# A Multi-Tone Sound Absorber Based on an Array of Shunted Loudspeakers
## 3 Simulations  
In this section, the acoustic properties of four parallel arranged shunted loudspeakers are investigated by both the proposed method and the finite element method. The same loudspeaker is adopted in each shunted loudspeaker, and the measured Thiele-Small (TS) parameters of the loudspeaker are listed in Table 1. The resonance frequencies of the shunted loudspeakers are chosen as 100 Hz, 200 Hz, 300 Hz, and 400 Hz, and the designed shunt circuit parameters are listed in Table 2.  
The sound absorption coefficient of the array composed of SL\({}_{1}\)-SL\({}_{4}\) can be calculated by the finite element model built in the commercial software (Comsol Multiphysics v5.3) as shown in Figure 2. The plane at \(z=1.8\) m is set as a plane wave incident surface and e incident sound pressure is 1 Pa. The bottom surface at \(z=0\) m is uniformly divided into four parts, which are defined as impedance boundaries. The size of each impedance boundary is the same as the effective area in Table 1. Set the impedance at \((0<x<a/2,\,a/2<y<a)\) of SL\({}_{1}\) unit boundary as \(Z_{1}\), the impedance at \((a/2<x<a,\,a/2<y<a)\) of SL\({}_{2}\) unit boundary as \(Z_{2}\), the impedance at \((0<x<a/2,\,0<y<a/2)\) of SL\({}_{3}\) unit boundary as \(Z_{3}\), and the impedance at \((a/2<x<a,\,0<y<a/2)\) of SL\({}_{4}\) unit boundary as \(Z_{4}\). The element size is selected as an extremely fine mesh size. The free tetrahedral mesh consists of 24,973 domain elements, 3344 boundary elements, and 315 edge elements, with the number of degrees of freedom as 36536. The absorption coefficient is calculated by  
\[\alpha=1-\frac{E_{r}}{E_{\rm in}}, \tag{16}\]  
where \(E_{\rm r}\) is the reflected energy and \(E_{\rm in}\) is the incident energy at the surface \(z=0\). \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & **Notation** & **Value** & **Unit** \\ \hline DC resistance & \(R_{\rm E}\) & 32.00 & \(\Omega\) \\ Inductance of coil & \(L_{\rm E}\) & 7.24 & mH \\ Moving mass & \(M_{\rm ms}\) & 15.25 & g \\ Mechanical resistance & \(R_{\rm ms}\) & 1.57 & kg/s \\ Mechanical compliance & \(C_{\rm ms}\) & 0.67 & mm/N \\ Force factor & \(Bl\) & 17.12 & T-m \\ Effective area & \(S_{0}\) & \(1.51\times 10^{-2}\) & m\({}^{2}\) \\ Back cavity volume & \(V\) & \(2.2\times 10^{-3}\) & m\({}^{3}\) \\ Cavity depth & \(D\) & 7.5 & cm \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Measured Thiele–Small (TS) parameters and dimensions of a closed-box loudspeaker unit. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \(\mathbf{SL_{1}}\) & \(\mathbf{SL_{2}}\) & \(\mathbf{SL_{3}}\) & \(\mathbf{SL_{4}}\) \\ \hline \(R\) (\(\Omega\)) & \(-31.95\) & \(-31.95\) & \(-31.95\) \\ \(L\) (mH) & \(-7.23\) & \(-7.23\) & \(-7.23\) & \(-7.23\) \\ \(L_{\rm s}\)The solid curve in Figure 3 shows the normal incidence absorption coefficient of the shunted loudspeaker array by using the proposed method in Section 2. Four resonance frequencies occur at 100 Hz, 200 Hz, 300 Hz, 400 Hz, where the sound absorption coefficients are all 1.00. The dashed curve in Figure 3 reveals the result by using the Finite Element Method (FEM), where the frequency deviations at the resonance frequencies are 0 Hz, 5 Hz, 6 Hz, and 4 Hz, respective due to a simulation error in Comsol. The dotted curves reflect the sound absorption coefficients of SL\({}_{1}\)-SL\({}_{4}\) units, where the resonance frequencies are 100 Hz, 200 Hz, 300 Hz, 400 Hz of each unit respectively. Comparing with the resonance frequencies of the array, it is revealed that each shunted loudspeaker almost works independently. Therefore, a multi-tone noise absorber can be designed by using multiple shunted loudspeakers with different resonance frequencies in the same plane, where each unit can be designed independently.