During drilling and blasting excavations of deep caverns, the transient unloading of in-situ stress on the excavation boundary can induce vibrations in surrounding rock mass. For circular tunnel excavation, the balance mechanism of strain energy, kinetic energy and work done by radical stress during the process of the transient unloading of in-situ stress is analyzed, then a new attenuation formula is established to analyze the attenuation law of vibrations induced by the transient unloading of in-situ stress based on the strain energy of excavated rock mass with the tool of unit analysis. Case studies of Jinping II and Pubugou hydropower stations are presented to show how to identify and separate the vibrations induced by the transient unloading of in-situ stress from the monitored vibration signals. Studies indicate that the amplitudes of vibrations induced by the transient unloading of in-situ stress are proportional to the square root of strain energy density of excavated rock mass, and the bigger volume of excavated rock mass is, the intenser vibrations are induced by the transient unloading of in-situ stress. Compared with the existing attenuation formula proposed by Lu and Hustrulid, the new attenuation formula has a wider range of application and higher precision.