## 2 Underwater Explosion
### Pressure Wave
\begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**No.**} & \multirow{2}{*}{**Author, Year**} & \multicolumn{2}{c}{**Pressure on the Front of the**} & \multirow{2}{*}{**Remarks**} \\  & & & **Shock Wave \(p(t)\)** & \\  & & & **Pressure Drop Time Constant** & \\ \hline \multirow{2}{*}{1.} & \multirow{2}{*}{[PERSON] [10] 1948} & \(p_{max}=52\left\{\frac{3\left|\overline{\mu}\right|}{r}\right\}^{1.13}\) & Dependencies determined as a result of TNT explosion measurements weighing 70–136 kg \\  & & & \(\theta=0.093\sqrt[3]{m}\left\{\frac{3m}{r}\right\}^{-0.22}\) & Time constant given by [PERSON] \\ \hline \multirow{2}{*}{2.} & \multirow{2}{*}{[PERSON] et al. [27] 1966} & \(p_{max}=52\left\{\frac{3\left|\overline{\mu}\right|}{r}\right\}^{1.19}\) & \multirow{2}{*}{0.05} & \multirow{2}{*}{[PERSON] formulas with J.S.} \\  & & & \(p(t)=p_{max}\)\({}^{-6}\) & & \\  & & \(\theta=0.07\sqrt[3]{m}\left\{\frac{3m}{r}\right\}^{-0.17}\) & & \\ \hline \multirow{2}{*}{4.} & \multirow{2}{*}{[PERSON] et al. [27] 1966} & \(p_{max}=44\left\{\frac{3m}{r}\right\}^{1.5}\) & \(6<\frac{r}{\sqrt[3]{m}\left\{\frac{3m}{r}\right\}}<12\) & \(\rho_{TNT}\)—TNT density (\(p_{TNT}\approx 160\) kg/m\({}^{3}\)) \\  & & \(p(t)=p_{max}\)\({}^{-6}\) & \(\theta<t<(5\)–\(10)\theta\) & \\ \hline \multirow{2}{*}{5.} & \multirow{2}{*}{[PERSON] [28]} & \multirow{2}{*}{\(p(t)=K1\left(\frac{3\left|\overline{\mu}\right|}{r}\right)^{A1-\frac{ 5}{2}}\)} & \multirow{2}{*}{[PERSON] [31]} & [PERSON] [28] & [PERSON] \& & [PERSON] et al. \\  & & & [31] & [28] & [PERSON] [29] \\  & & \(\theta=K2\cdot\sqrt[3]{m}\left\{\frac{3m}{r}\right\}^{A2}\) & \(K1=52.12\) & \(K1=148.93\) & \(K1=52.16\) & \(K1=29.9\) \\  & & & \(A1=1.18\) & \(A1=1.13\) & \(A1=1.13\) & \(A1=1.81\) \\  & & & \(K2=0.092\) & \(K2=0.058\) & \(K2=0.058\) & \(K2=0.058\) \\  & & & \(A2=-0.185\) & \(A2=-0.22\) & \(A2=-0.22\) & \(A2=-0.22\) \\ \hline \multirow{2}{*}{6.} & \multirow{2}{*}{[PERSON] \& [PERSON] [26] 2002} & \multirow{2}{*}{\(p_{max}=P_{\epsilon}\left(\frac{\mu}{r}\right)^{1+A}\)} & \multirow{2}{*}{1. [PERSON] [2]} & \multirow{2}{*}{2. [PERSON] [2]} & \multirow{2}{*}{3. [PERSON] [2]} \\  & & & \(p(t)=p_{max}\)\({}^{-\frac{1}{2}}\) & & \\ \cline{1-2}  & & & \(p_{\mathcal{C}}=1.42\) & \(p_{\mathcal{C}}=1.45\) & \(Pc=1.67\) \\ \cline{1-2}  & & & \(vc=0.992\) & \(vc=1.24\) & \(vc=1.01\) \\ \cline{1-2}  & & & \(A=0.13\) & \(A=0.13\) & \(A=0.18\) \\ \cline{1-2}  & & & \(B=0.18\) & \(B=0.23\) & \(B=0.185\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Formulas for calculating pressure values and pressure waveforms given by various authors. Using the equation for a pressure pulse (5), the energy carried by a shock wave can be compared and evaluated. By calculating the area under the curve, it is possible to determine the equivalent pressure of the shock wave described by various authors. Referring to the waveforms presented in Figure 3, the \(I^{+}\) value was calculated for the considered pressure waves in the 0-2 ms time range (Figure 5). Based on these calculations, differences between individual descriptions of pressure waves can be assessed. In this case, most calculations (except Ming) results with similar shock wave maximum pressure \(p_{max}\) (7-8 MPa), but the total energy carried by the explosion differed in some cases more than twice. This means that depending on the adopted shock wave model, different strength results of marine structures can be obtained despite similar pressure on the shock wave front. The above considerations should be taken into account in cases related to strength calculations of marine structures. Another important aspect is the decrease in the shock wave pressure due to the increase of distance. According to the equations given in Table 1, it can be calculated how the shock wave pressure values changes while increasing the distance from the detonation epicenter.