## 4 Vibration Data Analysis
### Spectrum Analysis  
Since the four sets of data represent acceleration vibration measurements taken at different time intervals and the train speeds are not entirely consistent, the time series of the data vary. To facilitate data processing, reduce errors, and ensure that each dataset was fully represented within its time frame, we set the data time range to 5 s. Additionally, we defined the vertical track direction as the Z direction, the horizontal perpendicular track direction as the X direction, and the track direction as the Y direction. The frequency bands were categorized as follows: 0-20 Hz was designated as low frequency, 20-40 Hz as medium frequency, and above 40 Hz as high frequency [24; 25]. Figure 4 illustrates the power spectrum data and their average values for each dataset. Figure 5 illustrates that the vibration frequencies at each measuring point predominantly fell within the 20-50 Hz range. Notably, the prominent frequencies across all directions were clustered around 25 Hz and 35 Hz. In the Z direction, extending from measurement point P2 to measurement point P6, the high-frequency components disappeared at P5 and reappeared at P6. This phenomenon occurred because the tunnel structure was located directly beneath measurement point P5. The presence of the tunnel caused reflection and refraction of the vibration waves, leading to a reduction in the high-frequency components in the Z direction at this location. In the X direction, extending from measurement point P2 to measurement point P6, the high-frequency components increased at P3. Similarly, in the Y direction, extending from P2 to P6, an increase in high-frequency components was observed at P3. This phenomenon of increased high-frequency components occurred at P3 in both the X and Y directions. This was also due to the presence of the tunnel structure directly beneath measurement points P5 and P2. The lining beneath the tunnel was in direct contact with the track slab, allowing vibrations in the X and Y directions to be transmitted directly through the tunnel lining to the surrounding soil. This resulted in faster propagation and less attenuation of vibration waves in the X and Y directions. However, for the transmission of vibration waves in the Z direction, the waves must first pass through the lining beneath the tunnel and then propagate upward to the lining above the tunnel before reaching the soil, leading to smaller vibrations in the Z direction. Additionally, the vibration waves underwent reflection and refraction between the two tunnels, causing the vibrations to overlap at measurement point P3, which resulted in the appearance of high-frequency components. In previous studies on surface vibrations in homogeneous soil layers, the amplitude of the power spectrum was found to decrease significantly as the distance from the vibration source increased. However, in Figure 5, as the distance from the train-running tunnel increases, the expected attenuation of the high-frequency components of the Z-direction power spectrum by the soil does not occur [19]. Figure 4: _Cont_. Conversely, the presence of the tunnel appears to amplify high-frequency components to some extent. This phenomenon can be attributed to the existence of the tunnel structure at measurement point P5. Vibration propagation can be understood as the flow of energy within a medium. High-frequency wave energy tends to experience less loss when traveling through high-stiffness media such as tunnel linings and steel materials, while greater energy dissipation occurs in soil layers. This redistribution of energy can lead to localized accumulation or reduction of energy at specific measurement points. High-stiffness materials, such as tunnel linings, can act as \"energy conduits\" for high-frequency waves, resulting in multiple reflections at interfaces that cause localized amplification. Additionally, the impedance differences at the soil-tunnel interface can induce further reflections and refractions.