## 4 Results and Discussion
### Hoods at Tunnel Exit  
The entrance hood will be located at the exit of the tunnel when the train runs in the reverse direction, the effects of which on the MPW is rarely reported. Therefore, the effects of the enlarged cross-section, oblique entrance, and holes on the side wall of the hood on the MPW are studied gradually based on the enlarged hood with a constant cross-section. Figure 11: The effects of number of holes at entrance hood on tunnel aerodynamics: (**a**) Pressure gradient at RP point; (**b**) MPW at RM point. Figure 12: The effects of position of holes on tunnel aerodynamics: (**a**) Pressure gradient at RP point; (**b**) MPW at RM point. As the blind hood at the tunnel exit has little effect on the aerodynamics inside the tunnel, the waveform of the compression wave and the MPW will not change with the cross-section area. The maximum values of the pressure gradient at RH point and the MPW at RM point when a hood installed at the tunnel exit with different areas and a constant length of 80 m is show in Figure 13.  
As is well known, when the compression wave arrives at the exit of the tunnel, part of the energy enters the hood and propagates forward, while the other part propagates backwards as a reflected expansion wave. The results show that the intensity of the reflected expansion wave increases gradually with the hood area, while the compression wave propagating forward decreases, resulting in a mitigation of the pressure gradient at the RH point. The MPW is generated from the energy radiating out from the compression wave, which propagates to the exit of the hood. According to the low frequency approximation theory, the amplitude of the MPW outside the exit can be expressed as:  
\[\Delta p_{m}(r,t)=\frac{2A}{\Omega cr}\bigg{(}\frac{\partial p}{\partial t} \bigg{)} \tag{10}\]  
where \(\Omega\) is the spatial angle at the hood exit, \(\partial p/\partial t\) represents the pressure gradient inside the hood, and \(r\) denotes the distance from the hood exit to the monitoring point. It is shown that that the amplitude of the MPW is proportional to the cross-section area of the hood and the pressure gradient at the hood exit, which increases as the area of the cross-section reduces. At the same time, the influence of the cross-section is dominant, resulting in an increase of the MPW intensity with the enlargement of the cross-section, which has an effect in contrast to the entrance hood. It is worth noting that the intensity of the MPW with a constant hood reduces compared to that without a hood due to a larger spatial angle. In addition, the change rate of the MPW slows down with the increasing area of the hood when \(A_{\text{t}}>155\) m\({}^{2}\). Considering the requirement of the entrance hood, \(A_{\text{t}}=155\) m\({}^{2}\) is taken in subsequent study. The pressure contour of the hood with an oblique exit (\(k=1.25\)) when the compression wave arrives is shown in Figure 14. The compression wave radiates out from the hood earlier with an enlarged radiating area, which means that the spatial angle can be increased by an oblique exit, resulting in a decrease in the amplitude of the MPW. Figure 13: The effects of cross-section area of exit hood on the maximum values of pressure gradient and MPW. The amplitude of the MPW at the RM point with different oblique rate of the hood exit is shown in Figure 15. As the oblique rate increases, the intensity of the MPW decreases gradually due to the increasing spatial angle. The amplitude of the MPW can be reduced to 68.36 Pa when \(k=1.25\), which drops by 47.46% compared to that without any hood. Therefore, \(k=1.25\) is taken in subsequent calculation to be consistent with the hood at the tunnel entrance. The pressure gradient of the compression wave in the exit hood can be further reduced by opening holes on the side wall, leading to a decrease in the MPW emitting from the hood.