## 2 Acoustic Performance Evaluation Indicators for an Expandable Duct Silencing System  
### Acoustic Performance Evaluation Indicators of Noise Reduction Systems  
The noise reduction objectives that the air conditioning duct noise reduction system must accomplish in different frequency bands are determined by the acoustic performance indicators. Indicators for assessing acoustic performance that are often employed are noise reduction, transmission loss, and insertion loss. Transmission loss can reflect the acoustic performance of silencers and this study mainly evaluates the acoustic performance of silencers from the perspective of transmission loss. 1. Transmission loss caused by cross-sectional mutation  
In perfect circumstances, sound waves will travel an unending distance without interference when they travel through a steady unchanging medium. However, sound transmission will be impeded when the propagation medium shifts or the air duct's cross-section abruptly changes. A portion of the incident sound wave is transmitted, which keeps on traveling through the air duct, while the remaining portion is reflected back, establishing a reflected sound wave, as shown in Figure 1.  
The calculation formula for the acoustic impedance \(Z\) of the air duct is  
\[Z=S\rho u \tag{1}\]  
The formula involves \(S\), representing the air duct's cross-sectional area, its unit is m\({}^{2}\); \(\rho\), denoting the medium's density, its unit is kg/m\({}^{3}\), and \(u\), representing the sound waves' velocity, its unit is m/s. The sound pressure \(p_{tr}\) of the transmitted sound wave is equal to the total of the incident wave sound pressure \(p_{in}\) and the reflected wave sound pressure  
Figure 1: Air ducts with variations in cross-section and medium. at the point where the cross-section of the air duct changes because the pressure is equal on both sides of the cross-section. \[p_{in}+p_{re}=p_{tr} \tag{2}\]  
Furthermore, the volumetric velocity at every location in the air duct is equal when sound waves flow through it. Consequentially, the volume velocities of the two sound waves on the left add up to the volume velocities on the right side of the cross-section. \[S_{1}\frac{p_{in}}{z_{1}}-S_{1}\frac{p_{re}}{z_{1}}=S_{2}\frac{p_{tr}}{z_{2}} \tag{3}\]  
The acoustic impedances of the two mediums are represented in the formula as \(z_{1}=\rho_{1}u_{1}\) and \(z_{2}=\rho_{2}u_{2}\). Equation (2) may be combined to modify the previous equation to  
\[\frac{z_{1}(p_{in}+p_{re})}{S_{1}(p_{in}-p_{re})}=\frac{z_{2}}{S_{2}} \tag{4}\]  
The ratio of the sound pressure amplitude \(p_{re}\) of the incoming wave to the amplitude \(p_{in}\) of the reflected wave is known as the reflection coefficient \(R\). When we replace it in the previous equation, we receive the following benefits:  
\[R=\frac{p_{re}}{p_{in}}=\frac{S_{1}z_{2}-S_{2}z_{1}}{S_{1}z_{2}+S_{2}z_{1}} \tag{5}\]  
The ratio of the incoming wave's sound pressure amplitude \(p_{in}\) to the transmitted wave's amplitude \(p_{tr}\) is known as the transmission coefficient (\(T\)). It may be produced by inserting it into Equations (2) and (4) as follows:  
\[T=\frac{p_{tr}}{p_{in}}=\frac{2S_{1}z_{2}}{S_{1}z_{2}+S_{2}z_{1}} \tag{6}\]  
The ratio of the incident sound power \(W_{in}\) to the reflected sound power \(W_{re}\) is commonly referred to as the sound power's reflection coefficient, or \(R_{w}\). \[R_{w}=\frac{W_{re}}{W_{in}}=\frac{S_{1}p_{re}u_{re}}{S_{1}p_{in}u_{in}}=R^{2} \tag{7}\]  
The ratio of the incident sound power \(W_{in}\) to the transmitted sound power \(W_{tr}\) is known as the sound power transfer coefficient, or \(T_{w}\). \[T_{w}=\frac{W_{tr}}{W_{in}}=\frac{S_{2}p_{tr}u_{tr}}{S_{1}p_{in}u_{in}}=\frac {S_{2}z_{1}}{S_{1}z_{2}}T^{2} \tag{8}\]  
\(R_{w}+T_{w}=1\) Respect the principle of energy conservation.