## 1 Introduction  
Rectangular ducts are used in the air-conditioning systems of high-speed trains+ to guide airflow. Some rectangular ducts adopt the oblique baffles (Figure 1) to form the varying cross-sections which improve the performance of the ducts in noise attenuation [1].  
The current studies on the duct acoustics of high-speed train are practiced based on simulation, such as the hybrid method of finite element and statistical energy analysis  
Figure 1: Rectangular ducts used in a Fuxing bullet train.  
(FE-SEA) [1; 2]. Most scholars have focused their research on ducts suitable for any vehicle based on a variety of methods. Both analytical and numerical methods have been widely used in the field of duct acoustics. Generally, the numerical methods, such as the finite element method (FEM) [3; 4; 5; 6], boundary element method (BEM) [7] and computational fluid dynamics (CFD) method [8; 9; 10], are popular for analyzing duct systems with complex geometries. [PERSON] et al. [4] proposed a spectral FEM approach to compute the transfer matrix (TM) of duct systems with arbitrary geometries. [PERSON] et al. [8] proposed the time domain CFD approaches to predict the acoustic performance of the duct systems without and with mean flow.  
On the other hand, the analytical method is more efficient in computation than the numerical methods. Two types of analytical methods have been used to investigate a duct system with varying cross-sections. The Wentzel-Kramers-Brillouin (WKB) method [11; 12; 13; 14; 15] utilizes the high frequency approximation that allows to neglect certain terms in the nonlinear governing equations of the media in a duct. [PERSON] et al. [12] used the WKB approximation to derive the exact solutions for one-dimensional (1D) ducts with area variations in the absence of mean flow. [PERSON] et al. [14; 15] derived a WKB-type solution to the generalized Helmholtz equation in 1D ducts with nonuniform cross-sectional areas and inhomogeneity in mean flow. The WKB approximation is less accurate than solving the full wave equations of the duct. The solutions to the wave equations of 1D ducts with varying cross-sections have been studied for decades [16; 17; 18; 19; 20; 21; 22; 23]. [PERSON] et al. [24] developed the 1D solution to a horn-like rectangular duct at frequencies lower than 250 Hz. However, the three-dimensional (3D) solutions to the wave equations in rectangular ducts with gradient cross-sections (RDGCs), which are more accurate at higher frequencies than the 1D solutions, are rarely seen.  
The objective of this paper is to develop the 3D analytical solutions to the wave equations of the expanding or shrinking RDGCs at frequencies up to 1600 Hz. The derived solutions are used to obtain the TMs and transmission losses (TLs) of the duct systems consisted of RDGCs, which have been validated with the measured results. Lastly, the effects of the RDGC geometries on the acoustic performance and flow efficiency of the duct systems are discussed.  
The organization of this paper is as follows: Section 2 develops the 3D analytical solutions to the wave equations of RDGCs. In Section 3, the TMs of the RDGCs are derived. In Section 4, several duct systems consisting of RDGCs are modeled to obtain the TMs. The TLs and pressure losses of the duct systems with different RDGC geometries are obtained and discussed in Section 5. Finally, the conclusions are presented in Section 6.