# Study on Vibration Effects and Optimal Delay Time for Tunnel Cut-Blasting Beneath Existing Railways
## 3 The Mechanism and Effect of Vibration Reduction Through Waveform Interference
### Analysis of the Vibration Reduction Effect  
As demonstrated in the previous section, the duration and period of single-hole blasting seismic waves remain largely unaffected by the number of blasts for a given point. Consequently, the superposition effect between columns of blasting seismic waves in multi-hole millisecond blasting can be analyzed using the parameters derived from single-hole blasting seismic waves. From Figure 4, the complete waveforms of the first detonation were extracted. Applying the calibration method described earlier, the main vibration phase duration (\(\Delta\)t\({}_{\text{m}}\)) and the main period (T) were determined for each node. Table 6 presents the relevant blasting seismic wave parameters for single-hole detonation at points A, B\({}_{1}\), and B\({}_{2}\). To compare the vibration reduction effects at various points under different millisecond time delays, the vibration reduction rate \(\varepsilon\) is defined as:  
\[\varepsilon=\frac{\left|P_{2}-P_{1}\right|}{P_{1}}\times 100\% \tag{11}\]  
where P\({}_{1}\) is PPV without millisecond blasting, and P\({}_{2}\) is PPV with millisecond blasting. To analyze vibration reduction under varying millisecond delay conditions, experimental cases with delays of 1, 2, 3, 5, 7, 9, 12, and 15 ms were conducted. Figure 8 illustrates the variation curves of vibration reduction rate versus time difference at each monitoring point. Table 6 provides key waveform parameters at point B\({}_{1}\): a main vibration phase duration (\(\Delta\)t\({}_{\text{m}}\)) of 7.3 ms and dominant period (T) of 1.8 ms. When the delay time (\(\Delta\)t) is 1 ms, it falls within the \(\Delta\)t < T range in Figure 8b and satisfies \(\frac{T}{3}<1\) ms \(<\frac{2T}{3}\). Therefore, point B\({}_{1}\) achieves vibration reduction through main-period superposition. Simultaneously, the vibration reduction effect for the 1 ms millisecond blasting is significantly greater than that for other delay times. However, as the delay time increases (T < \(\Delta\)t < \(\Delta\)t\({}_{\text{m}}\)), the main periods of seismic waves become staggered, while the main vibration phase remains superimposed. Consequently, the vibration reduction effect deteriorates with \begin{table}
\begin{tabular}{c c c} \hline \hline \multirow{2}{*}{**Point**} & **Main Vibration Duration** & **Main Period** \\  & **\(\Delta\)t\({}_{\text{m}}\) (ms)** & **T (ms)** \\ \hline A & 2.8 & 0.6 \\ B\({}_{1}\) & 7.3 & 1.8 \\ B\({}_{2}\) & 6.7 & 1.3 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Parameters for single-hole detonation at three points. an increasing delay time. When the delay time exceeds 7 ms (\(\Delta\)t \(>\)\(\Delta\)t\({}_{\rm m}\)), the vibration reduction rate no longer undergoes significant changes. Under this condition, point B1 achieves vibration reduction through waveform separation, with the PPV of millisecond blasting approximating that of a single wave. At Point B2, the main vibration phase duration (\(\Delta\)t\({}_{\rm m}\)) is 6.7 ms, and the dominant period (T) is 1.3 ms. When subjected to 1 ms time delays, it falls within the \(\Delta\)t \(<\) T range and satisfies \(\frac{2T}{3}<1\) ms \(<T\). Phase amplification in blasting seismic waves induces significant vibration reduction deterioration. For time delays of 2-5 ms (T \(<\)\(\Delta\)t \(<\)\(\Delta\)t\({}_{\rm m}\)), staggered principal wave cycles coupled with superimposed vibration phases yield unstable reduction rates, as exemplified by 2 ms delays achieving minimal reduction while 5 ms demonstrates maximal efficacy relative to comparable delays. Beyond the \(\Delta\)t\({}_{\rm m}\) threshold at 7 ms (\(\Delta\)t \(>\)\(\Delta\)t\({}_{\rm m}\)), waveform separation governs the vibration reduction mechanism, stabilizing the reduction rate despite increasing delay times, with resultant PPV values converging toward single-wave equivalence. For point A, the main vibration phase duration (\(\Delta\)t\({}_{\rm m}\)) is 2.8 ms and the main period (T) is 0.6 ms.