## 3 Analyses and Results
### Analytic Part: Calculation of the Periodic Resonance Frequency  
Firstly, the DBM model is used to describe the porous medium. This Section reports the analytical design procedure which, for fixed thickness and resonance frequency, returns the unit cell dimension and airflow resistivity of the model. In particular, it is illustrated how Equation (4) is inverted in order to estimate the required foam airflow resistivity \(\sigma\) for a desired resonance frequency. First of all, it should be pointed out that the resonance physics that is exploited herein, is linked to the well-known phenomenon that occurs when half the wavelength is equal to the periodicity dimension (namely, the unit cell dimension). So told, Equation (4) essentially involves four quantities: The wavenumber \(k\), the frequency \(f\) (and thus the angular frequency \(\omega=2\pi f\)), the speed of sound in air \(c_{0}\) (that, in this context, can be considered as a constant value), and the foam airflow resistivity \(\sigma\). Normally, such quantities are linked to each other such that the first one is the output, while the others are treated as inputs. In order to explicit the wavelength \(\lambda\), which is required for the aforementioned reason, the wavenumber \(k\) is expressed as \(k=\frac{2\pi}{\lambda}\); then, since in correspondence of a periodic resonance, it is verified that the dimension of periodicity, \(d\), is given by [25; 26]:  
\[d=\frac{\lambda}{2}, \tag{10}\]  
in that case one may write that \(k=\frac{\pi}{4}\). At this point, considering that the real part of a complex wavenumber is generally linked to the propagation of the wave, while the imaginary one is related to its dissipation, for the purpose of the present analysis only  
Figure 1: Flowchart of the design task concerning meta-materials described with the DBM model. As illustrated, the blue, red, and green sections are related to the activities executed in the analytical, numerical, and machine learning phases.  
the real part may be taken into account. Therefore, by neglecting the imaginary part of Equation (4) and applying the aforementioned substitutions, in correspondence of a periodic resonance one obtains:  
\[\sigma=36.1f\bigg{(}\frac{c_{0}}{2 df}-1\bigg{)}^{1.618}. \tag{11}\]  
The code used for this design phase, written in MATLAB, is based on Equation (11) and initially asks for two geometrical constraints: The total available thickness, and the minimum unit cell dimension, which could be restrained by manufacturing aspects, and thus is limited by both lower and upper boundaries; furthermore, the prompt also requires the input of a target resonance frequency, which could fall in the whole audible range. Then, starting from the combination of the two geometrical constraints, the scripts calculates the possible numbers (and thus dimensions) of unit cells along the thickness; for each of them, the required \(\sigma\) at the assigned resonance frequency is computed through Equation (11). In this configuration, the script may display one or more (or even none) possible combinations of unit cell dimensions and airflow resistivity that, from a practical point of view, may lead to the choice respectively of a specific periodic pattern and foam material, with the aim of obtaining an acoustic resonance at the desired frequency.