# Three-Dimensional Acoustic Analysis of a Rectangular Duct with Gradient Cross-Sections in High-Speed Trains: A Theoretical Derivation

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## Abstract

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## Featured Application

**The purpose of this work was to provide a detailed derivation process of the 3D analytical solution and TMs of the RDGCs based on the previous studies on the variable ducts and propose a certain reference for designing and improving the acoustic characteristics of the duct systems used in high-speed trains.**

## Abstract

## 1. Introduction

## 2. 3D Analytical Solutions to the Wave Equations of a RDGC

#### 2.1. 3D Solutions for a Straight Rectangular Duct

#### 2.2. 3D Solutions for a RDGC

## 3. Derivation of the TM for a RDGC

**T**($\left[\begin{array}{cc}A& B\\ C& D\end{array}\right]$) is used to describe the relationship as follows:

**T′**($\left[\begin{array}{cc}{A}^{\prime}& {B}^{\prime}\\ {C}^{\prime}& {D}^{\prime}\end{array}\right]$) of the shrinking RDGC is derived from the

**T**with a negative $\theta $.

## 4. The TMs and TLs of Rectangular Expansion Chambers (RECs)

#### 4.1. The TMs of the RECs with One or Double Baffles

**o**of the baffle coincides with that of the REC. All the RECs in the following sections have the same circular ducts and rectangular chamber as those in Figure 4.

^{U}. The TM (${T}_{2a}$) of the REC with double baffles distributed axially and the TM (${T}_{2t}$) with double baffles distributed transversely are given by

#### 4.2. Geometries of the RECs

#### 4.3. Calculation and Measurement of the TLs for the RECs

## 5. Results

#### 5.1. Experimental Validation of the Calculated Results

#### 5.2. TLs of the RECs

_{b}. Although the Type 2a has more TL peaks, it is worse in performance than the Type 2t at frequencies from 500 Hz to 1100 Hz. Generally, the Type 2t is better in acoustic performance than the other types, especially at frequencies from 600 Hz to 1100 Hz.

#### 5.3. Pressure Losses of the RECs

_{b}increases. In general, Type 2t has better performance in flow efficiency than the other types, especially at small values of θ and l

_{b}.

## 6. Conclusions

_{b}) of the baffle. The REC with double baffles distributed transversely (Type 2t) is better in acoustic performance than the other types at frequencies from 600 Hz to 1100 Hz. On the other hand, although the pressure losses of all types of RECs increase as θ or l

_{b}increases, Type 2t always has a lower pressure loss than other types. In summary, Type 2t generally has good performance in both acoustic attenuation and flow efficiency.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A. Derivation of the A, B, C and D in the TM

## Appendix B. FEM Methodology

^{3}. Then, the outlet was AML property, which could simulate the nonreflecting boundary condition. The inlet acoustic boundary condition was defined as the plane wave with 1 W sound power.

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**Figure 3.**(

**a**) A rectangular duct with expanding sections, (

**b**) a rectangular duct with shrinking sections.

Unit | TM | Annotations |
---|---|---|

I | ${T}_{\mathrm{I}}$ | —made of two uniform ducts and one sudden expansion section |

${II}^{W}$ | ${T}_{II}^{W}$ | —made of an expanding RDGC (II) and a shrinking RDGC (II′) |

III | ${T}_{III}$ | —made of two uniform ducts and one sudden contraction section |

m | 0 | 1 | 2 | |
---|---|---|---|---|

n | ||||

0 | 0 | 857.5 | 1715.0 | |

1 | 1143.3 | 1429.2 | 2061.2 | |

2 | 2286.7 | 2442.2 | 2858.3 |

Type 1 | Case 1-0 | The REC with one baffle | l_{b} = 0.50b, θ = 40° |

Case 1-1 | l_{b} = 0.50b, θ = 20° | ||

Case 1-2 | l_{b} = 0.50b, θ = 60° | ||

Case 1-3 | l_{b} = 0.30b, θ = 40° | ||

Case 1-4 | l_{b} = 0.40b, θ = 40° | ||

Type 2a | Case 2a-0 | The REC with double baffles distributed axially | l_{b} = 0.50b, θ = 40° |

Case 2a-1 | l_{b} = 0.50b, θ = 20° | ||

Case 2a-2 | l_{b} = 0.50b, θ = 60° | ||

Case 2a-3 | l_{b} = 0.30b, θ = 40° | ||

Case 2a-4 | l_{b} = 0.40b, θ = 40° | ||

Type 2t | Case 2t-0 | The REC with double baffles distributed transversely | l_{b} = 0.50b, θ = 40° |

Case 2t-1 | l_{b} = 0.50b, θ = 20° | ||

Case 2t-2 | l_{b} = 0.50b, θ = 60° | ||

Case 2t-3 | l_{b} = 0.30b, θ = 40° | ||

Case 2t-4 | l_{b} = 0.40b, θ = 40° |

Parameters | Values |
---|---|

Temperature | 24.5 °C |

Relative humidity | 29.2% |

Pressure | 101,300 Pa |

Case 1-0 | Case 2a-0 | Case 2t-0 |
---|---|---|

141 Pa | 140.2 Pa | 117.7 Pa |

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**MDPI and ACS Style**

Sun, Y.; Qiu, Y.; Liu, L.; Zheng, X.
Three-Dimensional Acoustic Analysis of a Rectangular Duct with Gradient Cross-Sections in High-Speed Trains: A Theoretical Derivation. *Appl. Sci.* **2022**, *12*, 5307.
https://doi.org/10.3390/app12115307

**AMA Style**

Sun Y, Qiu Y, Liu L, Zheng X.
Three-Dimensional Acoustic Analysis of a Rectangular Duct with Gradient Cross-Sections in High-Speed Trains: A Theoretical Derivation. *Applied Sciences*. 2022; 12(11):5307.
https://doi.org/10.3390/app12115307

**Chicago/Turabian Style**

Sun, Yanhong, Yi Qiu, Lianyun Liu, and Xu Zheng.
2022. "Three-Dimensional Acoustic Analysis of a Rectangular Duct with Gradient Cross-Sections in High-Speed Trains: A Theoretical Derivation" *Applied Sciences* 12, no. 11: 5307.
https://doi.org/10.3390/app12115307