# An Integration Strategy for Acoustic Metamaterials to Achieve Absorption by Design
## 1 Introduction  
Even in the 21 st century, acoustic noise can still be a pernicious problem. An obvious question facing the scientific and engineering community is: How can we do better to remediate this problem? While doing by trial and error and learning by experience clearly constitute a path forward, occasionally natural laws can offer us the ultimate limit on how well one can achieve. It is the purpose of this article to show that acoustic absorption is one such case, and knowing the limit represents a tremendous advantage since one can work backwards in designing strategies to achieve such limit. The relevant limit in the case of acoustic absorption is that imposed by the causality principle, which, combined with the resonant mode density determination based on the target absorption spectrum for a structure backed by a reflecting substrate, is shown to yield a self-consistent strategy for achieving absorptionby design. By considering the absorption through evanescent waves, we also address the practical issue of using a finite number of resonances to best achieve the target absorption spectrum. For absorption at least, our design approach presented in this article has overcome the Achilles' heel of narrow frequency bandwidth and dispersive character [1; 2] for acoustic metamaterials. Acoustic response of materials and/or structures must obey the causality principle, which states that the response at any moment of time can only depend on what occurred prior to that moment. In other words, the future cannot affect what happens now. Translated into mathematical language, this simple and intuitive statement can have deep consequences. In 1926-1927, [PERSON] and [PERSON] independently derived the well-known Kramers-Kronig relationship between the real and imaginary parts of the dielectric function for electromagnetic waves, based on the causality principle [3]. Another, much less-known consequence of the causality principle is that for any electromagnetic absorption spectrum, there is a minimum sample thickness [4; 5]. Adapted to acoustics, we have derived the following relationship for a flat structure sitting on a reflecting substrate [6]:  
\[d\geq\frac{1}{4\pi^{2}}\frac{B_{\text{eff}}}{B_{0}}\bigg{|}\int_{0}^{\infty} \ln[1-A(\lambda)]d\lambda\bigg{|}=d_{\text{min}}. \tag{1}\]  
Here, \(\lambda\)\((=2\pi v_{0}/\omega)\) denotes the sound wavelength in air, with \(v_{0}=340\) m/s being the speed of air-borne sound, \(\omega\) the angular frequency, \(A(\lambda)\) the absorption spectrum, \(B_{\text{eff}}\) the effective bulk modulus of the sound absorbing structure in the static limit, and \(B_{0}\) the bulk modulus of air. For the sake of making the present article self-contained, we give a detailed derivation of Equation (1) in Appendix A that first appeared in the Supplementary Information section of ref. [6]. Equation (1) tells us that the acoustic absorption spectrum cannot be independent from the sample thickness. A somewhat interesting aspect of Equation (1) is that only the bulk modulus appears in the causality constraint (in the electromagnetic case only the magnetic permeability appears in the causality constraint), whereas it is well known in acoustics that there are two material parameters: bulk modulus and mass density. This is attributed to the fact that in the derivation, it is necessary to take the zero-frequency limit. Hence, no inertial effect can be present. Some immediate implications of Equation (1) are that there cannot be 100% absorption over a finite frequency range by a finite-thickness structure, and that closer the absorption gets to 100%, the thicker the sample must be. Also, for the same frequency bandwidth, a lower central frequency implies a thicker minimum sample thickness, and vice versa. It should be mentioned that there have been numerous studies to broaden the frequency range of absorption for acoustic metamaterials, some of which were quite successful [7; 8; 9; 10].