## 5 Conclusions  
Based on the one-dimensional compressible unsteady non-isentropic flow model and the improved generalized Riemann variable characteristic method, this paper elucidates the propagation and evolution characteristics of different initial compression waves.  
1. When the initial compression waves propagate in very long tunnels, they undergo a steepening phase (characterized by steepening waveform, decreasing wavelength, and increasing pressure gradient) followed by an attenuation phase (characterized by elongation waveform, increasing wavelength, and decreasing pressure gradient). During the steepening phase, the nonlinear effects of the compression waves are stronger than the frictional effects, whereas during the attenuation phase, the opposite is true. Due to the frictional effects, the pressure amplitude decreases continuously during compression wave propagation.
2. The pressure gradient of the initial compression waves of the type A tunnel entrance hoods and type B tunnel entrance hoods decreased by 32.38% and 68.11%, respectively. This is compared to the case of no tunnel entrance hood. The tunnel entrance hoods not only reduce the pressure gradient of the initial compression waves but also extend the steepening critical distance.
3. Steady frictional effects primarily reduce the pressure amplitude of compression waves without affecting the steepening process of the compression waves. Unsteady frictional effects alter the waveform of the compression waves, reducing the pressure gradient and suppressing the steepening process of the compression waves. The decrease in ambient temperature increases the nonlinear effects of compression waves, reducing viscous dissipation and resulting in an increased steepening rate and attenuation ratio. The degree of the steepening of compression waves depends on the initial compression waves' pressure gradient and friction effects. As a train's speed increases, the steepening critical distance shortens.
4. The mitigating performance of different tunnel entrance hoods shows a trend of initially increasing and then decreasing with increasing propagation distance. The type A tunnel entrance and type B tunnel entrance hoods achieve maximum reduction rates of 63% and 95%, respectively, at distances of 3 km and 4 km. When assessing the performance of tunnel entrance hoods, it is essential to consider not only their impact on the pressure gradient of initial compression waves but also the propagation characteristics of the initial compression waves in the tunnel. This implies the need to find the optimal matching relationship between tunnel entrance hoods and tunnel length to ensure their mitigating performance is maximized.  
Writing--Review and Editing by [PERSON], [PERSON] and [PERSON]; Methodology by [PERSON]; Data Curation by [PERSON]; Visualization by [PERSON] and [PERSON]; Funding Acquisition by [PERSON] All authors have read and agreed to the published version of the manuscript.  
Figure 25: The variation pattern of steepening critical distances with train speeds.  
This work was financially supported by the National Key R&D Program of China (Grant No.2022 YFB2603400).  
**Institutional Review Board Statement**: Not applicable.  
**Informed Consent Statement**: Not applicable.  
**Data Availability Statement**: Data are contained within the article.  
**Conflicts of Interest**: The authors declare no conflicts of interest.