# Experimental Analysis of Ultrasonic Multiple Scattering Attenuation through the Air with Fine Dust
## 6 Discussion  
Owing to the scattering attenuation obtained by the experiment, the number of fine dust particles per unit volume can be quantified with a scattering cross-section, based on Equation (4). For practical purposes, the scattering cross-section of fine dust is usually calculated assuming a spherical particle shape. However, the shape of fine dust particles is neither spherical nor regular. Therefore, we further investigated the shape effect with respect to the scattering cross-section to understand the variation range in the estimation of number density. Analytical and numerical simulation studies were conducted to determine the scattering cross-section. The analytical solution of the scattering cross-section for a perfect circular particle can be calculated using Equation (11). \[\sigma_{t}=4a\frac{1}{ka}\left[\left|\frac{J_{0}^{\prime}(ka)}{H_{0}^{(2)^{ \prime}}(ka)}\right|^{2}+2\sum\left|\frac{J_{l}^{\prime}(ka)}{H_{0}^{(2)^{ \prime}}(ka)}\right|^{2}\right], \tag{11}\]  
where \(J_{0}\) and \(H_{0}^{(2)}\) are the Bessel functions of the first kind and Henkel function of the second kind, respectively. It is notable that the analytical expression correlates with the applied frequency range and size parameter in Equation (10). If the shape of the particle is not circular, the cross-section can be numerically calculated as:  
\[\sigma_{t}=\int_{0}^{2\pi}|f(\varphi)|^{2}d\varphi \tag{12}\]  
where \(f(\varphi)\) represents the scattered amplitude as a function of the azimuth (\(\varphi\)). The scattering cross-section is the total summation of the amplitude of the scattered wave, which can be measured by the sensors around the particle. A schematic illustration of the simulation is presented in Figure 10, where the scattered wave field is obtained by the sensors around the particle after the incident plane waves are propagated. The particle  
Figure 9: Coherent wave attenuation across a range of standard particle dosage: (**a**) at several frequencies, and (**b**) at averaged frequencies over a statistically valid region. shape was obtained from the scanning electron microscopy image of the actual fine dust sample, as illustrated in Figure 11a. The simulation was performed using a finite-difference time-domain (FDTD) simulation tool \(k\)-wave [25], and the simulation parameters used in this study are summarized in Table 1.  
In the numerical simulation, the scattering cross-sections of three different particle shapes were computed. A conventional desktop computer with a GPU (NVIDIA GeForce \begin{table}
\begin{tabular}{c c c} \hline \hline \multicolumn{3}{c}{**Material Properties**} \\ \hline  & **Wave Speed (m/s)** & **Mass Density (kg/m\({}^{3}\))** \\ \hline Air & 343 & 1.2754 \\ Fine dust particle & 343 & 500 \\ \hline \multicolumn{3}{c}{**Simulation parameters**} \\ \hline Number of grid points (\(\text{N}_{\text{x}}\times\text{N}_{\text{y}}\)) & 1500 \(\times\) 1500 \\ Grid spacing (dx and dy) & 0.057 μm \\ Time step (dt) & 0.02 ns \\ Time duration (\(\Gamma\)) & 4 μs \\ Number of sensing points & 3676 (sensor array radius 37.05 μm) \\ Sensor spacing along \(y\) axis & 1.65 μm \\ CFL condition & 0.1 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Summary of numerical simulation parameters. Figure 11: Analysis of scattering cross-section (**a**) particle shapes, (**b**) computed scattering cross-section. Figure 10: Schematic illustration of scattering cross-section calculation. RTX 2060 Super) was used for the simulation, and the average computation time per particle was approximately 16 h. To identify the directional effect, additional simulations were performed using 90\({}^{\circ}\) rotated particles (marked as vertical) relative to the incident wave. Figure 11b presents seven scattering cross-section values, including an analytical solution. The results obtained indicate that the analytical solution is within an acceptable range relative to SEM (1) vertical particle at the applied frequencies in this study.