## 4 Results and Discussion
### Dominant Frequency and Decay Rate of the Aerodynamic Pressure after the Train Exits  
After the train exits the enclosed housing for sound emission alleviation, as the pressure wave continues to propagate inside the housing for sound emission alleviation, the aerodynamic pressure does not immediately revert to zero but remains varying. Figure 17 shows the pressure curves of P3 on sections after the train traverses the exit of the housing for sound emission alleviation. The results show that the pressure curves, in particular at the central sections, become periodic waves, and the amplitude gradually decreases, similar to the free vibration decay curves. In addition, the pressure wave is much more periodic, and its amplitude is much higher in the central region than that close to both ends. These characteristics are different from those when the train is passing inside the housing for sound emission alleviation. Consequently, the dominant frequency and decay rate of the pressure curves are analyzed as follows. The dominant frequency of the aerodynamic pressure at the monitoring points in each section is obtained through spectral analyses. Figure 18 shows the calculated power spectrum density curves of the pressure of P3 on sections after the train traverses the exit of the housing for sound emission alleviation from 14.3-45 s and the sampling frequency is 500 Hz. In all sections, the dominant frequency is equal to 0.2 Hz. In fact, the dominant frequency can be theoretically derived using the propagation law of the pressure wave. When a compression wave reaches the monitoring point, the pressure reaches a positive extreme value; then, the expansion wave is reflected to be an expansion wave at one end of the housing for sound emission alleviation. When the expansion wave reaches the monitoring point, the extreme pressure is negative. After that, the expansion wave is reflected to be a compression wave again at another end of the housing for sound emission  
Figure 16: Pressure distribution on S6 when extreme pressure appears. Figure 15: Pressure distribution on S1 when extreme pressure appears. alleviation. When the new compression wave reaches the monitoring point, the pressure reaches the next positive extreme value. During this cycle, both compression and expansion waves travel the entire housing for sound emission alleviation once under the sound velocity. Consequently, the dominant frequency can be expressed as:  
\[f=\frac{v}{2l}=0.2\;\mathrm{Hz} \tag{11}\]  
where \(l\) is the length of the housing for sound emission alleviation and \(v\) is the sound velocity. This formula is the same as that proposed by [PERSON] et al. [47] in tunnel aerodynamics. Figure 17: Pressure curves of different measuring points for the train passing through the housing for sound emission alleviation. (**a**) S1. (**b**) S3. (**c**) S6. (**d**) S9. (**e**) S11. During the propagation, the strength of the aerodynamic pressure decreases due to the friction and refection effects. This dynamic process is important for the structure design. In order to evaluate the attenuation characteristic, the decay rate \(\delta\) of the aerodynamic pressure is defined as [49]:  
\[\delta=\ln(\frac{A_{K}}{A_{K+1}}) \tag{12}\]  
where \(A_{K}\) is a pressure extreme in the process of pressure decay, \(A_{K+1}\) is the next pressure extreme after a cycle. The definition of the decay rate of the aerodynamic pressure is similar to that of a damped free vibration. Figure 19 shows the pressure extremes envelope and the decay rate of the aerodynamic pressure fitted using Equation (11). The decay rate of the aerodynamic pressure on all sections is close, and the average decay rate is calculated as 0.262. The highest decay rate is 0.277 in the middle section. Both decay rate and the dominant frequency are key parameters for the structural design of the enclosed housing for sound emission alleviation because they are necessary when the engineer evaluates the fatigue life.