# Noise Analysis and Structural Optimization of Automobile Scroll Compressor Air Valve
## 4 Muffler Acoustic Calculation Model  
### Theoretical Studies  
The propagation of acoustic waves in a homogeneous, non-viscous medium is described by the Helmholtz Equation (1) [18]:  
\[\left\{\begin{aligned} \
abla^{2}P+k^{2}P=0\\ k=\frac{\omega}{c}\end{aligned}\right. \tag{1}\]  
where \(\
abla^{2}\) denotes the Laplace operator, \(k\) denotes the wave number, \(\omega\) denotes the angular frequency, and \(c\) and \(P\) denote the sound velocity and sound pressure, respectively. The solution for the region yields its sound pressure.  
The physical structure of the tube and chamber has been precisely delineated in Figure 10, and the acoustic specifications are as follows:  
\[\Delta L=10\lg\left[1+\frac{1}{4}\left(\frac{S_{1}}{S_{2}}-\frac{S_{2}}{S_{1} }\right)^{2}\sin^{2}(kL)\right] \tag{2}\]  
In Equation (2), \(\Delta L\) represents the amount of noise elimination achieved by the expansion chamber, the variables \(S_{1}\) and \(S_{2}\) represent indicate the cross-sectional surface area of the expansion chamber, as well as the cross-sectional surface area at the inlet and outlet zone, respectively, \(L\) represents the length of the expansion chamber, and \(k\) is the wave number.  
The expression for the expansion ratio \(m\) of a resistant muffler is defined as follows:  
\[m=S_{2}/S_{1} \tag{3}\]  
Figure 10: Schematic diagram of the structure of an ideal expansion muffler.  
Figure 9: Comparison between simulation and experimental results.  
The maximum level of silencing performance is achieved when the sine function reaches its maximum value of 1, resulting in superior noise elimination as follows:  
\[\Delta L_{\max}=10\lg\left[1+\frac{1}{4}(m-1/m)^{2}\right] \tag{4}\]  
And Equation (5) is as follows:  
\[kL=(2n-1)\pi/2 \tag{5}\]  
Equation (6) provides a method for calculating the band-in frequency:  
\[k=2\pi f/c \tag{6}\]  
At this point, the silencing frequency of the corresponding muffler is as follows:  
\[f_{\max}=\frac{2n+1}{4}\cdot\frac{c}{L} \tag{7}\]  
When the sinusoidal function attains a minimum value of 0, the muffler ceases to perform its muffler function and the sound wave, thus traversing the expansion section unattenuated. The corresponding frequency at this point is the passing frequency, the expression of which is as follows:  
\[f_{\min}=\frac{n}{2}\frac{c}{L} \tag{8}\]  
By utilizing Equations (7) and (8), the corresponding frequency range of the expansion muffler can be calculated as follows:  
\[\frac{n}{2}\frac{c}{L}\leq f\leq\frac{2n+1}{4}\cdot\frac{c}{L},n=0,1,2,\cdot \cdot\cdot\cdot\cdot. \tag{9}\]  
It is evident that the length of the expansion chamber \(L\) is a multiple of an odd number corresponding to the given frequency 1/4 wavelength.  
According to the established plane wave theory, the appropriate calculation frequency for noise reduction can be determined.  
\[f_{c}=1.8\frac{c}{\pi d_{0}} \tag{10}\]  
Furthermore, it is clear from Equation (2) that regardless of whether the pipe section is dilated or constricted, so long as the dilation ratio, m, remains identical, the effectiveness of noise mitigation throughout the structure will remain the same, with the primary determinant of the predominant frequency of noise reduction being the \(L\) dimension of the expansion muffler.