## 2 Materials and Methods
### Experimental Parameter Design  
The spatial structure of the wire is a cylinder with a proportionally hollow interior. The hollow ratio, \(a\), represents the ratio between the inner diameter, \(d\), and outer diameter, \(D\). The hollow structure of the wire and one of the physical objects are shown in Figure 2.  
The length of the wire is a constant in this study because the current density is invariable in the length direction. A shorter wire requires a lower test voltage when the capacitance value of the system is constant. Considering the perforation hole size of the oil well affected by a shockwave in engineering practice, \(L\) represents the length of the wire, which is selected as 30.00 mm. The shockwave maintains almost the same cylinder shape, irrespective of the wire cross-section. The electric explosion of metal wire is a physical phase transition process with a breakneck development speed. For the electric explosion of pure copper metal, the theoretical enthalpy value is the lowest energy required during the rapid transformation of pure copper wire from a condensed state to a plasma state. Table 3 shows the enthalpy value parameters of Cu(ref) in NIST-JANAF [29]. The theoretical total enthalpy value is 392.26 kJ-mol\({}^{-1}\). Considering the influence of energy dissipation and the spatial instability during the underwater wire electric explosion process, the Marx generator's pre-stored energy is set to three times the theoretical enthalpy value of pure copper wire. To obtain a high-power impulse current, the Marx generator is used to increase an impulse current and compress pulse discharge time. When the electric pulse is introduced into the electric explosive wire consumable, the current first forms on the surface due to the skin depth effect, and then the current density diffuses inward to the center with the evolution of the pulse. Therefore, slightly increasing the diameter of the wire and making it hollow can optimize the electrical explosion process. In this paper, the changes of energy deposition and shockwave pressure of the electric explosion of wire were studied with a different outer diameter, \(D\), and wire \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Test** & **First** & **Second** & **Approximate** & **Loop** & **Loop** \\
**Voltage** & **Current Peak** & **Current Peak** & **Period** & **Resistance** & **Inductance** \\
**U/k\(\
abla\)** & **\(\mathbf{I_{\mathrm{P}}}\)/kA** & **\(\mathbf{I_{\mathrm{P}}}\)/kA** & **\(\mathbf{I_{\mathrm{P}}}\)/kA** & **\(\mathbf{I_{\mathrm{P}}}\)/B** & **\(\mathbf{R_{\mathrm{b}}}\)/m\(\mathbf{\Omega}\)** & **\(\mathbf{L_{\mathrm{b}}}\)/μ**H** \\ \hline
3.00 & 20.80 & 12.21 & 15.9 & 107.2 \(\pm\) 3.2 & 1.6 \(\pm\) 0.1 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Basic performance parameters of the circuit (5-stage Marx). Figure 2: The hollow structure of the wire and one of physical objects. hollow ratio, \(a\), when three times the energy was injected. The driving energy, \(E_{store}\), and voltage, \(U\), required for different diameters and hollow ratios in the pure copper test are shown in Table 4. The driving energy, \(E_{store}\), is a physical quantity determined indirectly. In Equation (4), all parameters except the wire length, \(L\), are fixed, and \(\rho\) and \(M\) represent the density and molar mass of copper, respectively. Because the wire length is measured with Vernier calipers, the energy calculation results retain five significant digits. The voltage, \(U\), is a physical quantity estimated by driving energy and determined by measuring the instrument directly. The adjustment accuracy of the high-voltage DC power supply can only reach two decimal places. \[E_{store}=\frac{3\pi}{4}\frac{\left(1-a^{2}\right)D^{2}L\rho}{M}H_{total}= \frac{1}{2}CU^{2} \tag{4}\]  
In the electric explosion test, the conductivity of water has influence on the energy conversion process and the result of the final shockwave pressure test.