## 4 Results of Numerical Errors and Discussion  
In order to evaluate the SPH simulation results, the SPH numerical error of the sound pressure \(\varepsilon_{\text{pre}}\) is calculated. The error is defined by:  
\[\varepsilon_{\text{pre}}=\frac{1}{nP}\sqrt{\sum_{j=1}^{n}\left(\delta p_{j}^{ \text{L}}-\delta p(r_{j})\right)^{2}} \tag{18}\]  
where \(\varepsilon_{\text{pre}}\) is a non-dimensional parameter and \(n\) denotes the number of particles that are used to compute numerical error. \(\delta p_{j}^{\text{L}}\) is the simulation sound pressure of the particle \(j\), and \(\delta p(r_{j})\) is the theoretical sound pressure in the position of particle \(j\). \(P\) denotes the pressure that is used to non-dimensionalize the numerical error. In this paper, \(P\) is defined as the amplitude of an acoustic wave in the sound propagation and is defined as the difference of the sound pressure amplitudes of two acoustic waves in the sound interference, that is \(P=50\) Pa in the sound propagation and \(P=20\) Pa in the sound interference. The region that is used to compute the numerical error is from 0 m to 90 m in the sound propagation and from 0 m to 70 m in the sound interference. According to the computational parameters given in Table 1, we simulate the sound propagation and interference using the SPH algorithms with \(\alpha_{\Pi}\) changing from 0.00 to 0.08; meanwhile, \(\beta_{\Pi}\) is set as 0.50. Then, we calculate the SPH numerical errors of the sound pressure according to Equation (18), as shown in Table 2.  
From Table 2, it can be seen that the SPH algorithms with artificial viscosity can improve the accuracy of the simulation. Compared to the sound interference simulation, when the parameters of artificial viscosity term are the same, the SPH numerical errors of sound propagation are smaller. For sound propagation, the SPH numerical errors of sound pressure reach the minimum when \(\alpha_{\Pi}=0.06\), \(\beta_{\Pi}=0.5\). For sound interference, the SPH numerical errors of sound pressure reach the minimum when \(\alpha_{\Pi}=0.03\), \(\beta_{\Pi}=0.5\). Figure 4: Sound interference pressure of acoustic waves at \(t=0.15\) s.  
Since sound propagation and interference are similar, we mainly discuss the effects of artificial viscosity on different sound interference models in the present work. In the discussion, different computational parameters, namely the smoothing kernel function, particle spacing and the Courant number, are considered. Different sound interference cases with Gaussian, cubic spline and quintic spline kernel functions are simulated. The artificial viscosity terms with different values of \(\alpha_{\Pi}\) and \(\beta_{\Pi}\) are added, and the numerical errors of sound pressure are plotted in Figure 5.  
As shown in Figure 5, for the same particle spacing \(\Delta x\) and Courant number, the smoothing kernel functions have obvious effects on the SPH simulation results. The SPH simulation using the cubic spline kernel function has more accuracy results than that using the Gaussian kernel function or the quintic spline kernel function. The numerical errors of SPH simulation using the quintic spline kernel \begin{table}
\begin{tabular}{c c c c} \hline \hline \(\alpha_{\Pi}\) & \(\beta_{\Pi}\) & **Sound Propagation \(\varepsilon_{\text{pre}}\) (\(\times 10^{-4}\))** & **Sound Interference \(\varepsilon_{\text{pre}}\) (\(\times 10^{-4}\) )** \\ \hline
0.00 & 0.00 & 2.574247 & 3.674502 \\
0.00 & 0.50 & 2.571422 & 3.674424 \\
0.01 & 0.50 & 2.303623 & 3.615698 \\
0.02 & 0.50 & 2.177653 & 3.580848 \\
0.03 & 0.50 & 2.098490 & 3.565303 \\
0.04 & 0.50 & 2.059441 & 3.566019 \\
0.05 & 0.50 & 2.038529 & 3.580722 \\
0.06 & 0.50 & 2.034482 & 3.607429 \\
0.07 & 0.50 & 2.039658 & 3.644548 \\
0.08 & 0.50 & 2.041069 & 3.690689 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The SPH numerical errors of sound pressure. Figure 5: Numerical errors of sound pressure for sound interference models with different kinds of smoothing kernel functions.