## 3 Sound Field Simulation and Its Analysis Using Acoustic Parameters
### Simulation Test for the Reverberant Environment
#### 3.1.1 Simulation Setup  
The FDTD method is the numerical solution of the differential equation of a wave. The FDTD method is commonly used for nonstaggered compact schemes expressing only pressure [38] and [PERSON]'s staggered schemes expressing particle velocity and pressure [39]. In this study, the simulation was modeled as [PERSON]'s scheme to use a circular rigid body [40] and a perfectly matched layer (PML) boundary [41]. The circular rigid body boundary was used to model the silent intruder because the characteristics of a person and a PVC pipe were observed to be similar. The PML condition was used to describe the anechoic environment. The reverberation of the control space was controlled by adjusting the sound absorption coefficient at the boundary. Hence, the momentum equation with the impedance boundary condition was used, and it is expressed as follows:  
\[\mathrm{v}_{\mathrm{x}}^{[\mathrm{n}+0.5]}(\mathrm{u}+0.5,\mathrm{w})= \left(\frac{1-\lambda_{\mathrm{c}}\zeta}{1+\lambda_{\mathrm{c}}\zeta} \right)\mathrm{v}_{\mathrm{x}}^{[\mathrm{n}-0.5]}(\mathrm{u}+0.5,\mathrm{w}) +\frac{2\lambda_{\mathrm{c}}}{\rho_{0}\mathrm{c}(1+\lambda_{\mathrm{c}}\zeta )}\mathrm{P}^{[\mathrm{n}]}(\mathrm{u},\mathrm{w}) \tag{11}\]  
\[\mathrm{v}_{\mathrm{x}}^{[\mathrm{n}+0.5]}(\mathrm{u}-0.5,\mathrm{w})= \left(\frac{1-\lambda_{\mathrm{c}}\zeta}{1+\lambda_{\mathrm{c}}\zeta} \right)\mathrm{v}_{\mathrm{x}}^{[\mathrm{n}-0.5]}(\mathrm{u}-0.5,\mathrm{w}) -\frac{2\lambda_{\mathrm{c}}}{\rho_{0}\mathrm{c}(1+\lambda_{\mathrm{c}}\zeta )}\mathrm{P}^{[\mathrm{n}]}(\mathrm{u},\mathrm{w}) \tag{12}\]  
\[\mathrm{v}_{\mathrm{y}}^{[\mathrm{n}+0.5]}(\mathrm{u},\mathrm{w}+0.5)= \left(\frac{1-\lambda_{\mathrm{c}}\zeta}{1+\lambda_{\mathrm{c}}\zeta} \right)\mathrm{v}_{\mathrm{y}}^{[\mathrm{n}-0.5]}(\mathrm{u},\mathrm{w}+0.5) +\frac{2\lambda_{\mathrm{c}}}{\rho_{0}\mathrm{c}(1+\lambda_{\mathrm{c}}\zeta )}\mathrm{P}^{[\mathrm{n}]}(\mathrm{u},\mathrm{w}) \tag{13}\]  
\[\mathrm{v}_{\mathrm{y}}^{[\mathrm{n}+0.5]}(\mathrm{u},\mathrm{w}-0.5)= \left(\frac{1-\lambda_{\mathrm{c}}\zeta}{1+\lambda_{\mathrm{c}}\zeta} \right)\mathrm{v}_{\mathrm{y}}^{[\mathrm{n}-0.5]}(\mathrm{u},\mathrm{w}-0.5) -\frac{2\lambda_{\mathrm{c}}}{\rho_{0}\mathrm{c}(1+\lambda_{\mathrm{c}}\zeta )}\mathrm{P}^{[\mathrm{n}]}(\mathrm{u},\mathrm{w}) \tag{14}\]  
\[\zeta= \frac{1+\sqrt{1-\alpha}}{1-\sqrt{1-\alpha}} \tag{15}\]  
where p is the sound pressure; \(\mathrm{v}_{\mathrm{x}}\) and \(\mathrm{v}_{\mathrm{y}}\) are the particle velocities of the x and y axes, respectively; \(\rho_{0}\) is the air density; \(\mathrm{c}\) is the speed of sound; \(\lambda_{\mathrm{c}}\) is the courant number; \(\zeta\) is the specific acoustic impedance; \(\alpha\) is the absorption coefficient; n is the time index; and u and w are indices of the spatial point. In this study, this impedance boundary condition was derived by combining the asymmetric finite-difference approximation used in [39] and the locally reacting boundary used in a room simulationin [38]. The derivation is described in Appendix A. Therefore, we enabled the simulation of the reverberation environment in the Yee scheme using the change in \(\alpha\). The FDTD simulation utilized a 2 m \(\times\) 3 m control space (Figure 7) and a spatial resolution of 0.01 m. The sampling frequency (f\({}_{\text{s,FDTD}}\)) was 49 kHz. As the selection criteria of the parameters, a sampling rate that satisfied the courant condition was selected while the spatial resolution was fixed. The position of the silent intruder was set at representative positions (A, B, C, and D) as mentioned in Section 2.3.