## 2 Acoustic Performance Evaluation Indicators for an Expandable Duct Silencing System
### Acoustic Performance Evaluation Indicators of Noise Reduction Systems
The transmission loss may be computed using the following formula in the absence of a reflected sound wave at the muffler's outlet:  
\[Z=S\rho u \tag{1}\]0  
The incident sound power level and incident sound power at the air duct muffler's inlet are represented by the formula's \(L_{W_{in}}\) and \(W_{in}\) their units are dB and W, respectively, while the transmitted sound power level and transmitted sound power at the air duct muffler's outlet are represented by \(L_{W_{tr}}\) and \(W_{tr}\), respectively. Equation (8) may be substituted to determine that  
\[Z=S\rho u \tag{1}\]1  
It is undisputed that the air duct silencing system's transmission loss is independent of the noise source and only depends on the muffler's structural characteristics. In general, insertion loss is favored as the silencing system's acoustic performance assessment index when assessing its acoustic performance. Although the insertion loss measuring technique is rather straightforward, it presents some challenges for simulation prediction since it necessitates a thorough analysis of the impacts of the sound radiation impedance at the outlet, air duct attenuation impedance, and sound source impedance. Studying the muffler's noise reduction effect can provide an approximation of the acoustic performance of the duct muffler system, as the transmission loss is mostly connected to the muffler's structural features. The insertion loss is equal to the transmission loss when the noise source and the air duct silencing system's output are both non-reflective sound waves. The transmission loss and noise reduction level of the air duct muffler are equivalent when the acoustic impedance of the inlet and outlet ends are the same. Transmission loss of expansion muffler based on plane waves  
An example of a resistance muffler is an expansion chamber air duct muffler, which has a primary chamber and air ducts attached to both sides of it. Its effortless processing and straightforward structure are among its qualities. The expansion chamber's \(S_{2}\) cross-sectional area is greater than the air duct muffler's \(S_{1}\) and \(S_{3}\) intake and outlet ends combined. With the help of this function, the expansion chamber duct muffler reduces noise by reflecting and interfering with sound waves by expanding and contracting the duct section, as shown in \[Z=S\rho u \tag{1}\]2  
The expansion chamber duct muffler's one-dimensional transmission loss (\(TL\)) may be computed using the following formula:  
\[Z=S\rho u \tag{1}\]3  
The expansion ratio is given by the formula \(m=S_{2}/S_{1}\), while the wave number, which is based on the sound wave's frequency, is determined by \(k=2\pi/\lambda\). Figure 3 depicts the transmission loss curve with frequency variation when the expansion chamber duct muffler's length \(L\) is 20 cm. From the graph, it is clear that the expansion chamber duct muffler's transmission loss curve exhibits periodic variations. As frequency increases, the transmission loss also exhibits periodic oscillations between 0 and the maximum value. Furthermore, the amplitude of transmission loss is proportional to the expansion ratio \(m\) while the expansion chamber's length remains constant. The expansion chamber muffler's noise reduction increases with increasing \(m\). Reducing the cross-sectional size of the air duct \(S_{1}\) or raising the cross-sectional area of the expansion chamber \(S_{2}\) are two ways to raise the expansion ratio \(m\). As demonstrated by Figure 4, the frequency corresponding to the amplitude of the transmission loss will fluctuate while the expansion ratio m is constant. The amplitude of the transmission loss stays constant. \[Z=S\rho u \tag{1}\]4 Diagram showing the expansion chamber muffler structure.