# Numerical Analysis of Ultrasonic Multiple Scattering for Fine Dust Number Density Estimation
## 4 Numerical Simulations  
Two-dimensional acoustic plane wave propagation in air with fine dust particles was simulated to generate acoustic wavefield data and evaluate the proposed number density estimation approach. A finite-difference time-domain (FDTD) simulation tool, \(k\)-wave, was used in this study [26]. The simulation was performed using a desktop computer with a graphical processing unit (GPU) (NVIDIA GeForce RTX 2060 Super), and the total calculation time per model was approximately 23 h.  
The overall structure of the numerical simulation model is illustrated in Figure 3 and the numerical simulation parameters are summarized in Table 1. The simulated media represent 1425 \(\upmu\)m \(\times\) 57 \(\upmu\)m air spaces where 10 \(\upmu\)m fine dust particles were randomly spread out with different volume fractions (1, 2, 5, 10, and 15%). For each particle volume fraction case, two models with different particle placements were simulated. A spatially random distribution of dust particles is assumed to represent the air-particle system in reality. Perfectly matched layers (PMLs) [27] were added at the  
Figure 2: Schematic of the proposed number density estimation approach.  
two ends of the simulation space to eliminate boundary reflections, as shown in Figure 3. PMLs were not added to the two edges parallel to the wave propagation path to model a plane wave incidence condition. Plane acoustic waves were generated with an excitation signal that was a linear chirp, across 0 to 10 MHz (see Figure 3), applied to the left end of the air space. We chose the frequency band (0-10 MHz) that gives a _ka_ value below 1 to use the simplified analytical solution of the total scattering cross-section (Equation (13)). The corresponding acoustic wavefields were captured by a 35 \(\times\) 30 sensor array at each model, as shown in Figure 3. Here, each of the 35 acoustic wave signals along the _x_-axis represented a signal set (1-D scan data) for a single realization \(\mathrm{s}_{\mathrm{m}}(x_{i},t)\). To obtain an ensemble-averaged signal set \(\left\langle\mathrm{s}(x_{i},t)\right\rangle,\) multiple realizations were needed. For each volume fraction case with two medium models, 60 acoustic wave signal sets (30 signal sets per model) were obtained to give the number of ensemble averaging (\(M\)) of 60. Two separate models, each providing 35 signal sets, were simulated, instead of a unified model providing 60 signal sets, for each volume fraction case due to limited computation power.  
\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 & 1050 \\ \hline \multicolumn{3}{c}{**Simulation Parameters**} \\ \hline Number of grid points (\(\mathrm{N}_{\mathrm{x}}\times\mathrm{N}_{\mathrm{y}}\)) & 25,000 \(\times\) 1000 \\ Grid spacing (dx and dy) & 0.057 μm \\ Time step (dt) & 0.0166 ns \\ Time duration (T) & 8 μs \\ Number of sensing points at & 1050 \\ each medium & (35 \(\times\) 30) \\ Sensor spacing along \(x\) axis & 36 μm \\ Sensor spacing along \(y\) axis & 1.65 μm \\ Fine dust particle & 1, 2, 5, 10, and 15\% \\ volume fractions & \\ \hline \hline \end{tabular}
\end{table}
Table 1: Summary of the numerical simulation parameters.  
Figure 3: The overall structure of the numerical simulation model. At each volume fraction case, two models that had different scatterer distributions were simulated to obtain acoustic wave responses across multiple realizations.