## 6 Conclusions  
In this work we have experimentally, numerically and theoretically evaluated the possibilities of 2D arrays of Helmholtz resonators for sound absorption and diffusion. On the one hand, the array has been optimized in order to present perfect absorption for a targeted range of frequencies with wavelength much larger than the characteristic size of the building blocks, i.e., in the low frequency regime. While the geometrical parameters of the Helmholtz resonators are bounded due to the limits imposed by the finite size of the transversal section of the impedance tube, the distances between resonators are totally free (meaning that no lower and upper bounds are imposed for this parameter). It is important to notice that the optimal solution is given here by a discrete distribution of resonators. The 2D structure presents experimentally a quasi-perfect absorption of sound in the range of frequencies of interest in good agreement with the theoretical predictions. On the other hand, the diffusion coefficient of the structure has been evaluated in order to analyze the combined effect between multiple scattering in the array and the resonances at higher frequencies. The diffusion coefficient of the system is improved for a broad range of frequencies. In fact the comparison with the system of Helmholtz resonators without cavities shows that the multiple scattering effect plays an important role on the diffusion coefficient, as also shown in reference [14]. The combined effect of the two phenomena gives an overall ILR of 5 dB between 200 Hz and 1700 Hz. The results shown in this work go further and combine absorption and diffusion in the same structure, improving the noise mitigation performances of the systems shown in reference [14].  
Conceptualization, A.C., R.P., V.R.-G. and L.M.G.-R.; methodology, I.H.-D., A.C., R.P., V.R.-G. and L.M.G.-R.; software, I.H.D., A.C. and V.R.-G.; validation, I.H.D., A.C., R.P., V.R.-G., L.M.G.-R. and V.J.S.-M.; formal analysis, I.H.D., A.C., V.R.-G. and L.M.G.-R.; investigation, I.H.-D.; resources, I.H.-D. and R.P.; data curation, I.H.-D.; writing-original draft preparation, I.H.-D.; writing-review and editing, I.H.-D., A.C., R.P., V.R.-G., L.M.G.-R. and V.J.S.-M.; visualization, I.H.-D., A.C., V.R.-G. and L.M.G.-R.; supervision, A.C., R.P., V.R.-G., L.M.G.-R. and V.J.S.-M.; project administration, R.P., L.M.G.-R. and V.J.S.-M.; funding acquisition, R.P. and V.J.S.-M. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the European Space Agency under the Networking/Partnering Initiative (NPI) contract number 441-2015.  
In memoriam to [PERSON], Senior Advisor for the Structures, Mechanisms and Materials Division of ESA, a great man that always gave us his tireless support. AC acknowledges financial support from Generalitat Valenciana through the grant APOSTD/2018/229. VRG acknowledges the financial support from RFI Le Mans Acoustique (Region Pays de la Loire) in the framework of the project HYPERMETA funded under the program Etoiles Montantes of the Region Pays de la Loire. Authors acknowledge the support of the European Space Agency under contract 441-2015 Co- Sponsored PhD \"Acoustic Reduction Methods for the Launch Pad\" and project TRP ESA AO/1-9479/18/NL/LvH \"Launch Sound Level Reduction\". This article is  
Figure 8: ILR of the system with cavities.  
based upon work from COST Action DENORMS CA15125, supported by COST (European Cooperation in Science and Technology).  
**Conflicts of Interest:** The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.