## 2 Mathematical Formulation
### Impedance Matrix Equation
#### 2.1.2 Array of Transducers
\(\mathbf{F_{s}}\) is the harmonic load vector for the structure. \(\mathbf{G}\) is a matrix that is used to find nodes on the surface (wet surface) where the elastic structure is in contact with the acoustic medium. \(\mathbf{Q}\) is a transformation matrix that converts the outward unit normal force of the nodes on the wet surface to the force of the node in the fixed coordinate system (the global coordinate system). \(\mathbf{S}\) is a diagonal matrix where the diagonal elements are the areas of each element on the wet surface of the elastic structure. \(\mathbf{p}\) is the surface pressure acting on the radiating surface. In this study, Equation (20) has been developed in the form necessary to obtain the acoustic radiation impedance of an array. When an elastic surrounding structure of arbitrary shape is placed around the radiating surface of the array as shown in Figure 3b, the finite element modeling of the surrounding structure is given by the following discretized governing equation:  
\[\mathbf{Z_{s}v_{s}=F_{s}-G_{2}Q_{2}S_{2}P_{2}-G_{3}Q_{3}S_{3}P_{3}} \tag{21}\]  
where \(\mathbf{p}_{2}(s_{2}\times 1)\) is the surface pressure acting on the radiating surface when the elastic structure and fluid 1 come into contact, as shown in Figure 3b. \(\mathbf{p}_{3}(s_{3}\times 1)\) is the surface pressure acting on the radiating surface when the elastic structure and fluid 2 come into contact. \(\mathbf{S_{2}}(s_{2}\times s_{2})\) and \(\mathbf{S_{3}}(s_{3}\times s_{3})\) are the area matrices of the inner and outer surfaces of the elastic surrounding structure, respectively. \(\mathbf{G}_{2}(s_{0}\times 3s_{2})\) and \(\mathbf{G}_{3}(s_{0}\times 3s_{3})\) are transformation matrixes of the inner and outer surfaces of the elastic surrounding structure, respectively. \(\mathbf{Q}_{2}(3s_{2}\times s_{2})\) and \(\mathbf{Q}_{3}(3s_{3}\times s_{3})\) are transformation matrixes of the inner and outer surfaces of the elastic surrounding structure, respectively. Using this transformation matrix, the velocities \(\mathbf{v_{n2}}(s_{2}\times 1)\) and \(\mathbf{v_{n3}}(s_{2}\times 1)\) in the normal direction of the radiating surface can be expressed as the following equation in the fixed coordinate system:  
\[\mathbf{v_{n2}=Q_{2}^{T}v_{2}\quad,\qquad v_{n3}=Q_{3}^{T}v_{3}} \tag{22}\]  
where \(\mathbf{v}_{2}\) and \(\mathbf{v}_{3}\) are the velocity vectors corresponding to the DOFs of the structure on the inner and outer wet surfaces of the structure, respectively. As these velocity vectors are  
Figure 3: Schematic diagram of a model with fluid-structure interaction: (**a**) an elastic structure immersed in an acoustic medium and (**b**) the radiating surface of the array and its elastic surrounding structure. included in \(\mathbf{v_{s}}\), the velocity vector \(\mathbf{v_{s}}\) and the corresponding harmonic load vector \(\mathbf{F_{s}}\) can be expressed as follows:  
\[\mathbf{v_{s}}\mathbf{=}[\mathbf{v_{2}},\mathbf{v_{in}},\mathbf{v_{3}}]^{\text {T}},\quad\quad\mathbf{F_{s}}\mathbf{=}[\mathbf{F_{2}},\mathbf{F_{in}},\mathbf{ F_{3}}]^{\text{T}} \tag{23}\]  
The subscript \(\mathbf{in}\) denotes that the value is not on the wet surface. \(s_{0},\ s_{1},\ s_{2}\) and \(s_{3}\), indicating the sizes of each matrix, are the DOFs of the structure, the DOF of the fluid on the outer surface of the array, the DOFs of fluid 1 on the inner surface of the surrounding structure, and the DOFs of fluid 2 on the outer surface of the surrounding structure, respectively. Next, fluid 1 between the array and the surrounding structure and fluid 2 outside the surrounding structure satisfy the Helmholtz equation. The radiating surface is divided into boundary elements, and the fluid is modeled and formulated. Using the acoustic BEM, the relation between the surface pressure and the normal surface velocity of the radiating surface is given by the following equation:  
\[\left[ \begin{array}{cc}\mathbf{B_{1,11}}&\mathbf{B_{1,12}}\\ \mathbf{B_{1,21}}&\mathbf{B_{1,22}}\end{array}