## Appendix A  
The simulation was modeled as the Yee scheme of the FDTD method (Figure 11). The wave equation is expressed as a two-dimensional linear acoustic domain [39]. \[\mathrm{v}_{\mathrm{x}}^{[\mathrm{n}+0.5]}(\mathrm{u}+0.5,\mathrm{w}) = \mathrm{v}_{\mathrm{x}}^{[\mathrm{n}-0.5]}(\mathrm{u}+0.5,\mathrm{ w})-\frac{\delta\mathrm{t}}{\rho_{0}\delta\mathrm{s}}\times[\mathrm{p}^{[ \mathrm{n}]}(\mathrm{u}+1,\mathrm{w})-\mathrm{p}^{[\mathrm{n}]}(\mathrm{u}, \mathrm{w})] \tag{11}\] \[\mathrm{v}_{\mathrm{y}}^{[\mathrm{n}+0.5]}(\mathrm{u},\mathrm{w}+0. 5) = \mathrm{v}_{\mathrm{y}}^{[\mathrm{n}-0.5]}(\mathrm{u},\mathrm{w}+0. 5)-\frac{\delta\mathrm{t}}{\rho_{0}\delta\mathrm{s}}\times[\mathrm{p}^{[ \mathrm{n}]}(\mathrm{u},\mathrm{w}+1)-\mathrm{p}^{[\mathrm{n}]}(\mathrm{u}, \mathrm{w})] \tag{12}\]\[ \begin{array}{ll}\text{p}^{[n+1]}(\text{u},\text{w})=\text{p}^{[n]}(\text{u}, \text{w})&-\frac{\partial\text{v}^{2}\delta\text{t}}{\delta\text{g}}[\text{v}_{ \text{x}}^{[\text{n}+0.5]}(\text{u}+0.5\,,\text{w})-\text{v}_{\text{x}}^{[ \text{n}+0.5]}(\text{u}-0.5\,,\text{w})]\\ &-\frac{\partial\text{v}^{2}\delta\text{t}}{\delta\text{s}}[\text{v}_{\text{y}}^ {[\text{n}+0.5]}(\text{u},\text{w}+0.5)-\text{v}_{\text{y}}^{[\text{n}+0.5]}( \text{u},\text{w}-0.5\,)]\end{array}  \tag{10}\]  
where p is the sound pressure, vx and vy are the particle velocities of the x and y axes, respectively, \(\delta\)s is the spatial discretization step, \(\delta\)t is the time discretization step, u and w are indices of the spatial point, \(\rho_{0}\) is the air density, and c is the speed of sound. The equation of the boundary condition causing reflection is expressed using the asymmetric finite-difference approximation expressed in [39]. \[\frac{\partial\text{p}}{\partial\text{x}}\bigg{|}^{[\text{n}]}(\text{u}+0.5\,, \text{w})=\frac{2}{\delta\text{s}}\Big{[}\text{p}^{[\text{n}]}(\text{u}+0.5\,, \text{w})-\text{p}^{[\text{n}]}(\text{u},\text{w})\Big{]} \tag{11}\]  
In Figure 11, the boundary of the right side on the x-axis is expressed in Equation (10). No term of \(\text{p}^{[\text{n}]}(\text{u}+1,\text{w})\) exists in the momentum Equation (11). Therefore, when the approximation of Equation (11) is introduced into Equation (11), it is expressed as follows:  
\[\text{v}_{\text{x}}^{[\text{n}+0.5]}(\text{u}+0.5,\text{w})=\text{v}_{\text{x} }^{[\text{n}-0.5]}(\text{u}+0.5,\text{w})-\frac{2\delta\text{t}}{\rho_{0} \delta\text{s}}\times[\text{p}^{[\text{n}]}(\text{u}+0.5\,,\text{w})-\text{p}^ {[\text{n}]}(\text{u},\text{w})] \tag{12}\]  
In Equation (12), \(\text{p}^{[\text{n}]}(\text{u}+0.5\,,\text{w})\) represents the velocity point in the x direction as a spatial point that does not exist in Figure 11. The point \((\text{u}+0.5\,,\text{w})\) is an impedance boundary of the FDTD domain, which is suitable for expressing the locally reacting boundary affected only by a normal velocity because of the lattice structure of the Yee scheme. Equation (11) represents the acoustic impedance with the locally reacting boundary. \[Z=\!\!\Big{(}\frac{\text{p}}{\text{v}_{\text{n}}}\Big{)}_{\text{surface}} \tag{13}\]  
where P is the acoustic pressure, and vn is the normal velocity. \(\text{p}^{[\text{n}]}(\text{u}+0.5\,,\text{w})\) in (29) is replaced with Zv\({}^{[\text{n}]}(\text{u}+0.5\,,\text{w})\) based on Equation (13). The linear interpolation to represent vx\({}^{[\text{n}]}_{\text{x}}\) as vx\({}^{[\text{n}-0.5]}\) and vx\({}^{[\text{n}+0.5]}_{\text{x}}\) is as follows:  
\[ \begin{array}{ll}\text{v}_{\text{x}}^{[\text{n}+0.5]}(\text{u}+0.5,\text{w}) =\text{v}_{\text{x}}^{[\text{n}-0.5]}(\text{u}+0.5,\text{w})-\text{p}^{[ \text{n}]}(\text{u},\text{w})]\\ \text{v}_{\text{x}}^{[\text{n}+0.5]}(\text{u}+0.5,\text{w})=\text{v}_{\text{x}} ^{[\text{n}-0.5]}(\text{u}+0.5,\text{w})\\ -\frac{2\delta\text{t}}{\rho_{0}\delta\text{s}}\times\Big{[}\frac{Z}{2}[\text{v }^{[\text{n}-0.5]}(\text{u}+0.5\,,\text{w})\ +\ \text{v}^{[\text{n}+0.5]}(\text{u}+0.5\,,\text{w})\Big{]}-\text{p}^{[\text{n}]}( \text{u},\text{w})\Big{]}\end{array} .