# Numerical Analysis of Ultrasonic Multiple Scattering for Fine Dust Number Density Estimation
## 2 Theoretical Background on Multiple Scattering of Acoustic Waves  
Air with fine dust can be considered a discrete random medium where fine dust particles are randomly distributed within a background medium (i.e., air). When acoustic waves propagate through a discrete random medium, they interact with scatterers. The wave propagation lies in a different regime depending on the wave propagating distance (\(L\)) and scattering mean free path (\(l_{S}\)), as illustrated in Figure 1. Here, \(l_{S}\) is the characteristic length of the exponential decay of a coherent wave that propagates in its initial direction after taking ensemble averaging over multiple realizations of disorder. Figure 1: Conceptual illustration of multiple scattering of acoustic waves. An example time-domain signal is displayed below the conceptual illustration of each regime. The scattering mean free path (\(l_{S}\)) is the characteristic length of the exponential decay of a coherent wave that propagates in its initial direction after taking ensemble averaging over multiple realizations of disorder. In the single-scattering regime (Regime 1), where \(L\) is comparable to or less than \(l_{S}\), unscattered and singly scattered wave components (presumably coherent waves) are dominant in the measured response. In this regime, the medium can be considered quasi-homogeneous, but with a renormalized wave speed. When the value of \(L\) is equivalent to a few mean free paths, single scattering becomes unimportant, and the wave propagation lies in the intermediate multiple scattering regime (Regime 2). In this regime, multiply scattered wave components are developed, but coherent waves are still observable with sufficient ensemble averaging. As \(L\) further increases to become much greater than \(l_{S}\), most wave energy is diffusively transferred by incoherent waves (Regime 3). In this regime, the transfer of the average wave intensity can be approximated by the diffusion equation. In this study, we focus on using coherent waves under Regimes 1 and 2 to estimate the number density of fine dust particles in the air, considering that acoustic wave propagation in the air-fine dust system typically lies in Regimes 1 and 2. Moreover, the independent scattering approximation (ISA) method, the basis of the proposed approach, works for Regimes 1 and 2 but not for Regime 3. For a discrete random medium (e.g., air with fine dust), the solution of acoustic wave motion \(\mathrm{G}(\omega,\ \mathbf{r},\mathbf{r}^{\prime})\) at the angular frequency \(\omega\) and the position vector \(\mathbf{r}\) given a point source location \(\mathbf{r}^{\prime}\) can be represented as the summation of the incident wave and all possible waves scattered by the scatterers (e.g., random particles) [16]. It is given by:  
\[\mathrm{G}\big{(}\omega,\ \mathbf{r},\mathbf{r}^{\prime}\big{)}=G_{0}\big{(}\omega,\mathbf{r} -\mathbf{r}^{\prime}\big{)}+\int\ G_{0}(\omega,\mathbf{r}-\mathbf{r}_{1})V(\mathbf{r}_{1}) \mathrm{G}\big{(}\omega,\ \mathbf{r}_{1},\mathbf{r}^{\prime}\big{)}d\mathbf{r}_{1}, \tag{1}\]  
where \(G_{0}(\omega,\mathbf{r}-\mathbf{r}^{\prime})\) is the Green's function (solution) of the homogeneous background medium (a non-scattering medium such as air without particles) at the frequency (\(\omega\)) given the source position (\(\mathbf{r}^{\prime}\)) and the receiver position (\(\mathbf{r}\)), and \(V(\mathbf{r}_{1})\) is the scattering potential at position (\(\mathbf{r}_{1}\)). \(\mathrm{G}(\omega,\ \mathbf{r},\mathbf{r}^{\prime})\) represents the complete wave response solution including incident and all the scattered waves for the discrete random medium. Note that \(G_{0}\) is a nonlocal operator that depends only on the distance between the source and receiver (\(|\mathbf{r}-\mathbf{r}^{\prime}|\)), and it is expressed in the wavenumber (\(\mathbf{k}\)) domain as:  
\[G_{0}(\omega,\ \mathbf{k})=\frac{1}{{k_{0}}^{2}(\omega)-k^{2}}, \tag{2}\]  
where \(k_{0}(\omega)=\omega/c_{0}\), and \(c_{0}\) is the wave speed in the homogeneous background medium.