## 1 Introduction  
The drill and blast method is a common way to excavate an underground roadway or tunnel [1]. blasting in urban tunnels can induce vibrations in the ground and adjacent buildings [2]. The peak velocity of particle vibration (PPV) should not exceed the safety levels set by the construction specifications or the management. This will impact the construction's progress when adopting the short-advance method or even when switching to the mechanical excavation method when the vibration velocity is required to be low. Thus, in densely populated and built-up areas, designers adjust blasting parameters and anticipate possible vibrations [3, 4] to meet the needs of safety and maintain efficient production activities. Vibrations from cut blasting are the focus of predictions. Many methods can predict blast vibrations [5]. The most commonly used are the charge-distance-velocity empirical equations and their improvements [6, 7, 8]. For example, [PERSON] et al. [9] determined a critical particle velocity that was incorporated into a vibration attenuation model to obtain permissible charge weights to limit the blast damage to within a short distance from the charge. [PERSON] et al. [10] proved the Sadovsky model is a more satisfactory model for predicting blast-induced vibrations as compared to empirical models. Its advantage is that it is simple and easy to use. Its disadvantage is that it can only predict the PPV and cannot obtain the complete waveform. Because delay errors and other factors are not considered, more caution is required in predicting delay blasting. There is also the linear superposition method [11, 12, 13]. For example, [PERSON] et al. [14] developed an innovative and simplified analytical approach to signature hole analysis to predict the production of blast-induced ground vibrations. This method relies on the observed blast vibration waves. Its advantage is that the measured vibration data imply the influence of the blast source and geological conditions, which can reflect the changes in the blast vibration wave more realistically. Its disadvantage is that it assumes that the vibration waveforms caused by the initiation of blast holes charged with the same explosive are identical, which is inconsistent with the knowledge from blasting practices. Each blast hole has different free surface conditions during initiation due to different positions and initiation time sequences. These conditions significantly affect its rock-breaking capacity and the induced vibration wave. In cut blasting [15; 16], the first few holes have only one free surface and easily cause the largest amplitude [17]. The subsequent holes usually have more than two free surfaces to make the blast much easier. Even if they use the same blasting parameters, the vibration waves caused by the earlier- and later-initiated holes differ significantly. The vibration frequency of the later-initiated hole is higher and the amplitude is lower [18]. Therefore, in using the linear superposition method to predict the vibration of cut blasting, it is necessary to clarify the formation time of the new free surface. This helps frame the duration that the linear superposition method holds the assumption and adjusts the blasting parameters after forming the new free surface based on the vibration-reduction effect. The role of the free surface is in the two aspects of rock breaking and vibration reduction [19; 20]. [PERSON] et al. [21] studied short-delay blasting with a single free surface in underground mines using electronic detonators. The authors stated that if the delay intervals were longer than the formation time of a new free surface, there would be two free surfaces for the later-detonated blast hole: the initial free surface and the new free surface produced by the former detonated blast hole. [PERSON] [22] developed an analytical model of nonplanar radiation and studied the effect of the free surface on the body waves generated by the explosion; the author pointed out that the angle of incidence played an important role in the process.