# Research on the Impact of a Fluid Field on an Acoustic Field in Herschel-Quincke Tube
## 4 Results and Discussion
### The Pressure Field with Uniform Mean Flow  
The second case is the infinite tube with the uniform mean flow. The Mach number varies from 0.1 to 0.3 and the cross-section ratio between main duct and bypass duct \(\frac{A_{1}}{A_{2}}\) is from 1.0 to 1.8. The acoustic resonance in the HQ tube is calculated in this part. Both the influence of cross-section and the Mach number on the acoustic resonance are considered in this section.  
The distribution of the dynamic acoustic pressure in the tube is plotted. Figure 4a is the pressure field when \(\frac{A_{1}}{A_{2}}=1.5\), \(M=0.2\) at the frequency \(f=0.39\). It can be seen that the resonance mode is concentrated on the main duct area. Figure 4b shows the acoustic resonance of \(f=0.32\) when \(\frac{A_{1}}{A_{2}}=1.2\), \(M=0.3\). In this case, most of the energy is concentrated in the bypass tube. Both the resonances in Figure 4a,b are symmetric about y-axis. The maximum pressure amplitude of downstream and upstream interface is also calculated (see Figure 5).  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Cross-section** & 1.0 & 1.2 & 1.5 & 1.8 \\
**Resonance** & & & & \\
**frequency** & 0.46 & 0.43 & 0.41 & 0.37 \\ \hline \hline \end{tabular}
\end{table}
Table 1: The resonance frequency for different cross-section ratios: \(M=0.0\).  
Figure 3: The acoustic resonance in HQ tube: \(\frac{A_{1}}{A_{2}}=1.2\), \(f=0.43\), \(M=0.0\).  
Figure 5 shows the maximum pressure amplitude for different Mach numbers with the cross-section ratio \(\frac{A_{1}}{A_{2}}=1.5\). The picture Figure 4(a) depicts that the maximum pressure appears at the frequency \(f=0.41\) when the Mach number \(M=0.0\). Figure 4(b) indicates the resonance frequency of \(f=0.39\) and \(M=0.1\). We can see from Figure 4(c) that the peak value of the resonance occurs when \(f=0.39\) and \(M=0.2\). Figure 4(d) reveals the resonance with \(f=0.37\) and \(M=0.3\). These four curb figures depict a phenomenon whereby the resonance frequency decreases with increasing Mach number. Table 2 shows the peak of acoustic pressure with the change in Mach number.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Mach number** & 0.0 & 0.1 & 0.2 & 0.3 \\
**Acoustic pressure** & 2.2 & 2.0 & 3.7 & 3.1 \\ \hline \hline \end{tabular}
\end{table}
Table 2: The resonance frequency for different Mach numbers: \(\frac{A_{1}}{A_{2}}=1.5\).  
Figure 4: The acoustic resonance in HQ tube.  
Figure 5: The maximum pressure amplitude for different Mach numbers.  
Furthermore, the change in resonance frequency with the variation in cross-section and Mach number is also investigated. As can be seen from Figure 6, the two curves show the fluctuation of resonance frequency. The horizontal axis is the cross-section and the vertical axis represents the resonance frequency. The black solid line and black dashed line represent the results where the Mach number is equal to 0.1 and 0.2, respectively. It can be seen that the resonance frequency decreases with the increase in cross-section ratio whether \(M=0.1\) or \(M=0.2\). However, for the definite cross-section ratio, there is no definite relationship between the resonance frequency and Mach number.