## 3 Sound Propagation and Interference Simulation
### Numerical Simulations and Model Verification  
Table 1 lists the computational parameters that are used in the fundamental SPH algorithms of sound propagation and interference. The artificial viscosity term (Equation (12)) is added to the fundamental SPH algorithms to simulate sound propagation and interference. With the purpose of validating the accuracy of our algorithm, the SPH results with and without artificial viscosity are compared to the theoretical solution in Figure 1. The SPH results with artificial viscosity were calculated on the condition that the \(\alpha\)n and \(\beta\)n are set to be \(0.06\) and \(0.50\), respectively. As shown in Figure 1, the theoretical solution is plotted using a solid line, and the SPH results are represented by scattered points. The scattered points are plotted at intervals of \(20\) for clearly identifying the SPH results. It can be observed that the two SPH simulation results both possess similar smoothing trends compared to the theoretical solution. Several peaks and valleys appear in the graph between 0 m and 80 m, and the sound pressure amplitudes of all three computational cases agree well with each other. Both the SPH simulation results (with or without artificial viscosity) are in good agreement with the theoretical solution. In order to provide a better comparison, three magnified portions of Figure 1 are provided in Figure 2a-c, separately. Figure 2a,b describes the first and second peaks, respectively, showing the portions of the peaks bounded by the black frames in Figure 1. Since the two valleys in Figure 1 are nearly identical, only one of them is shown in detail in Figure 2c. The SPH simulation result in the region \(26\leq x\leq 38\) m are plotted at intervals of four in Figure 2a. From the figure, it can be seen that the sound pressure of theoretical solutions firstly increases and then decreases with increasing x in a smoothly changing wave. However, the standard SPH algorithm gives a different result, as can be seen in the black frame in the middle of the figure. It is clear that the sound pressure obtained with the standard SPH algorithm fluctuates dramatically around the theoretical value, where two unphysical peaks and one unphysical valley can be seen. In comparison, the SPH algorithm with artificial viscosity obtains a much better result, that is the simu \begin{table}
\begin{tabular}{c c} \hline
**Simulation Parameters** & **Values** \\ \hline Smoothing length & 0.13 m \\ Kernel type & cubic spline \\ Courant number & 0.04 \\ Particle spacing \(\Delta x\) & 0.04 m \\ Sound speed & 340 m/s \\ Density & 1.0 kg/m\({}^{3}\) \\ \hline \end{tabular}
\end{table} 
Table 1: Parameters for the SPH algorithms of sound propagation and interference. Figure 1: Sound propagation pressure of acoustic wave at \(t=0.25\) s.  
by the artificial viscosity are closer to the theoretical solution. Furthermore, all of the points are stable with this result, and no obvious unphysical peak values are obtained. Similarly, the simulation results in Figure 2b altered with artificial viscosity are better than those obtained using the standard SPH algorithm, as can be seen from the values in the two black frames. Therefore, both Figure 2a and Figure 2b show that the SPH algorithm with artificial viscosity can reduce unphysical oscillations in sound propagation simulation. In Figure 2c, the scattered points of the SPH results in the region between 48 m and 58 m are plotted at intervals of four. From the figure, it can also be seen that the SPH algorithm with artificial viscosity can obtain a smoother result by effectively reducing unphysical oscillations. Figure 3 shows the changes of sound pressure during sound interference. The sound interference results are calculated on the condition that the \(\alpha_{\Pi}\) and \(\beta_{\Pi}\) are set to 0.06 and 0.50, respectively. Figure 3a gives the sources of acoustic waves in a particle form. In Figure 3b-d, the two acoustic waves propagate along two opposite directions, and there is no interference.