# A Multi-Tone Sound Absorber Based on an Array of Shunted Loudspeakers
## 2 Theory  
The schematic of the shunted loudspeaker array is presented in Figure 1a, where four shunted loudspeakers with equal areas are installed at the left terminal \(z=0\) of a square impedance duct. A plane wave is incident normally on the right terminal at \(z=L\) with a constant particle velocity \(v_{0}(\omega)\). Each shunt loudspeaker (named as SL\({}_{1}\) to SL\({}_{4}\)) consists of a closed-box loudspeaker and a shunt circuit connected to the loudspeaker's terminals. The length of the duct cross section is \(a\), and the length of the front face of each shunted loudspeaker unit is \(a/2\) respectively. The shunt circuit is shown in Figure 1b, where the negative resistance \(-R_{\mathrm{e}}\) and negative inductance \(-L_{\mathrm{e}}\) are used to counteract the D.C. resistance \(R_{\mathrm{E}}\) and voice coil inductance \(L_{\mathrm{E}}\) of the loudspeaker respectively. The capacitance \(C_{\mathrm{s}}\) and inductance \(L_{\mathrm{s}}\) can be switched according to the design targets. Figure 1: _Cont._  
The specific acoustic impedance of the \(i\)th shunted loudspeaker (\(i=1,2,3,4\)) is [4; 19]  
\[Z_{i}=\frac{R_{ms}}{S_{0}}+\mathrm{j}\omega\frac{M_{ms}}{S_{0}}+\frac{1}{\mathrm{ j}\omega C_{ms}S_{0}}+\frac{S_{0}}{\mathrm{j}\omega C_{\mathrm{ab}}}+\frac{B^{2}l^{2} }{S_{0}(R_{E}+\mathrm{j}\omega L_{E}+Z_{\mathrm{s}})}, \tag{1}\]  
where \(\omega\) is the angular frequency, \(R_{\mathrm{ms}}\) is the mechanical resistance of the driver suspension losses of the loudspeaker, \(S_{0}\) is the effective surface area of the driver cone of the loudspeaker, \(M_{\mathrm{ms}}\) is the mechanical mass of the driver cone (including reactive air load) of the loudspeaker, \(C_{\mathrm{ms}}\) is the mechanical compliance of the driver suspension of the loudspeaker, \(C_{\mathrm{ab}}=V/\rho_{0}c_{0}{}^{2}\) is the equivalent acoustic capacitance due to the back cavity of the loudspeaker, \(V\) is the volume of the back cavity, \(\rho_{0}\) and \(c_{0}\) are the air density and sound velocity respectively, \(B\) is the magnetic flux density of the loudspeaker driver, \(l\) is the voice coil length, and \(Z_{\mathrm{s}}\) is the electrical impedance of the shunt circuit. Omitting the time harmonic factor \(e^{\mathrm{j}\omega t}\), the acoustic field inside the impedance duct shown in Figure 1a can be expanded using the modal solution as  
\[p(x,y,z,\omega)= \sum_{q=0}^{\infty}\psi_{q}(x,y,k_{q})\left(A_{q}\mathrm{e}^{ \mathrm{j}\sqrt{k_{0}^{2}-k_{q}^{2}}z}+B_{q}\mathrm{e}^{-\mathrm{j}\sqrt{k_{0} ^{2}-k_{q}^{2}}z}\right), \tag{2}\]  
where \(q\) is the mode index, \(k_{0}=\omega/c_{0}\) and \(k_{q}=q\pi/a\) are the total wavenumber and the lateral wavenumber of the \(q\)th mode respectively, \(\psi_{q}\big{(}x,y,k_{q}\big{)}=\cos(k_{q}x)\cos(k_{q}y)/\sqrt{S}\) is the \(q\)th mode function, \(S=a^{2}\) is the section area of the duct, \(A_{q}\) and \(B_{q}\) are the \(q\)th mode coefficients of the incident wave and reflected wave respectively. The boundary conditions at \(z=0\) and \(z=L\) are  
\[v_{z}(x,y,z,\omega)\big{|}_{z=L} =\left.\frac{1}{-\mathrm{j}\omega\rho_{0}}\frac{\partial p(x,y,z, \omega)}{\partial z}\right|_{z=L}=v_{0}(\omega), \tag{3}\] \[\text{and }\left.\left[\frac{\partial p(x,y,z,\omega)}{\partial z}- \mathrm{j}k_{0}Z_{0}(x,y,\omega)p(x,y,z,\omega)\right]\right|_{z=0}=0, \tag{4}\]  
where \(v_{z}(x,y,z,\omega)\) is the velocity in the \(z\) direction, and the specific acoustic impedance is  
\[Z_{0}(x,y,\omega)= \begin{cases}Z_{1}&x\in(0,a/2),y\in(a/2,a)\\ Z_{2}&x\in(a/2,a),y\in(a/2,a)\\ Z_{3}&x\in(0,a/2),y\in(0,a/2)\\ Z_{4}&x\in(a/2,a),y\in(0,a/2)\end{cases} . \tag{5}\]  
Figure 1: (**a**) A schematic of the shunted loudspeaker array set at the end of an impedance duct; (**b**) the equivalent circuit of a shunt loudspeaker unit.