## 1 Introduction
It effectively addresses the issues of excessive axial size and limited sound power, ultimately improving the penetration distance of the seismic waves through the subsurface strata between wells. Figure 1: Schematic diagram of crosswell acoustic seismic source. (**a**) the traditional structure; (**b**) the new structure. In practical applications, the outer diameter of the metal substrate is often larger than the outer diameter of the piezoelectric ceramic. This size difference allows for easy mounting of the trilaminar bending ring onto the liquid cavity. The resonance frequency of the bending ring transducer plays a crucial role in driving the vibration of the liquid cavity within the piezoelectric circular tube. When the resonance frequency of the bending ring transducer is closer to the resonance frequency of the liquid cavity, it results in a stronger amplification effect of the radiation sound field [12]. While several studies have investigated bending disc or ring transducers [13; 14; 15; 16; 17; 18; 19; 20; 21; 22; 23; 24; 25; 26], most of them focused on deriving the resonant frequency and antiresonant frequency of disc or ring transducers with equal diameters. Consequently, when calculating the resonant frequencies of trilaminar bending ring transducers with unequal diameters, there can be a significant margin of error. Furthermore, previous studies have paid limited attention to the effective electromechanical coupling coefficient of the transducer. However, the effective electromechanical coupling coefficient of the bending ring transducer is closely linked to the amplification effect of the radiation sound field. Optimizing this coefficient is of great importance for the design of the crosswell seismic acoustic source. Therefore, the existing studies have certain limitations. They fail to accurately calculate the resonant frequency of trilaminar bending ring transducers with unequal diameters and do not optimize the effective electromechanical coupling coefficient of this specific model. In this research, we have addressed these limitations and conducted further investigations. These advancements constitute one of the innovative contributions of this study. The aim of this study is to derive an analytical expression for the resonant frequency, antiresonant frequency, and effective electromechanical coupling coefficient of a trilaminar bending ring transducer with unequal diameters. The derivation is based on [PERSON]'s hypotheses and classical electroelasticity theory, while the accuracy of the theoretical derivations is validated through finite element simulation (FEM) and experiment tests. Additionally, this paper optimizes the effective electromechanical coupling coefficients of the first-order bending vibration for the trilaminar bending ring transducer with unequal diameters by adjusting the thickness ratio and radius ratio between the metal substrate and the piezoelectric ceramic under different boundary conditions. Specifically, the study focuses on the variations of the effective electromechanical coupling coefficient with changes in the radius ratio (\(r_{1}/r_{2}\)) under the free inner and fixed outer boundary conditions. Notably, the study presents the theoretical derivations and optimization process and provides a physical explanation for the observed variations.