## 4 Optimization of Effective Electromechanical Coupling Coefficients  
The effective electromechanical coupling coefficient is a crucial factor in evaluating the electromechanical conversion capability of a transducer. Generally, a larger effective electromechanical coupling coefficient indicates a higher conversion capability. Particularly, when used as the driving source for the vibration of the liquid cavity in the piezoelectric circular tube, the effective electromechanical coupling coefficient of the bending ring transducer is closely related to the amplification effect of the radiated sound field. In the development of bending ring transducers, typically, only the first-order bending vibration modes of the transducers are of concern. Therefore, this section aims to optimize the effective electromechanical coupling coefficients (\(k_{\mathit{eff}}\)) of the first-order bending vibration modes of the trilaminar bending ring with unequal diameters under different boundary conditions. The variation law of the \(k_{\mathit{eff}}\) for the radius ratio is analyzed, with particular emphasis on the case of free inner and fixed outer boundary conditions, and the physical mechanism contributing to this variation is investigated. To maintain the overall size of the crosswell seismic acoustic source and facilitate the assembly of the trilaminar bending ring with unequal diameters onto a piezoelectric tube, the total diameter (or diameter of the metal substrate) of the trilaminar bending ring with unequal diameters is kept constant. With the outer radius of the trilaminar bending ring with unequal diameters of 60 mm, the outer radius of the ceramic 50 mm, and the total thickness of 7.5 mm remaining unchanged, the effective electromechanical coupling coefficient (\(k_{\mathit{eff}}\)) of the first-order bending vibration versus thickness ratio (\(h_{\mathit{nn}}/h\)) and radius ratio (\(r_{0}/r_{1}\)) under different boundary conditions are calculated, and the results are shown in Figure 8.  
Figure 8: The \(k_{\mathit{eff}}\) of first-order bending resonance versus thickness ratio (\(h_{\mathit{nn}}/h\)) and radius ratio (\(r_{0}/r_{1}\)) under different boundary conditions. (**a**) the free inner and fixed outer; (**b**) the fixed inner and free outer; (**c**) the free inner and free outer; (**d**) the free inner and supportive outer. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Resonant Frequency**} & **Theoretical Calculation** & **FEM Simulation** & **Experimental** & **Error 1** & **Error 2** \\  & **Value (Hz)** & **Value (Hz)** & **Test Value (Hz)** & & \\ \hline \(f_{71}\) & 375.9 & 374.5 & 376.5 & 0.37\% & 0.16\% \\ \(f_{73}\) & 2049.9 & 2043.7 & 2002.8 & 0.3\% & 2.3\% \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Comparison of theoretical, simulation, and test resonant frequency results of the prototype. The results shown in Figure 8a,c,d demonstrate that the value of \(k_{\mathit{eff}}\) grows along with the decrease in the inner radius \(r_{0}\) under the free inner and fixed outer, the free inner and free outer, and the free inner and supportive outer boundary conditions. Figure 8b shows that under the fixed inner and free outer boundary condition, with the increase in inner radius (\(r_{0}\)), \(k_{\mathit{eff}}\) reaches its peak slowly, then rapidly decreases, and the peak appears at different locations for different thickness ratios.