## 2 Underwater Explosion
### Gas Bubble  
Another significant phenomenon always created during an underwater explosion is a gas bubble. It contains a mixture of explosive gases that fill the volume of the explosive prior to detonation. In the initial phase, these highly compressed gases expand until the hydrostatic pressure of the water resists its inertia, causing it to shrink. This is called pulsation. The duration of pulsation depends on the mass of the explosive \(m\) and the depth of detonation \(H\) and is described by the relationship [1; 10]:  
\[t_{puls}=0.3\frac{\sqrt[3]{m}}{1+0.1H}\text{, }s \tag{2}\]  
where:  
\(m\)--explosive mass, kg  
\(H\)--depth, mThe pulsation continues until the bubble comes to the surface. The second pulsation time is about 70% and the third about 50% of the first pulsation [1].  
An important parameter that classifies the type of explosion is the gas bubble radius. If the gas bubble is in contact with the hull surface we are talking about a close contact explosion. The effects of such explosion are difficult to predict and calculate using numerical methods. For this reason, most publications refer to non-contact explosions. However, there are some empirical formulas that allow to calculate the diameter of a gas bubble formed after the explosion. The maximum radius of the first pulsation gas bubble describes the relationship [1]:  
\[R_{pmax}=1.53\sqrt[]{\frac{m}{1+0.1H}},\ m \tag{3}\]  
Based on Formula (3), the gas bubble radius was calculated depending on the depth of detonation. The results are shown in Figure 2.  
The graph (Figure 2) shows that at a shallower depth than 10 m, the bubble radius exceeds the depth of detonation, which leads to the release of explosive gases into the atmosphere. In addition, even for an explosive charge of 500 kg at the same depth, the gas bubble does not exceed 10 m. This leads to the conclusion that detonating a charge at a distance of more than 10 m, there is a non-contact explosion, regardless of explosives mass. In this case, only the shock wave impact and waveform are considered. It is worth noting that some torpedoes are characterized by the minimum detonation depth parameter, resulting from this underwater explosion property [20]. If torpedo detonates at a shallower depth, the gas bubble will reach the surface of the water practically immediately after the  
Figure 1: Diagram of the underwater explosion.  
detonation, which will direct the explosion energy towards the surface and reduce its energy due to the shorter contact time with the aquatic environment and jet phenomenon [9; 21]. A bubble oscillation close to the free surface creates a free surface spike. This phenomenon have been widely examined by [21; 22; 23; 24; 25]. According to experiments, the maximum height of the spray dome is significantly related to the charge and initial depth of the explosive. The empirical formulas for the maximum height of the spray dome formed by a relatively shallow underwater explosion caused by TNT explosive were proposed by [24; 25]:  
\[\left\{\begin{array}{l}\frac{Y_{max}}{m^{\frac{1}{3}}}=32.4\left(\frac{H}{m ^{\frac{1}{4}}}\right)^{0.1}\ \ \ 0.0037<\frac{H}{m^{\frac{1}{4}}}<0.74\\ \frac{Y_{max}}{m^{\frac{1}{3}}}=21.7\left(\frac{H}{m^{\frac{1}{4}}}\right)^{-1.24}\ \ \ 0.74<\frac{H}{m^{\frac{1}{4}}}<1.56\end{array}\right. \tag{4}\]  
where \(m\) is the charge mass (kg), \(Y_{max}\) represents the maximum height of the spray dome (m) and \(H\) is the detonation depth (m).