## 2 Method
### Synthesizing the Required Pressure Field by Finite Control Points  
For simplicity, here we only consider the two-dimensional (2D) domain problem and it is convenient to extend our method to the three-dimensional (3D) case. In a fluid medium without any body forces (Figure 1b), the acoustic pressure within an arbitrarily shaped domain \(D\) is related to the pressure and velocity on the boundary \(S\) as described by Kirchhoff-Helmholtz integral equation [54]. Discretizing the boundary into \(N\) elements, the pressure \(p\) at any location \(\mathbf{r}_{j}\) within \(D\) can be given by  
\[p\big{(}\mathbf{r}_{j}\big{)}=\sum_{q=1}^{N}\Bigg{(}p\big{(}\mathbf{r}_{q} \big{)}\frac{\partial G\big{(}\mathbf{r}_{q},\mathbf{r}_{j}\big{)}}{\partial \mathbf{n}\big{(}\mathbf{r}_{q}\big{)}}-G\big{(}\mathbf{r}_{q},\mathbf{r}_{j} \big{)}\frac{\partial p\big{(}\mathbf{r}_{q}\big{)}}{\partial\mathbf{n}\big{(} \mathbf{r}_{q}\big{)}}\Bigg{)}\Delta S_{q}, \tag{4}\]  
where \(\mathbf{r}_{q}\), \(\mathbf{n}\big{(}\mathbf{r}_{q}\big{)}\), and \(\Delta S_{q}\) indicate the center position, normal direction, and length of the \(q\)th boundary element, respectively. \(G\big{(}\mathbf{r}_{q},\mathbf{r}_{j}\big{)}\), the free-space Green's function, which represents the wave emitted from a point source located at \(\mathbf{r}_{q}\) and measured at location \(\mathbf{r}_{j}\) in a free space, has the following form  
\[G\big{(}\mathbf{r}_{k},\mathbf{r}_{j}\big{)}=\frac{i}{4}H_{0}^{(1)}\big{(}k \|\mathbf{r}_{k}-\mathbf{r}_{j}\|\big{)}, \tag{5}\]  
where \(H_{0}^{(1)}(\cdot)\) is the zero order Hankel function of the first kind, \(k=2\pi/\lambda\) is the wavenumber, \(\lambda\) is the wavelength of the incident wave, \(i\) is the imaginary unit, and \(\|\cdot\|\) is the operator of Euclidean distance. When \(\mathbf{r}_{j}\) is located on the boundary \(S\), the Kirchhoff-Helmholtz equation becomes [54]  
\[\frac{1}{2}p\big{(}\mathbf{r}_{j}\big{)}=\sum_{q=1}^{N}\Bigg{(}p\big{(} \mathbf{r}_{q}\big{)}\frac{\partial G\big{(}\mathbf{r}_{q},\mathbf{r}_{j} \big{)}}{\partial\mathbf{n}\big{(}\mathbf{r}_{q}\big{)}}-G\big{(}\mathbf{r}_ {q},\mathbf{r}_{j}\big{)}\frac{\partial p\big{(}\mathbf{r}_{q}\big{)}}{ \partial\mathbf{n}\big{(}\mathbf{r}_{q}\big{)}}\Bigg{)}\Delta S_{q}. \tag{6}\]  
Then, the normal derivative of \(p\big{(}\mathbf{r}_{q}\big{)}\) can be expressed by the pressure on the boundary elements. Combining Equations (4) and (6), we can see that \(p\big{(}\mathbf{r}_{j}\big{)}\) is only determined by the pressure of the discretized boundary elements. For equally discretized boundary elements, when the interval between the adjacent element centers satisfies Nyquist condition, the complete boundary information can be reproduced. Thus, any required sound field can be synthesized by a finite number of control points distributed on the boundary. Supposing that a circular area with radius \(r_{0}\), as illustrated in Figure 1c, is the target region with required pressure distribution. Mode-matching is performed to derive the pressure of the control  
Figure 1: (**a**) The acoustic radiation torque on an asymmetric particle in the acoustic field is dependent on the angular position \(\sigma\); (**b**) The geometric parameter definitions for discretized Kirchhoff–Helmholtz integral equation within an arbitrary shaped 2D domain; (**c**) The principles of sound field synthesis technique. Multi plane wave transducers are homogenously mounted on a circular and the desired acoustic field is synthesized within the range \(S\). points with the following procedures.