## 1 Introduction
To make the scattered objects invisible to the incident acoustic waves, it is necessary to control both the forward-scattered sound and backward-scattered sound simultaneously. Multi-Zone Active Noise Control of the Scattered Sound from an Infinite Rigid Cylinder with Uniformly Placed Secondary Loudspeakers  
When a plane wave is incident vertically on an infinite rigid cylinder, the scattered acoustic transfer function (from the primary source to the error point) at a point with cylindrical coordinates \((\rho,\theta)\) can be expressed as [5; 25]:  
\[Z_{pc\_sca}(\rho,\theta)=-\mathrm{i}w\rho_{v}\sum_{n=0}^{\infty}\epsilon_{n}( -\mathrm{i})^{n}\frac{J_{n}^{\prime}(ka)}{\mathrm{H}_{n}^{(1)\prime}(ka)} \mathrm{H}_{n}^{(1)}(k\rho)\cos(n\theta) \tag{1}\]where \(k=\frac{w}{c}\) denotes the wave number, \(w\) is the angular frequency, \(\rho_{o}\) is the fluid density, \(c\) is the speed of sound, \(a\) is the radius of an infinite rigid cylinder, \(\int_{n}(\cdot)\) represents the Bessel function of the first type, \(n\) is order, \(\int_{n}^{\prime}(\cdot)\) denotes its derivative, \(\mathrm{H}_{n}^{(1)}(\cdot)\) denotes the Hankel function of the first type, \(\mathrm{H}_{n}^{(1)\prime}(\cdot)\) represents its derivative, \(\varepsilon_{n=0}=1\), and \(\varepsilon_{n\
eq 0}=2\). When an infinite line of monopoles parallel to the cylinder is incident at cylindrical coordinates \((\rho_{s},\theta_{s})\) and the infinite line of monopoles is treated as a secondary loudspeaker, the transfer function from the secondary loudspeaker to the error point with cylindrical coordinates \((\rho,\theta)\) can be written as [5; 25]:  
\[Z_{se}(\rho,\theta)=\mathrm{i}w\rho_{v}\sum_{n=0}^{\infty}\varepsilon_{n}[ ]_{n}(k\rho_{inf})-\frac{\int_{n}^{\prime}(ka)}{\mathrm{H}_{n}^{(1)\prime}(ka) }\mathrm{H}_{n}^{(1)}(k\rho_{inf})]\mathrm{H}_{n}^{(1)}(k\rho_{sup})\cos[n( \theta-\theta_{s})] \tag{2}\]  
where \(\rho_{inf}=\mathrm{inf}(\rho,\rho_{s})\), \(\rho_{sup}=\mathrm{sup}(\rho,\rho_{s})\). In the free field, Equations (1) and (2) can be adequately calculated for the scattered sound within \(ka=100\). The truncation order of (1) must be greater than \(ka\), and the truncation order of (2) must be greater than \(10 ka\)[5]. As shown in Figure 1a, one plane wave is incident, \(NL\) secondary loudspeakers are uniformly placed at a distance of \(r_{s}\) from the cylinder, \(N_{e1}\) error points are placed in the backward-scattered sound control area of the cylinder that is denoted as target area 1, \(N_{e2}\) error points are placed in the forward-scattered sound control area of the cylinder that is denoted as target area 2, and \(N_{ob}\) observing points are uniformly placed at a distance of \(r_{far}\) from the cylinder that is denoted as far-field observation area. The scattered sound pressure matrix of the error area (including target area 1 and target area 2) is expressed as:  
\[\mathbf{P}_{err\_ANCOff}=\left[ \begin{array}{c}\mathbf{Z}_{pe1\_sca}\\ \mathbf{Z}_{pe2\_sca}\end{array} \right]\mathbf{q}_{p} \tag{3}\]  
where \(\mathbf{Z}_{pe1\_sca}\) is the matrix (dimension \(N_{e1}\times 1\)) of scattered acoustic transfer functions of target area 1 when one plane wave is incident, \(\mathbf{Z}_{pe2\_sca}\) is the matrix (dimension \(N_{e2}\times 1\)) of scattered acoustic transfer functions of target area 2 when one plane wave is incident, and \(\mathbf{q}_{p}\) is the strength vector (dimension \(1\times 1\)) of one incident plane wave. Figure 1: Schematic of: (**a**) the RLS algorithm with the uniformly placed secondary loudspeakers; and (**b**) the new algorithm. The black crosses denote the constraint points. When \(NL\) secondary sources are activated, the sound pressure matrix of the error area (including target area 1 and target area 2) can be written as:  
\[\mathbf{P}_{err\_ANCon}=\left[ \begin{array}{c}\mathbf{Z}_{sel\_uni}\\ \mathbf{Z}_{so2\_uni}\end{array} \right]\mathbf{q}_{s}+\left[ \begin{array}{c} \mathbf{Z}_{pe1\_sca}\\ \mathbf{Z}_{pe2\_sca}\end{array}