## 3 Results and Discussion
### Analysis of the Effect of Helmholtz Resonators on Micro-Pressure Waves  
#### 3.2.1 Analysis of Helmholtz Resonator Resonance Conditions  
When a high-speed maglev train enters a tunnel, similar to a piston entering a cylinder, the airflow is restricted by the tunnel walls, and the air at rest at the front of the train is violently compressed, resulting in a sudden increase in air pressure and formation of a compression wave that propagates forward along the tunnel at the speed of sound. The initial compression wave data measured in the numerical simulations are subjected to a dot-time Fourier transform to analyze its dominant frequency. The results of the Fourier transform are shown in Figure 10.  
Figure 9: Pressure–time history curve of different measuring points in the same section. As shown in the diagram, there are three main frequencies of the initial compressional wave: 5, 15, and 25 Hz. Specifically, 15 Hz corresponds to the main frequency carrying the highest energy. When the main frequency of the compressional wave is in the Helmholtz resonator band, resonance occurs. The resonance phenomenon consumes a large amount of energy and reduces the compressional wave gradient, thereby relieving the micro-pressure wave amplitude. According to [PERSON] [25] and [PERSON] et al. [26], the resonant frequency of a two-cavity Helmholtz resonator can be derived using Equation (2) as follows:  
\[f_{0}=\frac{c}{2\pi}\sqrt{\frac{\rho}{V}\sum_{i=1}^{2}\frac{S_{i}}{M_{i}}} \tag{2}\]  
where \(f_{0}\) denotes the resonant frequency of the Helmholtz resonator, \(c\) denotes the speed of sound, \(\rho\) denotes the density of air, \(V\) denotes the cavity volume, \(S_{i}\) denotes the cross-sectional area of the neck channel, and \(M_{i}\) denotes the equivalent air mass (0.12). The resonant frequency of the Helmholtz resonator is calculated as 14.63 Hz. As the incident wave frequency is in the resonant band of the Helmholtz resonator, it is necessary to calculate the resonant bandwidth. Before calculating the bandwidth, the neck aperture should be judged appropriately as follows:  
\[\frac{0.01}{\sqrt{f_{0}}}{<\mathrm{r}<\frac{10}{f_{0}}} \tag{3}\]  
where \(\mathrm{r}\) denotes the radius of the neck channel of the Helmholtz resonator, and the size can be calculated to satisfy the moderate condition. According to [PERSON] [27] et al. and [PERSON] [28], the bandwidth calculation method is shown by the Helmholtz resonator acoustic resistance, which is first calculated using Equation (4). \[R_{a}=\frac{\sqrt{2\rho\eta\omega}}{\pi r^{2}}\left(\frac{l}{r}+2\right) \tag{4}\]  
where \(\eta\) denotes the shear viscosity coefficient of air, which is considered as \(1.86\times 10^{-5}\), and \(\omega\) denotes the angular frequency, which can be calculated as: \(\omega=2\pi f_{0}\). Furthermore, 1 denotes the neck channel length of the Helmholtz resonator. Figure 10: Main frequency distribution of the initial compressional wave. Combined with the theoretical study by [PERSON] et al. [29], Equation (5) is applied to calculate the acoustic resistivity ratio, and Equation (6) is applied to calculate the quality factor of the sound absorption structure of the Helmholtz resonator as follows:  
\[X_{S}=\frac{R_{a}S}{\rho c} \tag{5}\]  
\[Q_{R}=\frac{c}{2\pi(1+X_{S})Df_{0}} \tag{6}\]  
where \(X_{S}\) denotes the acoustic resistivity ratio, \(Q_{R}\) denotes the quality factor, \(D\) denotes the rigid length of the Helmholtz resonator cavity (i.e., the distance from the cavity mouth to the bottom of the cavity), and S denotes the sum of the cross-sectional areas of the multiple neck channels. Finally, the bandwidth is calculated using Equation (7) as follows:  
\[z=\sqrt{(\frac{1}{Q_{R}})^{2}+1}\pm\frac{1}{2Q_{R}} \tag{7}\]  
where \(z\) denotes the range of frequency bands in the central resonant frequency. The width of the resonant band is calculated between 13.76 Hz and 16.13 Hz. The initial compressional wave has three main frequencies.