## 1 Introduction  
At present, underwater vehicles such as submarines and unmanned underwater vehicles (UUVs) are being widely used in national defense, marine development, and other fields. They are playing an increasingly important role in underwater attack, underwater defense, underwater detection, and underwater rescue. To evaluate their acoustic stealth performance and the influence of self-noise on the detection performance of local sonar, real-time and accurate prediction of radiated noise from underwater vehicles is increasingly needed. Therefore, predicting radiated noise from underwater structures has always been a concern of scholars [1; 2; 3; 4]. Currently, the common method of predicting radiated sound from underwater structures is the prediction method using monitoring structure vibration. Another difficult problem for scholars is how to effectively reduce the number of measuring points used in the prediction and how to improve the efficiency of sound radiation prediction for the convenience of engineering applications. The prediction of radiated sound from a vibrating structure is one of the important research fields in structural acoustics. Several classical and recognized calculation methods have been developed, such as the boundary element method (BEM) [1], the finite element method (FEM) [5], a method of combining BEM and FEM (FEM-BEM) [6; 7; 8] and the equivalent source method [9; 10]. However, these conventional approaches have their own disadvantages. The FEM and the FEM-BEM methods require a known excitation force, which is often difficult to obtain in practical engineering applications. The BEM and the equivalent source method only need to know the surface vibration of structures, but they both require the vibration information of many measuring points on the structure surface. Insufficient measuring points are the key factors that restrict the prediction accuracy in practical applications. Moreover, the prediction efficiency of these two methods is low, due to the processing of many measuring points. Thus, many researchers have made other attempts to find a better way to predict the radiated sound field. To improve the prediction efficiency, scholars have proposed some prediction methods based on acoustic radiation modes (ARM) [2]. In the course of carrying out research, [PERSON] [11] identified a set of orthonormal boundary velocity patterns through the singular value decomposition, viewed as the original idea of ARM [12; 13]. To reduce the computational cost, many papers have proposed different and simpler algorithms to calculate ARM [14]. Some methods of applying ARM to the calculation of radiated sound power have been also proposed. [PERSON] and [PERSON] [15] obtained two formulations for calculating the radiated sound power based on the ARM and the structural mode, respectively, and they studied the number of sensors. Some new developments have been achieved in the application of ARM. [PERSON] et al. [16] developed a modal decomposition technique to analyze the sound power contributed by the individual mode of a structure submerged in a heavy fluid. [PERSON] et al. [17] proposed a new application for the ARM: computing the sound power of structures with convex shapes based on the mapped ARM and the mapping relationship. In general, the advantage of ARM is that it improves the prediction efficiency by applying the theory of modes. However, the eigenmode of structural vibration is a structural vibration mode, thereby making it inefficient at decomposing the structure vibration by using ARM. Moreover, no general calculation software currently exists for ARM. In this situation, the calculation of ARM of a general structure is cumbersome. Researchers have also tried other approaches to solve the problem [18; 19]. [PERSON] and [PERSON] [3] pointed out that statistical energy analysis (SEA) can be applied to predict the radiated sound power of a structure in the high-frequency band with high modal density.