## 3 Validation
### Analysis Condition  
This section describes how to determine the input vector \(\mathbf{F_{i}}\) of the impedance matrix equation. It is assumed that no external forces are applied to the array as the example analysis model is a projector array. The conditions are expressed by Equation (36).  
\[\begin{array}{l}F_{1,i}=0\\ F_{2,i}=0\\ F_{3,i}=0\end{array}\qquad\text{where}\ \ i=1,\cdots,96 \tag{36}\]  
Phase shifting the applied voltages is done to steer a projected beam. Normally, about 120\({}^{\circ}\) of the cylindrical surfaces is used for beam steering [36]. Thus, only 8 of the 24 staves are used in this model, which means that 32 of 96 transducers are operated, as shown in Figure 6.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Component** & **Dimensions** & **Symbol** & **Value** \\ \hline \multirow{2}{*}{Piston Head} & Cross-sectional area & \(A_{\text{Head}}\) & 0.04 [m\({}^{2}\)] \\  & Longitudinal length & \(L_{\text{Head}}\) & 0.06 [m] \\ \hline \multirow{4}{*}{PZT Ceramic} & Cross-sectional area & \(A_{\text{PZT}}\) & 3.77 \(\times\) 10\({}^{-3}\) [m\({}^{2}\)] \\  & Longitudinal length & \(L_{\text{PZT}}\) & 0.08 [m] \\  & Number of rings & \(n_{\text{PZT}}\) & 10 \\  & Thickness of each ring & \(t_{\text{PZT}}\) & 8.00 \(\times\) 10\({}^{-3}\) [m] \\ \hline \multirow{2}{*}{Tail Mass} & Cross-sectional area & \(A_{\text{Tail}}\) & 7.12 \(\times\) 10\({}^{-3}\) [m\({}^{2}\)] \\  & Longitudinal length & \(L_{\text{Tail}}\) & 0.192 [m] \\ \hline \multirow{2}{*}{Stress Rod} & Cross-sectional area & \(A_{\text{st}}\) & 7.80 \(\times\) 10\({}^{-5}\) [m\({}^{2}\)] \\  & Longitudinal length & \(L_{\text{st}}\) & 0.272 [m] \\ \hline \multirow{2}{*}{Array} & Height & \(H_{array}\) & 0.88 [m] \\  & Diameter & \(D_{array}\) & 1.68 [m] \\ \hline \multirow{4}{*}{Acoustic Window} & Thickness & \(t_{airindow}\) & 0.025 [m] \\  & & \(a_{window}\) & 1.25 [m] \\ \cline{1-1}  & half-length of the & \(b_{window}\) & 1.25 [m] \\ \cline{1-1}  & principal axes & \(c_{window}\) & 0.75 [m] \\ \hline \hline \end{tabular}
\end{table}
Table 2: Dimensions of the analysis model.  
Figure 6: Phased cylindrical array (gray: the angular sector used for beam steering): (**a**) Top view. (**b**) Side view.  
The phase shifts must be proportional to the perpendicular distance from each transducer to the plane. The beam can also be steered in the elevation angle \(\theta\) by phase shifting. The applied voltage of each transducer has the phase shown in Equation (37).  
\[\begin{array}{l}V_{i}=\sqrt{2}V_{rms}\mathrm{e}^{\mathrm{i}\mathrm{i}\mathrm{ k}d_{i}}\ \left[\mathrm{V}\right]\qquad\text{where }i=1,\cdots,32\\ V_{i}=0\ \left[\mathrm{V}\right]\qquad\text{where }i=33,\cdots,96\end{array} \tag{37}\]  
where \(d_{i}\) is the perpendicular distance from each transducer to the plane, and \(V_{rms}\) is the RMS value of the applied voltage.  
By changing the Equation (19) as following equation, and fill the input force vector, \(\left[\begin{array}{cccc}\mathbf{F}_{1}&\cdots&\mathbf{F}_{n}\end{array} \right]^{\mathrm{T}}\), with Equations (36) and (37), the responses of the array can be obtained from the output vector, \(\left[\begin{array}{cccc}\mathbf{u}_{1}&\cdots&\mathbf{u}_{n}\end{array} \right]^{\mathrm{T}}\).  
\[\left\{\begin{array}{c}\mathbf{u}_{1}\\ \vdots\\ \mathbf{u}_{n}\end{array}\right\}=\left[\begin{array}{cccc}\mathbf{\tilde{Z} }_{11}&\cdots&\mathbf{\tilde{Z}}_{1n}\\ \vdots&\ddots&\vdots\\ \mathbf{\tilde{Z}}_{n1}&\cdots&\mathbf{\tilde{Z}}_{nn}\end{array}\right]^{-1} \left\{\begin{array}{c}\mathbf{F}_{1}\\ \vdots\\ \mathbf{F}_{n}\end{array}\right\} \tag{38}\]