarrow}{r}_{ik})\) is the pressure produced by the \(k\)th transducer at the designated point on \(i\)th transducer, and \(r_{ik}\) is the distance from the center of the \(k\)th transducer to the designated point. This force, \(F_{r,i}\), can be expressed in terms of the radiation impedance, \(Z_{r,i}\), of the \(i\)th transducer by dividing by the velocity of that transducer head, \(u_{1,i}\), as was done for a single transducer in Equation (10):  
\[Z_{r,i}=\frac{F_{r,i}}{u_{1,i}}=\frac{1}{u_{1,i}}\sum\limits_{k=1}^{n}\iint \limits_{S}p_{k}(\overset{\rightarrow}{r}_{ik})dS_{i}=\frac{1}{u_{1,i}}\sum \limits_{k=1}^{n}u_{1,i}Z_{r,ik}\qquad\text{where}\;\;Z_{r,ik}=\frac{1}{u_{1,i} }\iint\limits_{S}p_{k}(\overset{\rightarrow}{r}_{ik})dS_{i} \tag{15}\]  
Here, \(Z_{r,ik}\) is defined as the mutual radiation impedance between the \(i\)th and \(k\)th transducer, and \(u_{1,k}\) is the velocity of \(k\)th transducer head. Equation (15) shows that the coupling between transducers occurs in the acoustic region and can be expressed as mutual radiation impedance. Generally, the transducers that make up an array are all the same transducers, so when constructing the impedance matrix of an array, all transducers have the same electrical and mechanical impedance. Thus, the impedance matrix equation of the \(i\)th transducer can be expressed as  
\[\mathbf{Z_{i}u_{i}=F_{i}} \tag{16}\]  
where \(\mathbf{Z_{i}}\) is the impedance matrix of the \(i\)th transducer, \(\mathbf{u_{i}}\) is a vector of velocities and current of the \(i\)th transducer, and \(\mathbf{F_{i}}\) is a vector of the forces and voltage of the \(i\)th transducer. They can be written as  
\[\mathbf{Z_{i}}=\left[ \begin{array}{cccc}Z_{r,i}+Z_{1}+_{2}Z_{c}+_{4}Z_{c}+N ^{2}Z_{0}&-_{2}Z_{b}-N^{2}Z_{0}&-_{4}Z_{b}&-NZ_{0}\\ -_{2}Z_{b}-N^{2}Z_{0}&{}_{2}Z_{c}+_{3}Z_{c}+N^{2}Z_{0}&-_{3}Z_{b}+NZ_{0}\\ -_{4}Z_{b}&-_{3}Z_{b}&{}_{3}Z_{c}+_{4}Z_{c}&0\\ -NZ_{0}&+NZ_{0}&0&Z_{0}\end{array} \right] \tag{17}\]  
\[\mathbf{u_{i}}=\left[ \begin{array}{cccc}u_{1,i}&u_{2,i}&u_{3,i}&I&i\end{array}  \right]^{\mathsf{T}},\qquad\mathbf{F_{i}}=\left[ \begin{array}{cccc}F_{1,i} &F_{2,i}&F_{3,i}&V_{i}\end{array} \right]^{\mathsf{T}} \tag{18}\]  
where \(u_{1,i}\), \(u_{2,i}\) and \(u_{3,i}\) are the velocities, and \(l_{i}\) is the current of the \(i\)th transducer. \(F_{1,i}\), \(F_{2,i}\) and \(F_{3,i}\) are the forces, and \(V_{i}\) is the driving voltage of the \(i\)th transducer. Substituting Equation (15) into \(Z_{r,i}\) of Equation (17), the impedance matrix equation for the array of transducers can be derived as  
\[\begin{bmatrix}\tilde{\mathbf{Z}}_{1}&\cdots&\tilde{\mathbf{Z}}_{1 }&\left[ \begin{array}{c}\mathbf{u}_{1}\\ \vdots\\ \vdots\\ \tilde{\mathbf{Z}}_{1}&\cdots&\tilde{\mathbf{Z}}_{2n}\end{array} \right] \left[ \begin{array}{c}\mathbf{u}_{1}\\ \vdots\\ \mathbf{u}_{n}\end{array} \right]=\left[ \begin{array}{c}\mathbf{F}_{1}\\ \vdots\\ \mathbf{F}_{n}\end{array} \right]&\sigma&\sum\limits_{k=1  
When an elastic structure of arbitrary shape is immersed in the fluid as shown in Figure 3a, the finite element modeling of the structure is given by the following discretized governing equation [34]:  
\[\mathbf{Z_{s}v_{s}=F_{s}-GQSp}\qquad\textit{ where }\ \ \mathbf{Z_{s}=j\omega M+C+\frac{K}{j\omega}} \tag{20}\]  
where \(\mathbf{M}\), \(\mathbf{C}\) and \(\mathbf{K}\) are the mass, damping, and stiffness matrices of the elastic structure obtained by the FEM, respectively. \(\mathbf{v_{s}}\) is a velocity vector corresponding to all DOFs of the structure.