## 1 Introduction  
An underwater explosion (also referred to as UNDEX) represents a detonation below the surface of a water body. At an early stage of the UNDEX, the water medium in the vicinity of the explosion is compressed due to the expansion of the explosives at high temperature and pressure. As a result, a gas bubble forms and an initial shock wave is generated that propagates very fast in the water (approximately \(1\,\mathrm{m}\) within \(500\,\mathrm{\SIUnitSymbolMicro s}\)). Although the pressure peak of the first shock wave is high, its duration is extremely short, usually tens to hundreds of microseconds. Thus, the first shock wave is very hazardous to the local parts of solid structures located in the water. After the first shock wave, the high-pressure explosives lead to the expansion of the bubble, consequently resulting in a bubble expansion-contraction cycle. This cycle can be repeated up to ten times. As a result, pressure pulses are formed in the water, which can also destroy nearby solid structures. Although the peak value of the pulses is only 10-20% of that of the first shock wave, as the frequencies of the pulses are close to the inertial frequency of the underwater structure, the pressure pulses can be devastating to the whole structure. The theory of UNDEX has been widely utilized in many engineering disciplines, such as the underwater cutting of metal mechanical parts, development of protection of ships or submarines, military technology, and oil perforating during the exploitation process. For a detailed description of the UNDEX theory and its applications, the readers may refer to [1; 2; 3]. The previous studies of UNDEXs can be categorized as three major types: theoretical research, experimental study, and numerical simulation. Among the early theoretical studies of UNDEXs, a pioneer work was made by [PERSON] [4], in which most of the physical phenomena occurring during the explosion and the rules related to the UNDEX were presented. Apart from that, based on the similarity theory, [PERSON] and [PERSON] [5] deduced a relation between the properties of the shock wave generated after the detonation and the physical parameters during the explosion. The oscillations of the bubble caused by the UNDEX were also theoretically investigated by [PERSON] [6]. Later, in the year 1962, the theoretical solutions to the propagation and attenuation of the explosion wave in the water were obtained [7]. Moreover, the instability of the bubble during the rising process after the explosion was studied by [PERSON] and [PERSON] [8]. Recently, the theoretical research of the UNDEX focused on the intensity of the shock wave during the collapse of bubbles near a wall boundary [2], as well as the \"whipping effect\" on the ship caused by the explosion [9]. Analytical predictions were also made to identify the dominant factors that critically control the UNDEX [10]. However, since the UNDEX is a very complicated problem that is associated with the theory of detonation, shock dynamics, interface tracking, etc., it is usually difficult to obtain a theoretical solution unless many assumptions and simplifications are made. Consequently, the obtained solution is sometimes too ideal and, thus, far away from the reality. In contrast to that, by intentionally setting up appropriate experiments for the investigation of UNDEXs, researchers were able to obtain more reliable results from observations. Starting from the 1940s, a series of underwater nuclear tests such as the Baker test were performed [11]. In these UNDEX tests, the heights of the water waves formed by the explosion were measured. In addition, the influence of the explosion on the weather was also recorded. During the 1980 to 1990s, due to the technique development of the bubble generation and optical measurement, researchers were able to observe and record the behavior of the bubbles during the explosion such as the abovementioned multiple oscillations [12].