## 3 Proposed Method
### Sound Field Control Method Optimization Model  
Reference [30] points out that sound can be described by sound pressure and particle velocity. Reference [31] points out that when loudspeakers are placed unevenly, the loudspeaker signal obtained by controlling the sound pressure of a single region is less stable than the loudspeaker signal obtained by controlling the particle velocity of a single region. Inspired by these conclusions and combined with traditional sound field control methods, this paper proposes a sound field control method that minimizes the error of sound field reproduction in the bright zone, minimizes array effort, and controls the sound pressure and particle velocity in the dark zone at the same time. It is expected to improve the sound field reproduction effect in the bright zone, improve the stability of a loudspeaker array system, and reduce the acoustic energy radiation in the dark zone. The proposed method can be formulated as a constrained optimization problem as follows:  
\[ \begin{split}\underset{\mathbf{q}}{min}&\kappa\|\mathbf{p}_{ \mathbf{br}}-\mathbf{p}_{\mathbf{bo}}\|_{2}^{2}+(1-\kappa)\|\mathbf{p}_{\mathbf{dr}}\|_{2}^{2}+ \sigma\|\mathbf{q}\|_{2}^{2}\\ & s.t.&\|\mathbf{u}_{\mathbf{dr}}\|_{2}\leq\zeta\end{split}  \tag{20}\]where \(\kappa\) and \(\sigma\) are weight factors and \(0<\kappa<1\), \(0<\sigma<1\), \(\zeta\) indicates the threshold and is greater than zero, \(\|\mathbf{p_{br}}-\mathbf{p_{bo}}\|_{2}^{2}\) stands for the power of sound pressure error between the original sound field pressure and the reproduced sound field pressure in the bright zone, \(\|\mathbf{p_{dr}}\|_{2}^{2}\) stands for dark zone sound energy, \(\|\mathbf{q}\|_{2}^{2}\) stands for sound source power or control effort, and \(\mathbf{u_{dr}}=\mathbf{U_{dr}}\mathbf{q}\) stands for the radial particle velocity of the dark zone:  
\[\mathbf{U_{dr}}=\left( \begin{array}{cccc}u_{r}(\overrightarrow{l_{1}}, \overrightarrow{d_{1}})&u_{r}(\overrightarrow{l_{2}},\overrightarrow{d_{1}}) &\ldots&u_{r}(\overrightarrow{l_{M}},\overrightarrow{d_{1}})\\ u_{r}(\overrightarrow{l_{1}},\overrightarrow{d_{2}})&u_{r}( \overrightarrow{l_{2}},\overrightarrow{d_{2}})&\ldots&u_{r}( \overrightarrow{l_{M}},\overrightarrow{d_{2}})\\ \ldots&\ldots&\ldots&\ldots\\ u_{r}(\overrightarrow{l_{1}},\overrightarrow{d_{t}})&u_{r}( \overrightarrow{l_{2}},\overrightarrow{d_{t}})&\ldots&u_{r}( \overrightarrow{l_{M}},\overrightarrow{d_{t}})\end{array} \right) \tag{21}\]  
\[u_{r}(\overrightarrow{l_{n}},\overrightarrow{d})=\overrightarrow{u}( \overrightarrow{l_{n}},\overrightarrow{d})\cdot\overrightarrow{v}_{r}( \overrightarrow{d}) \tag{22}\]  
\[\overrightarrow{u}(\overrightarrow{l_{n}},\overrightarrow{d})=\frac{ike -\tilde{u}r}{4\pi r}\bigg{(}1+\frac{1}{\tilde{u}r}\bigg{)}\frac{( \overrightarrow{d}-\overrightarrow{l_{n}})}{r} \tag{23}\]  
where \(r\) is the distance between point \(\overrightarrow{d}\) and \(\overrightarrow{l_{n}}\) and \(r=|\overrightarrow{d}-\overrightarrow{l_{n}}|\), \(k\) is the wave number, \(i=\sqrt{-1}\), \(\overrightarrow{u}(\overrightarrow{l_{n}},\overrightarrow{d})\) is the particle velocity transmission function between a loudspeaker at \(\overrightarrow{l_{n}}\) and point \(\overrightarrow{d}\)[31], \(u_{r}(\overrightarrow{l_{n}},\overrightarrow{d})\) stands for the radial particle velocity transmission function, and \(\overrightarrow{v}_{r}(\overrightarrow{d})\) is the unit radial vector perpendicular to the surface of the dark zone and is inward. The cost function of the proposed method includes the error of reconstructed sound pressure in the bright zone, acoustic energy in the dark zone, and sound source power. The three weight factors \(\kappa\), \(\sigma\), and \(\zeta\) have their own functions in the model. \(\kappa\) is used to modulate the sound pressure error in the bright zone and acoustic energy in the dark zone, \(\sigma\) is used to adjust sound source power, and \(\zeta\) is used to control the value of particle velocity in the dark zone.