# An Integration Strategy for Acoustic Metamaterials to Achieve Absorption by Design
## Appendix A Causality Constraint on Sound Absorbing Structures  
Consider a layer of composite material backed by a rigid reflective wall (Figure 11a). In response to an incident sound wave, the reflected sound pressure \(p_{r}(t)\) is the superposition of the direct reflection of the incoming sound pressure at that instant, \(p_{i}(t)\) plus those in response to the incoming sound wave at earlier time, \(p_{i}(t-\tau)\), with \(\tau>0\). Hence,  
\[p_{r}(t)=\int_{0}^{\infty}K(\tau)p_{i}(t-\tau)d\tau \tag{1}\]  
where \(K(\tau)\) is the response kernel in the time domain. By carrying out Fourier transform \(p_{i/r}(\omega)=\int_{-\infty}^{\infty}p_{i/r}(t)e^{i\omega t}dt\), the reflection coefficient for each frequency may be expressed as  
\[R(\omega)\equiv\frac{p_{r}(\omega)}{p_{i}(\omega)}=\int_{0}^{\infty}K(\tau) e^{i\omega\tau}d\tau \tag{2}\]  
From Equation (2), \(R(\omega)\) is an analytic function of complex \(\omega\) in the upper half of the complex \(\omega\) plane. In terms of the wavelength \(\lambda=2\pi v_{0}/\omega\), where \(v_{0}\) is the speed of sounds in air, that means \(R(\lambda)\) has no singularities in the lower half-plane of complex \(\lambda\), but may have zeros that represent total absorptions of incoming energy. Here, the imaginary part of \(\lambda\) reflects dissipation. To determine the constraint on the reflection coefficient \(R(\lambda)\) by the causality principle, we introduce an ancillary function \(\widetilde{R}(\lambda)\) after [PERSON] and [PERSON] [4; 5],  
\[\widetilde{R}(\lambda)\equiv R(\lambda){\prod_{n}}\frac{\lambda- \lambda_{n}^{*}}{\lambda-\lambda_{n}} \tag{3}\]  
where \(\lambda_{n}\), satisfying \(R(\lambda_{n})=0\), are the zeros located in the lower half-plane of complex \(\lambda\), and \(*\) stands for complex conjugation. Since \(\widetilde{R}\) has neither zeros nor poles at \(\text{Im}(\lambda)<0\), \(\text{ln}\,\widetilde{R}\) is an analytic function in the lower half-plane of complex \(\lambda\), and the Cauchy theorem is valid. That is, the integral over a closed contour \(C\) in the lower half-plane of complex \(\lambda\) should yield zero, where the contour consistsof the real axis of and the semi-circle \(C_{\infty}\) that lies in the lower half-plane and has infinite radius as shown in Figure A1b. Hence  
\[\int_{C}\ln\widetilde{R}d\lambda=\int_{-\infty}^{+\infty}\ln\widetilde{R}d \lambda+\int_{C_{\infty}}\ln\widetilde{R}d\lambda=0 \tag{10}\]  
Note that \(|\widetilde{R}|\ =\ |R|\) at real wavelengths and \(\ln|R|\) is an even function of \(\lambda\) according to its definition Equation (11). Taking the real part of Equation (10) yields  
\[\text{Re}\int_{C}\ln\widetilde{R}d\lambda=2\int_{0}^{\infty}\ln|R|d\lambda+ \text{Re}\int_{C_{\infty}}\ln Rd\lambda\ +\ \text{Re}\sum_{n}\int_{C_{\infty}}\ln\frac{\lambda-\lambda_{n}^{*}}{ \lambda-\lambda_{n}}d\lambda=0 \tag{11}\]  
To calculate the second integral on the right-hand-side of Equation (11), we consider the infinite-wavelength limit of \(R\), i.e., the static limit. The reflection from a composite material layer can be characterized by an effective bulk modulus \(B_{\text{eff}}\) relating to its surface responses [12]. The surface displacement \(u\) under a pressure \(p\) is therefore given by the relationship (pressure) = (effective bulk modulus) \(\times\) (strain), or \(u=pd/B_{\text{eff}}\) with \(d\) being the sample thickness. The resulting surface impedance is given by \(Z=ip/(\omega u)=iZ_{0}B_{\text{eff}}\lambda/(2\pi B_{0}d)\) with \(Z_{0}=B_{0}/v_{0}\) being the air impedance and \(B_{0}\) the bulk modulus of air.