# An Integration Strategy for Acoustic Metamaterials to Achieve Absorption by Design
## Appendix C Derivation of Self-Energy Due to Cross-Channel Coupling by Evanescent Waves  
Since the surface impedance of the metamaterial unit is laterally inhomogeneous, it follows that the sound pressure field \(p(\mathbf{x})\), where \(\mathbf{x}\) denotes the lateral coordinate at the plane \(z=0\), must necessarily be inhomogeneous as well. By decomposing the pressure field as \(p(\mathbf{x})=\overline{p}+\delta p(\mathbf{x})\), where \(\overline{p}\) is the surface-averaged value, it has been shown in the main text that \(\delta p(\mathbf{x})\) is only coupled to the evanescent waves that decay exponential away from \(z=0\). In contrast, \(\overline{p}\) couples to the far-field propagating modes. Therefore, the measured surface impedance should be given by \(Z=\overline{p}/\overline{v}\) with \(\overline{v}=d\overline{u}/dt\) being the surface-averaged \(z\) component of the air displacement velocity. The reflection coefficient is given by \(R=(Z-Z_{0})/(Z+Z_{0})\). We expand \(\delta p({\bf x})\) in terms of the normalized Fourier basis function \(f_{\mathbf{\alpha}}({\bf x})=\exp[i{\bf k_{\mathbf{\alpha}}}\cdot{\bf x}]/L\), where \(\mathbf{\alpha}=(\alpha_{x},\alpha_{y})\) is discretized by the condition that the area integral of \(f_{\mathbf{\alpha}}\) over the surface of the metamaterial unit must vanish, due to the fact that the same condition applies to \(\delta p({\bf x})\). That means \(|{\bf k_{\mathbf{\alpha}}}|=(2\pi/L)\sqrt{\alpha_{x}^{2}+\alpha_{y}^{2}}\), with \(\alpha_{x},\alpha_{y}=\pm 1,\pm 2,\cdots\):  
\[\delta p({\bf x},z)=\sum\limits_{\mathbf{\alpha}}\delta p_{\mathbf{\alpha}}f_{\mathbf{ \alpha}}({\bf x})e^{-\sqrt{k_{\mathbf{\alpha}}^{2}-k_{0}^{2}}} \tag{100}\]  
where \(\delta p_{\mathbf{\alpha}}\) denotes the expansion coefficient, and \(k_{0}=2\pi/\lambda\). The exponential variation of \(\delta p({\bf x},z)\) means that it can couple to the \(z\) component of the air displacement velocity through [PERSON]'s law, \(\partial\delta p/\partial z=-i\omega\rho_{0}\delta v\), so that  
\[\delta v({\bf x})=\delta v({\bf x},z=0)=\frac{-i}{\omega\rho_{0}}\sum\limits_{ \mathbf{\alpha}}\delta p_{\mathbf{\alpha}}\sqrt{|{\bf k_{\mathbf{\alpha}}}|^{2}-k_{0}^{2} }f_{\mathbf{\alpha}}({\bf x}) \tag{101}\]  
By multiplying both sides of Equation (101) by \(f_{\mathbf{\alpha}}^{*}({\bf x})\) and integrating over the surface of the metamaterial unit's surface, we can solve for \(\delta p_{\mathbf{\alpha}}\):  
\[\delta p_{\mathbf{\alpha}}=i\omega\rho_{0}\frac{\int\limits_{surface}v({\bf x})f_ {\mathbf{\alpha}}^{*}({\bf x})d{\bf x}}{\sqrt{|{\bf k_{\mathbf{\alpha}}}|^{2}-k_{0}^{2 }}} \tag{102}\]  
where \(v({\bf x})=\overline{v}+\delta v({\bf x})\). It should be noted that in the above, the integral of \(v({\bf x})f_{\mathbf{\alpha}}^{*}({\bf x})\) is the same as the integral of \(\delta v({\bf x})f_{\mathbf{\alpha}}^{*}({\bf x})\), since the integral of \(\overline{v}f_{\mathbf{\alpha}}({\bf x})\) is zero. By substituting Equation (102) into Equation (100) and then interchanging the order of summation and integration, we obtain  
\[\delta p({\bf x})=\delta p({\bf x},z=0)=i\omega\rho_{0}\int\limits_{surface} \Lambda({\bf x},{\bf x}^{\prime})v({\bf x}^{\prime})d{\bf x}^{\prime} \tag{103}\]  
where \(\Lambda({\bf x},{\bf x}^{\prime})\equiv\sum\limits_{\mathbf{\alpha}}f_{\mathbf{ \alpha}}^{*}({\bf x}^{\prime})f_{\mathbf{\alpha}}({\bf x})/\sqrt{|{\bf k_{\mathbf{ \alpha}}}|^{2}-k_{0}^{2}}\). Since \(|{\bf k_{\mathbf{\alpha}}}|\gg k_{0}\), we can approximate \(\sqrt{|{\bf k_{\mathbf{\alpha}}}|^{2}-k_{0}^{2}}\) by \(|{\bf k_{\mathbf{\alpha}}}|\). By discretizing the 2D coordinate \({\bf x}\) by its 16 values, \({\bf x}_{n}\), that denotes the center position of the \(n^{\text{th}}\) FP channel, and replacing \(d{\bf x}^{\prime}\) by \(L^{2}/16\) and the integral by summation, we have:  
\[\delta p_{n}=i\omega\rho_{0}\sum\limits_{m=1}^{16}\Lambda_{mm}v_{m} \tag{104}\]  
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