## 2 Vibration Control Methods
When the number of wave arrivals within such a period is significant, the probability of increased vibration amplitude becomes higher, and the monitoring point is identified as a high-probability vibration hazard location. \[ \begin{array}{l}t_{k}=(k-1)\Delta t\\ A_{m,k}=\{t(i)|t_{k}\leq t(i)<t_{k}+\Delta t\,\}\\ n_{m,k}=n(A_{m,k})\\ n_{m}=\max(n_{m,k})\\ \end{array}  \tag{3}\]  
where \(\Delta t\) represents the length of the time interval; \(t_{k}\) is the starting time of the \(k\)-th time interval, \(k\geq 1\); \(A_{m,k}\) denotes the set of all arrival wave time data at the \(m\)-th monitoring point within the \(k\)-th time interval; \(n_{m,k}\) is the number of elements in the set \(A_{m,k}\), which corresponds to the number of blast holes; \(n_{m}\) is the maximum number of arrival waves at the \(m\)-th monitoring point within the same time interval. The initiation times of the above blast holes and subsequent holes are increased by a fixed duration to separate the vibration waves arriving simultaneously at the monitoring points, thereby significantly reducing the vibration caused by multi-hole blasting and addressing vibration safety concerns. Figure 2 illustrates the principle of separating vibration waves that arrive simultaneously at the monitoring points. Taking two single-hole vibration curves arriving at the monitoring points within the \(\Delta t\) period as an example, the figure depicts two extreme scenarios. Figure 2a shows two vibration curves arriving at the monitoring points at the same time. According to the principle of peak and trough superposition, translating one of the vibration curves backward by \(1/2T\) achieves the optimal vibration reduction effect. Figure 2b shows two vibration waves arriving at the monitoring points at the beginning and end of the same period, respectively. Translating the latter vibration curve backward by \(1/2T-\Delta t\) achieves the optimal vibration reduction effect. Therefore, for vibration waves arriving within the \(\Delta t\) period, the range of translation on the \(x\)-axis is \([1/2T-\Delta t,1/2T]\). To enhance the convenience of the method, this paper stipulates the use of \(1/2T\) as the additional time length required to separate vibration waves arriving simultaneously at the monitoring points. It is worth noting that by taking advantage of the differences in the charge amount, distance, and site factor among different blast holes, the vibration curves of single blast holes at any position, which are reconstructed by adjusting the measured vibration curves of single blast holes, possess the same periodicity. On this premise, according to the principle of half-period vibration reduction, it can be determined that moving the waveform arrival time by \(1/2T\) can achieve a relatively good vibration reduction effect. _Problem 2: Vibration hazards around the blast area caused by spatiotemporal variabilities between monitoring points and blast holes._  
Figure 1: Schematic of vibration waves arriving at the monitoring point during a fixed period. (**a**) Single-hole wave. (**b**) Multiple single-hole waves arriving within the same time window. Figure 2: Schematic of vibration wave separation during clustering arrival periods. (**a**) Simultaneous arrival. (**b**) \(\Delta t\)-interval arrival. The distances from each blast hole to different monitoring points are not the same, leading to varying arrival time sequences of the waves at each monitoring point. In a given blasting time sequence, most monitoring points are typically in a normal vibration state, while a specific monitoring point may experience hazards. The location of such monitoring points will shift with changes in the blasting time sequence. In practical blasting vibration control, a limited number of monitoring points are insufficient to comprehensively capture vibration conditions in all directions surrounding the blast area, making it impossible to accurately identify the location of potential vibration hazards.