## 2 SPH Algorithm for Acoustic Simulation  
Functions in the SPH method are written as a particle approximation form. The continuous SPH integral representation for \(f(\mathbf{r})\) can be written as:  
\[\left\langle f(\mathbf{r})\right\rangle=\int f(\mathbf{r}^{\prime})\mathcal{W}(\mathbf{r} \cdot\mathbf{r}^{\prime},h)\,\mathrm{d}\mathbf{r}^{\prime} \tag{1}\]where \(f\) is a function of the vector \(r\), \(\Omega\) is the volume of the integral, \(W\) is the smoothing kernel function and \(h\) is the smoothing length. In the SPH convention, the kernel approximation operator is marked by the angle bracket \(<>\). The particle approximation for the spatial derivative can be described as:  
\[\left\langle\
abla\cdot f(\mathbf{r}_{i})\right\rangle=\sum_{j=1}^{N }\frac{m_{j}}{\rho_{j}}f(\mathbf{r}_{j})\cdot\
abla_{i}W_{ij} \tag{2}\]  
\[\
abla_{i}W_{ij}=\frac{\mathbf{r}_{i}-\mathbf{r}_{j}}{r_{ij} }\frac{\partial W_{ij}}{\partial r_{ij}}=\frac{\mathbf{r}_{ij}}{r_{ ij}}\frac{\partial W_{ij}}{\partial r_{ij}} \tag{3}\]  
where \(r_{i}\) and \(r_{j}\) indicate the position of particles \(i\) and \(j\) separately, \(\mathbf{r}_{ij}=\mathbf{r}_{i}-\mathbf{r}_{j}\), \(r_{ij}\) is the distance between particle \(i\) and particle \(j\), \(N\) is the number of particles in the computational domain, \(m_{j}\) is the mass of particle \(j\) and \(\rho_{j}\) is the density of particle \(j\). \(W_{ij}=W(r_{ij}\), \(h)\) is the smoothing kernel function. \(h\) is the smoothing length that defines the influence area of the smoothing function \(W_{ij}\). In order to evaluate the effects of the kernel function, three different smoothing kernel functions are used to simulate sound propagation and interference, which are the Gaussian kernel function [2], the cubic spline kernel function [23] and the quintic spline kernel function [24]. The Gaussian kernel function can be written as:  
\[W(r_{ij},h)=\alpha_{{}_{D}}\exp(-q^{2}) \tag{4}\]  
where \(q=r_{ij}/h\), \(\alpha_{{}_{D}}\) is \(1/(\pi^{1/2}h)\), \(1/(\pi h^{2})\) and \(1/(\pi^{3/2}h^{3})\) in one-, two- and three-dimensional space, respectively. As discussed in [25], the Gaussian kernel function is sufficiently smooth even for high orders of derivatives, and it is very stable and accurate, especially for disordered particles. However, it is not really compact, as it never goes to zero theoretically, unless \(q\) approaches infinity. The cubic spline kernel function is given by:  
\[W(r_{ij},h)=\alpha_{{}_{D}}\left\{ \begin{array}{cl}\frac{2}{3}-q ^{2}+\frac{1}{2}q^{3}&0\leq q\leq 1\\ \frac{1}{6}(2-q)^{3}&1\leq q\leq 2\\ 0&q\geq 2\end{array} \right. \tag{5}\]  
where \(q=r_{ij}/h\), \(\alpha_{{}_{D}}\) is \(1/h\), \(15/(7\pi h^{2})\) and \(3/(2\pi h^{3})\) in one-, two- and three-dimensional space, respectively. As described in [PERSON]'s thesis [25, 26], so far, the cubic spline kernel has been the most widely-used smoothing function in the SPH literature, since it resembles a Gaussian function while having a narrower compact support (it is equal to zero for \(q>2\)). However, the second derivative of the cubic spline is a piecewise linear function, so the stability properties are inferior compared to other smoothing kernel functions. The quintic spline kernel function is:\[W(r_{\bar{y}},h)=\alpha_{D}\times\left[ \begin{array}{cc}(3-q)^{5}-6(2-q)^{5}+15(1 -q)^{5}&0\leq q<1\\ (3-q)^{5}-6(2-q)^{5}&1\leq q<2\\ (3-q)^{5}&2\leq q<3\\ 0&q>3\end{array} \right. \tag{6}\]  
where \(q=r_{\bar{y}}/h\), \(\alpha_{D}\) is \(1/(120\ h)\), \(7/(478\ \pi h^{2})\) and \(3/(359\ \pi h^{3})\) in one-, two- and three-dimensional space, respectively. According to the discussion on the quintic spline kernel function by [PERSON] _et al._[27], the quintic spline kernel function is a high order spline function and is not sensitive to the disorder particle distribution. The compact support domain of the quintic spline kernel function is narrower than the Gaussian kernel function.