## 1 Introduction  
The sound prediction method of underwater structures has been widely concerned by scholars in various countries as it could provide important prior information for self-noise monitoring and active noise controlling of submerged vehicles. Especially, the sound radiation of underwater vehicles does not often satisfy the free-field condition, which means that the reflection sounds from boundaries and scattering sounds could make a significant contribution to the sound radiation, according to the study of [PERSON] [1]. As analytical methods could hardly be applied for complex structures and acoustical environments, several numerical calculation methods have been developed to be applied for the sound prediction in non-free fields, such as the finite element method (FEM), the boundary element method (BEM), the combination of FEM and BEM (FEM-BEM), and the element radiation superposition method (ERSM) [2, 3, 4, 5, 6]. However, the methods have their own disadvantages, which are analyzed sufficiently in the study of [PERSON] [7]. For example, the details of excitation force are required in advance for FEM and FEM-BEM, which is impractical in engineering applications. Though BEM could accomplish the prediction with measurement data, the singular integrals could be an obstacle for fast prediction. Though ERSM has an obvious advantage in computational efficiency, its usage could hardly be flexible in complex acoustic environments. [PERSON] et al. first proposed the equivalent source method (ESM) to calculate the sound radiation of structure [8]. [PERSON] and [PERSON] et al. analyzed the stability of the equivalent source method and the factors affecting the calculation accuracy of ESM and found that ESM not only avoids the problem of singular integral, but also has higher computational efficiency compared with the commonly used numerical methods such as BEM [9; 10]. Using the complex vector radius method, [PERSON] et al. found that the non-uniqueness of the solution of ESM at singular frequencies could be avoided, which broadens the application of ESM [11]. Moreover, by modifying the transfer function, scholars have studied the sound field prediction of structures in the non-free field, and obtained many significant conclusions [12; 13; 14]. When ESM is used to predict the sound field of structure, the input is often chosen to be the vibration of the structure surface. In the study of [15], [PERSON] combined ESM with FEM to solve the difficulty of obtaining the vibration of the structure in a complex environment, and the sound prediction of an elastic cylindrical shell in idealized waveguide is completed. However, there was no further study on the measurement point arrangement in the above research. Most previous studies chose uniform measurement point arrangement for convenience. The denser the measurement points, the higher the prediction accuracy and the computational cost [7]. In the studies of [PERSON] and [PERSON] et al., they found that the measurement point arrangements are closely linked with the structural vibration mode (SVM) [7; 16]. [PERSON] proposed a method based on the dominantly radiated structural mode (DRSM) to calculate the sound power of a cylindrical shell [17]. As the mutual coupling among the modes at low frequencies could be ignored, the number of measurement points could be reduced. However, the method is only efficient in predicting the sound power of the structure. Meanwhile, for a structure with a complex shape or big size, the structural vibration mode could be quite complex, which limits the application of SVM. [PERSON] and [PERSON] put forward the concept of the acoustic radiation mode (ARM) [18; 19]. From the perspective of sound power, the sound pressure in the sound field is expressed as a set of orthogonal basis functions. Each order of ARM is independent, without coupling among SVMs. Therefore, the control and calculation of sound power could be very simple.