## 2 Numerical Model
### Acoustic Wave Propagation Model  
After the sound pressure waveform of the plasma explosion is obtained from the previous section, the acoustic wave propagation process can be simulated using the sound wave equation. When sound waves propagate in the air, they are longitudinal waves, and the standard form of the wave equation is  
\[\frac{\partial^{2}p}{\partial t^{2}}=c^{2}(\frac{1}{r}\frac{\partial}{\partial r }(r\frac{\partial p}{\partial r})+\frac{\partial^{2}p}{\partial z^{2}})+S(t) \tag{13}\]  
where \(p\) is the sound pressure; \(x\) and \(y\) represent the horizontal and vertical coordinates of the acoustic calculation domain, respectively; \(c\) is the velocity of the sound wave; and \(S(t)\) is the sound source function, which is obtained from explosion model of plasma.  
In a real situation, the propagation of the sound wave is influenced by the environment, so it is necessary to consider the effect of ground reflection and air absorption. The sound wave generated by the corona propagates outward in the form of a spherical wave. Therefore, we establish a 2D axisymmetric model of acoustic wave propagation. The upper part of the model is air, and the lower part is concrete. The acoustic source is set on a circular boundary near the symmetry axis. There are five monitoring points set up around the source, with the interval of 50 cm, and the Perfectly Matched Layer (PML) absorption boundary with 5 cm thick is set around it. The acoustic source propagation model is shown in Figure 4.  
Figure 3: Two-dimensional (2D) simulation model of explosion shock wave.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \(\rho_{0}\) (kg/m\({}^{3}\)) & \(\gamma\) & **Temperature (K)** & **Specific Heat (J/kg.K)** & \(e\) (J/kg) \\ \hline
1.225 & 1.4 & 288.2 & \(717.6\times 10^{3}\) & \(2.068\times 10^{5}\) \\ \hline \hline \end{tabular}
\end{table}
Table 3: Parameters of the air.  
There are two main absorption mechanisms of sound wave by actual air. One is the \"classical absorption\" caused by the viscosity and thermal conductivity of air, and the other is the \"relaxation absorption\" caused by the transfer of internal translational energy to the internal energy of air molecules [27]. According to the calculation method of air absorption loss specified in the ANSI S1.26-2014 standard, the attenuation coefficient of the sound wave is related to air temperature, humidity, pressure and sound frequency [28]. At standard atmospheric pressure, when the temperature is 20 \({}^{\circ}\)C, the attenuation coefficients of air with humidity of 10%, 50%, 70% and 100% are shown in Figure 5. It can be seen that the attenuation effect of air on high-frequency sound waves is stronger, and the change of humidity also shows great impact on the attenuation coefficient.  
In order to take the attenuation effect of air on sound waves at different frequencies into account, the calculation process of the model is divided into three steps. First, the sound pressure distribution of the frequency domain at the sound source position is obtained through Fast Fourier Transform (FFT). Then, the propagation model considering air absorption and attenuation is used to calculate the frequency domain distribution of the space sound pressure. Finally, the time-domain distribution of the sound pressure is obtained through Inverse Fast Fourier Transform (IFFT). The entire flow chart is shown in Figure 6.  
Figure 4: Spatial propagation model of plasma explosion sound source.  
Figure 5: Attenuation coefficient of air at different humidity.