## 1 Introduction  
Transducer arrays are usually applied for ultrasonic imaging [1, 2, 3, 4, 5] as they can detect objects in an acoustic medium by combining the electrical signals from each transducer. Therefore, it is important to precisely predict the array performance while taking into account the multi-physical behavior in the transducer array. FEM has been widely used for analyzing transducer array systems because of its high accuracy and the detailed information it provides for the design of transducer arrays. For example, [PERSON] et al. [6] and [PERSON] et al. [7] used FEM to study the design of low-frequency transducer arrays. Although FEM analysis is useful for final tuning, it is inefficient in the concept design stage because of its high computational complexity. To help with this problem, simpler models of transducer array systems have been studied, such as EC models. The EC model of the transducer array can efficiently predict the multi-physical response of an array using a small matrix equation. There have been many studies on how to represent various types of transducers as EC models [8, 9, 10, 11, 12, 13, 14, 15, 16, 17]. For example, [PERSON] extensively described the EC representation of various types of electromechanical transducers by using lumped parameter systems [8]. Later, he extended the theory and described the EC representation using distributed parameter systems [9]. In addition, the acoustic interaction between the acoustic field and transducer should be considered to apply the EC model to an array of transducers operating underwater, as in the case of sonar. The acoustic interaction effect is often characterized by the acoustic radiation impedance. For the calculation of acoustic radiation impedance, it is typical to use an analytical formulation or a numeric formulation. The analytical formulations have an advantage of computational efficiency. Many researchers have theoretically calculated the acoustic radiation impedance of pistons in an infinite rigid baffle [18; 19; 20; 21; 22; 23; 24; 25; 26; 27; 28; 29; 30]. Then, the underwater transducer array was tried to model using EC model with the acoustic radiation impedance of pistons in an infinite rigid baffle calculated by the theoretical formulation [10; 11; 12]. In many cases of actual sonar arrays, the acoustic radiation impedance usually cannot be obtained analytically as sonar arrays are finite and are sometimes not large compared to the wavelength in water [23]. To analyze the finite transducer arrays using the EC model, studies have used approaches based on Green's function to solve the Helmholtz integral equation [13; 14; 15; 16]. For example, [PERSON] [13] applied this method to an array of circular pistons with a finite planar baffle to calculate the transmission characteristics of a sonar array. [PERSON] [14] applied this method to an array of rectangular pistons with a finite planar baffle to analyze the effects of acoustic interaction on the vibration velocity of transducers. [PERSON] et al. [15] developed a model based on this method to study the acoustic behavior of a capacitive micromachined ultrasonic transducer (CMUT) array. However, the acoustic radiation impedance of an actual array having a complex shape is mostly coupled with the influence of surrounding structures. Actual sonars are affected by the acoustic effects of surrounding structures, such as baffles, enclosing domes, and nearby reflecting surfaces [16]. Exceptionally, [PERSON] [16] proposed an analysis method for array systems that reflects the influence of surrounding structures. He calculated the acoustic radiation impedance using only the Helmholtz integral equation formula, and so the surrounding structures were considered as rigid reflectors. Therefore, the influence of the vibration mode of the surrounding structure cannot be considered. In addition, this approach neglects the effects of sound waves that are transmitted through the acoustic window.