## 2 Underwater Explosion
### Pressure Wave  
The expanding gas bubble acts on the surrounding water layer, creating a spherical shock wave. In the initial phase, shockwave travels at a speed of \(v\approx\) 5000-8000 m/s. Then the water molecules act on the adjacent layers of water, losing their speed and move further at the sound of speed in water, which value reaches about \(c_{0}\) = 1500 m/s. The pressure wave profile and its value have been described by many researchers. The main of them is [PERSON], whose publications form the basis of most research on the subject. The problem is also described by other authors [1; 10; 26; 27; 28; 29]  
Underwater explosion phenomena is wide field of study with deeper aspects to be considered, some of its basic yet most important parameters are hereby summarized and briefly explained, intending to allow the comprehension of fundamental notions of the subject based on a literature survey. The article attempts to collect most of the descriptions of various researchers and implement them for CAE (Computer Aided Engineering) purposes. A similar approach has been proposed in [30]. Table 1 presents formulas necessary to calculate shock waves caused by underwater TNT explosion pressure values and their waveforms. Knowledge of the waveforms shown in Figure 3 allows the total shock wave energy to be calculated. The energy value is important due to the fact that short-term blast damage depends not only on overpressure, but also on positive phase duration. It is also important that the descriptions refer only to the positive pressure pulse wave, excluding the negative pressure phase and pulsation. However, in the methodology of calculations related to the strength analysis  
Figure 2: Size of the gas bubble radius depends on the mass of the explosive charge and depth. of warships [4, 5], it is assumed that if a ship withstands the first shock wave, she will be able to survive the subsequent ones caused by pulsations. For this reason, this manuscript refers only to the description of the first pressure wave caused by an underwater explosion. The course of the pressure wave over the years has been studied by various researchers. On the basis of empirical research, they developed formulas (Table 1) for pressure values on the shock wave front \(p_{max}\) and its time course \(p\) (\(t\)), which depend mainly on:  
* \(m\)--charge mass, kg (some formulas refer to the diameter of the explosive charge--ac)
* \(r\)--distance from epicenter, m
* \(\theta\)--exponent time constant, ms
* \(t\)--time, ms  
The differences in the formulas in the Table 1 most often result from the study of explosions caused by different-sized explosives. For example, [PERSON] studied explosives in the 70-136 kg range at a depth of 3-10 m while [PERSON] & [PERSON] based their mathematical descriptions mainly on [PERSON] and [PERSON] [33] studies, which tested explosive charges in the range 0.2 kg to 5.4 kg at a depth 76-152 m. The detonation of various explosives took place at different depths, which may cause discrepancies. Unfortunately, not all researchers provide the exact parameters of their research, so when using the mathematical description of the pressure wave, the awareness of these differences should be preserved.