# 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
### Principle of Vibration Reduction Through Waveform Interference  
Millisecond blasting is one of the most effective techniques for mitigating blast-induced vibrations. Employing appropriate millisecond delays can significantly reduce the adverse effects of blasting seismic waves on existing railway lines. The vibration reduction mechanisms in millisecond blasting primarily involve two phenomena: waveform separation and waveform interference. Waveform interference is achieved by short millisecond delay intervals, which induce a specific phase difference between successive blast waves. This phase difference enables the superposition of the seismic waves to achieve vibration reduction through destructive interference. Based on wave interference theory [32], the superposition mechanisms of blasting seismic waves are governed by the delay interval \(\Delta\)t relative to vibration characteristics (including the main period T and main vibration duration \(\Delta\)t\({}_{\text{m}}\)). When \(\Delta\)t < T, the constant phase difference in the dominant frequency band causes full overlap of the main-period waveforms (PPV points), generating significant constructive/destructive interference (Figure 6) with pronounced amplification or attenuation effects. Conversely, for intervals where T \(\leq\Delta\)t \(\leq\Delta\)t\({}_{\text{m}}\), the main-period (PPV) oscillations separate temporally, while the main vibration phases (amplitudes \(\geq\) 50% PPV) partially overlap (Figure 7). This partial spatiotemporal decoupling substantially reduces wave interaction efficiency. Consequently,  
Figure 5: Schematic diagram of each phase. the vibration reduction achieved through main-period superposition (\(\Delta\)t < T) demonstrates superior efficacy due to coherent full-phase modulation of energy-intensive components. When superposition of the dominant-period waveform components occurs, the resulting blasting seismic wave exhibits both constructive interference maxima and destructive interference minima. Consequently, it is imperative to select an appropriate inter-hole delay interval (\(\Delta\)t) in milliseconds to achieve vibration reduction. Let us consider two sinusoidal wave trains propagating in the same homogeneous medium. The displacements of these wave trains are expressed as:  
\[A_{1}=\sin(\omega\cdot t-\varphi_{1}) \tag{6}\]  
\[A_{2}=\sin(\omega\cdot t-\varphi_{2}) \tag{7}\]  
where \(A_{1}\) and \(A_{2}\) are the wave amplitudes, \(\omega\) is the angular frequency, and \(\varphi_{1}\) and \(\varphi_{2}\) are the initial phase angles. When the two columns of waves are superimposed:  
\[A=A_{1}+A_{2}=\sin(\omega\cdot t-\varphi_{1})+\sin(\omega\cdot t-\varphi_{2}) \tag{8}\]  
Treating the trigonometric functions of Equation (3):  
\[A=2\sin(\omega\cdot t-\frac{\varphi_{1}+\varphi_{2}}{2})\cdot\cos(\frac{ \varphi_{2}-\varphi_{1}}{2}) \tag{9}\]  
Figure 6: Superposition of main period (\(\Delta\)t < T). (a) phase increase; (b) phase decrease. Figure 7: Stagger of main period (T < \(\Delta\)t < \(\Delta\)t\({}_{\rm m}\)): (a) phase increase; (b) phase decrease. When t \(\in\) (0, \(\infty\)), it always satisfies  
\[-1<sin\bigg{(}\omega t+\frac{\varphi_{1}+\varphi_{2}}{2}\bigg{)}<1 \tag{10}\]  
Therefore, if the amplitude of the superimposed wave is less than that of any individual wave, the amplitude superposition coefficient A satisfies the constraint: \(-1\leq\Delta\leq 1\). It needs \(-\frac{1}{2}<\cos\Big{(}\frac{\varphi_{2}-\varphi_{1}}{2}\Big{)}<\frac{1}{2}\). Therefore, when \(\frac{2}{3}\pi<\varphi_{2}-\varphi_{1}<\frac{4}{3}\pi\), the peak value of the wave after the superposition of the two columns is smaller than the peak value of the single column. Then the two columns can achieve the effect of vibration reduction through waveform interference. Especially when \(\varphi_{2}-\varphi_{1}=\pi\), destructive interference occurs as wave peaks align with troughs, resulting in complete cancellation of the synthesized wave amplitude (A = 0).