## 2 Proposed Virtual Sound Source Construction Method
### Wave Field Synthesis in Proposed Method  
This section describes the generation of driving functions in the proposed method. As described above, the proposed method utilizes two-dimensional WFS in the \(xy\)-plane. Figure 1: Overview of VSS construction utilizing the proposed method. Because the secondary sources are located continuously along the \(x\)-axis, the sound pressure observed at a certain position \(\mathbf{r}\) can be expressed as  
\[\beta(\mathbf{r},f)=\int_{-\infty}^{\infty}\hat{\mathcal{D}}_{\rm 2D}(\mathbf{r}_{ \rm o},f)\hat{\mathcal{G}}_{\rm 2D}(\mathbf{r}-\mathbf{r}_{\rm o},f)dx_{\rm o}, \tag{6}\]  
where \(f\) denotes the frequency of the sound source, \(\hat{\mathcal{D}}_{\rm 2D}(\mathbf{r}_{\rm o},f)\) denotes the two-dimensional driving function for the secondary sound source at \(\mathbf{r}_{\rm o}=(x_{\rm o},y_{\rm o})\) in the frequency domain, and \(\hat{\mathcal{G}}_{\rm 2D}(\mathbf{r}-\mathbf{r}_{\rm o},f)\) denotes the two-dimensional Green's function between \(\mathbf{r}\) and \(\mathbf{r}_{\rm o}\) in the frequency domain. The Green's function can be expressed as  
\[\hat{\mathcal{G}}_{\rm 2D}(\mathbf{r}-\mathbf{r}_{\rm o},f)=\frac{j}{4}H_{0}^{(2)}(k| \mathbf{r}-\mathbf{r}_{\rm o}|), \tag{7}\]  
where \(j=\sqrt{-1}\) denotes the imaginary unit, \(H_{0}^{(2)}(\cdot)\) denotes the zero-order Hankel function of the second kind, and \(k\) denotes the wavenumber. In the proposed method, we construct focused sound sources [8; 22] as VSSs as shown in Figure 1. Figure 2 gives a comparison of the wavefront of a real monopole sound source and that of a focused VSS constructed with WFS. The arrows in red indicate the propagation direction, the dash lines in purple indicate the wavefronts, and the dotted lines in black indicate the boundary of tao half-spaces. The sound field converges in one half-space towards the focal point while it diverges in the other half-space. In the half-space where the sound field diverges, the wavefront is approximately the same as that of real monopole sound source. In the case of focused VSS, the driving function is given by the following expression [22]:  
\[\hat{\mathcal{D}}_{\rm 2D}(\mathbf{r}_{\rm o},f)=-g_{\rm o}\frac{jk(y_{\rm o }-y_{\rm S})}{2|\mathbf{r}_{\rm o}-\mathbf{r}_{\rm S}|}H_{1}^{(1)}(k|\mathbf{r}_{\rm o}- \mathbf{r}_{\rm S}|), \tag{8}\] \[g_{\rm o}=\sqrt{2\pi|y_{\rm Ref}-y_{\rm o}|}, \tag{9}\]  
where \(H_{1}^{(1)}(\cdot)\) denotes the first-order Hankel function of the first kind, and \(y_{\rm Ref}\) denotes the distance from the secondary sound source to the reference line in the \(y\)-axis direction. To utilize a loudspeaker array as secondary sound sources, it is necessary to apply discretization and finite-length truncation to the above driving function [27; 28]. In our method, these two instances of processing are carried out in the same way as those in [22]. After discretization, the driving function in the frequency domain is given as  
\[\mathcal{D}_{\rm 2D}(\mathbf{r}_{i},f)=\hat{\mathcal{D}}_{\rm 2D}(\mathbf{r}_{i},f) \cdot\frac{1}{\Delta x}\sum_{\mu=-\infty}^{\infty}\delta(x-\mu\Delta x), \tag{10}\]  
Figure 2: Comparison of a real sound source and a focused VSS constructed by WFS. where \(\delta(\cdot)\) denotes the Dirac delta function and \(\Delta x\) denotes the interval of the secondary sources on the \(x\)-axis. After finite-length truncation, the driving function in the frequency domain can be given as  
\[\mathcal{D}_{i}(f)=\mathcal{D}_{\text{2D}}(\mathbf{r}_{i},f)\cdot \begin{cases}1&1\leq i \leq I,\\ 0&\text{otherwise}.\end{cases}