# A Multi-Tone Sound Absorber Based on an Array of Shunted Loudspeakers
## 2 Theory
Substituting \(p(x,y,z,\omega)\) in Equation (2) into the boundary condition in Equations (3) and (4), the following is obtained:  
\[\frac{1}{\omega\rho_{0}}k_{z}^{q}(-A_{q}\mathrm{e}^{\mathrm{i}k_{z}^{q}L}+B_{q} \mathrm{e}^{-\mathrm{i}k_{z}^{q}L})=\sqrt{S}v_{0}(\omega)\delta_{0q}, \tag{6}\]  
\[\sum_{q=0}^{\infty}\psi_{q}\big{(}x,y,k_{q}\big{)}\Big{\{}[-k_{z}^{q}+k_{0}Z_{0 }(x,y,\omega)]A_{q}+[k_{z}^{q}+k_{0}Z_{0}(x,y,\omega)]B_{q}\Big{\}}=0, \tag{7}\]  
where \(k_{z}^{q}=\sqrt{k_{0}^{2}-k_{q}^{2}}\) is the wavenumber of the \(q\)th mode along the \(z\) direction and \(\delta_{0q}\) is the Kronecker delta function. Exploiting the orthogonality of the normal modes, Equation (7) can be simplified as  
\[k_{z}^{\mu}(-A_{\mu}+B_{\mu})+k_{0}\sum_{q=0}^{\infty}Z_{0}^{q\mu}(A_{q}+B_{q} )=0, \tag{8}\]  
\[Z_{0}^{q\mu}\equiv\iint\limits_{S}Z_{0}(x,y,\omega)\psi_{q}(x,y,k_{\lambda}) \psi_{\mu}^{*}(x,y,k_{\mu})\mathrm{d}S,\forall\mu\geq 0, \tag{9}\]  
where \(\mu\) and \(\lambda\) are the mode indexes, \(k_{\mu}=\mu\pi/a\) and \(k_{\lambda}=\lambda\pi/a\) are the lateral wavenumbers of the \(\mu\)th and \(\lambda\)th modes respectively, \(A_{\mu}\) and \(B_{\mu}\) are the \(\mu\)th mode coefficients of the incident wave and reflected wave respectively, \(k_{\mu}=\mu\pi/a\) is the lateral wavenumber of the \(\mu\)th mode, \(k_{z}^{\mu}=\sqrt{k_{0}^{2}-k_{\mu}^{2}}\) is the wavenumber of the \(\mu\)th mode along \(z\) direction, \(\psi_{\mu}\big{(}x,y,k_{\mu}\big{)}=\cos\big{(}k_{\mu}x\big{)}\cos\big{(}k_{ \mu}y\big{)}/\sqrt{S}\) is the \(\mu\)th mode function. The factor \(Z_{0}{}^{q\mu}\) mathematically measures the coupling of different modes caused by the inhomogeneity of the acoustical impedance at \(z=0\). The mode coupling makes it hard to obtain an analytical solution in a compact closed form. However, it is possible to approach the exact solution through iterations. Assume the inhomogeneity is quite weak such that all the high-order modes with \(q\) not being zero can be neglected, the plane-wave mode \(\psi_{0}(x,y,k_{0})\) is preserved and the coefficients \(A_{0}\) and \(B_{0}\) are derived as  
\[A_{0}\approx\rho_{0}c_{0}\sqrt{S}\Big{(}1+Z_{0}^{00}\Big{)}v_{0}(\omega)/2/ \Big{(}-\mathrm{j}\mathrm{sin}(k_{0}L)-Z_{0}^{00}\cos(k_{0}L)\Big{)}, \tag{10}\]  
\[B_{0}\approx A_{0}\Big{(}1-Z_{0}^{00}\Big{)}/\Big{(}1+Z_{0}^{00}\Big{)}, \tag{11}\]  
where the factor \(Z_{0}{}^{00}\) is calculated in Equation (9) when \(q\) and \(\mu\) are taken as zero. Assume that the inhomogeneity of the shunted loudspeaker array causes the plane-wave mode to be coupled to high-order modes while the high-order modes themselves are not coupled with each other, Equation (8) can be simplified as  
\[\Big{(}-k_{z}^{\mu}+k_{0}Z_{0}^{\mu\mu}\Big{)}A_{\mu}+\Big{(}k_{z}^{\mu}+k_{0} Z_{0}^{\mu\mu}\Big{)}B_{\mu}=-k_{0}Z_{0}^{0\mu}(A_{0}+B_{0}),\forall\mu\geq 0, \tag{12}\]  
where the factors \(Z_{0}{}^{\mu\mu}\) and \(Z_{0}{}^{0\mu}\) are calculated in Equation (9) when \(q\) is taken as \(\mu\) and \(0\) respectively. Substituting \(A_{0}\), \(B_{0}\) in Equations (10) and (11) into Equation (12), the coefficients of high-order modes can be calculated The normal incidence sound absorption coefficient \(\alpha\) can be calculated by  
\[\alpha=1-\frac{p_{r}^{2}}{p_{in}^{2}}=1-\frac{\left(\sum\limits_{q=0}^{\infty}\psi_ {q}\big{(}x,y,k_{q}\big{)}\,A_{q}\right)^{2}}{\left(\sum\limits_{q=0}^{\infty} \psi_{q}\big{(}x,y,k_{q}\big{)}\,B_{q}\right)^{2}}, \tag{15}\]  
where \(p_{r}\) and \(p_{\rm in}\) are reflected pressure and incident pressure at \(z=0\). It is clear that the more coupling measures \(Z_{0}{}^{q\mu}\) that are adopted; the more precise can be the results obtained.