# An Integration Strategy for Acoustic Metamaterials to Achieve Absorption by Design
## Appendix A Causality Constraint on Sound Absorbing Structures
Therefore, the reflection coefficient \(R=(Z-Z_{0})/(Z+Z_{0})\) is given by  
\[R=\frac{1+i2\pi dB_{0}/(\lambda B_{\text{eff}})}{1-i2\pi dB_{0}/(\lambda B_{ \text{eff}})} \tag{12}\]  
Since \(\lim_{|\lambda|\to\infty}\ln R=i4\pi dB_{0}/(\lambda B_{\text{eff}})\), the contour integral is therefore given by  
\[\int_{C_{\infty}}\ln Rd\lambda=\lim_{|\lambda|\to 0}\int_{0}^{-\pi}i\lambda\ln Rd \theta=4\pi^{2}dB_{0}/B_{\text{eff}} \tag{13}\]  
where \(\theta\) is the argument of complex \(\lambda\). By taking the limit of \(|\lambda|\to\infty\) in the above contour integral, one is essentially counting all the poles of \(\ln R\) in the lower half of the complex \(\lambda\) plane, with the imaginary part of each pole being relevant to the absorption of each resonance of the system. This is evident from the fact that in our previous work [12], it has been shown that the static limit the effective bulk modulus \(B_{\text{eff}}(\lambda\to\infty)=\rho_{0}d^{2}(\sum_{n}\alpha_{n}/\Omega_{n})^ {-1}\) with \(\Omega_{n}\) being the \(n\)th resonance frequency of the system and \(\alpha_{n}\) the relevant oscillator strength defined in the main text. Hence, taking the limit of \(|\lambda|\to\infty\) implies all the absorptions related to the resonances of the system are taken into account. In fact, for the designed structures shown in this work, if we let \(d=\overline{d}\) as defined in Equation (10) below, then the above formula for \(B_{\text{eff}}(\lambda\to\infty)\) is accurately equal to \(B_{0}/\varphi\) with porosity \(\varphi\equiv V_{\text{air}}/V_{\text{tot}}\) being the volume fraction of the air phase. This is in agreement with [PERSON]'s formula for the composite effective bulk modulus in the static limit, given by \(B_{\text{eff}}^{-1}=\varphi B_{0}^{-1}+(1-\varphi)B_{\text{solid}}^{-1}\). Since \(B_{\text{solid}}>>B_{0}\)\(B_{\text{eff}}=B_{0}/\varphi\) follows. In addition, for samples with identical FP channels either straight or folded, \(\varphi=\phi=S_{\text{air}}/S_{\text{tot}}\) where \(S_{\text{air}}\) is the area of FP channels' total surface cross sectional area and \(S_{\text{tot}}\) being the total area of the sample surface exposed to incident sound. Hence in this work we have \(B_{\text{eff}}=B_{0}/\phi\). For the third integral on the right-hand-side of Equation (A5), since \(\lim\limits_{|\lambda|\rightarrow\infty}\ln[(\lambda-\lambda_{n}^{*})/( \lambda-\lambda_{n})]=i2\text{Im}(\lambda_{n})/\lambda\), we have  
\[\int_{\text{C}_{\infty}}\ln\frac{\lambda-\lambda_{n}^{*}}{\lambda-\lambda_{n} }d\lambda=\lim\limits_{|\lambda|\rightarrow\infty}\int_{0}^{-\pi}i\lambda\ln \frac{\lambda-\lambda_{n}^{*}}{\lambda-\lambda_{n}}d\theta=2\pi\text{Im}( \lambda_{n})\] (A8)  
Substitution of Equations (A7) and (A8) into Equation (A5) yields  
\[-\int_{0}^{\infty}\ln|R(\lambda)|d\lambda=2\pi^{2}d(B_{0}/B_{\text{eff}})+\pi \sum\limits_{n}\text{Im}(\lambda_{n})\] (A9)  
As \([1-A(\lambda)]=|R(\lambda)|^{2}\), where \(A(\lambda)\) stands for the absorption coefficient, and all \(\lambda_{n}\) are in the lower half-plane, i.e., \(\text{Im}(\lambda_{n})<0\), we therefore have the inequality  
\[d_{\text{min}}=\frac{1}{4\pi^{2}}\frac{B_{\text{eff}}}{B_{0}}\left|\int_{0}^{ \infty}\ln[1-A(\lambda)]d\lambda\right|\leq d\] (A10)  
It follows from Equation (A9) that the equality in (A10) is attained when \(R(\lambda)\) has no zeros in the lower half-plane of complex \(\lambda\). Such \(R(\lambda)\) corresponds to the minimum phase-shift frequency dependence [4; 5] for which the variation of the phase of the reflection coefficient with \(\lambda\) does not exceed \(2\pi\), in the domain \(0<\lambda<\infty\).