## 1 Introduction  
Present-day urbanization and traffic growth could seriously threaten human health, inducing annoyance, sleep disturbance, or even ischemic heart diseases [1], therefore the interest on environment noise reduction is quickly growing. In this context, porous media for acoustic purposes are materials made of channels, cracks, or cavities, in which sound waves pass through the foam and lose energy due to viscous and thermal effects [2, 3]. However, porous media are not so efficient at low frequencies as they are at high ones [4]. Such limitation is generally bypassed through the use of multi-layer arrangements [5]; in any case, the effects of these solutions always rely on the allowable thickness or on the total mass of the soundproofing configuration [6, 7]. In order to overcome such a constraint, low frequency performance of acoustic packages can be significantly enhanced by resorting to the use of porous media with embedded periodic inclusions acting as local resonators [8, 9, 10]. Such a configuration represents a meta-material and provides proper dynamic filtering effects to the material, which could be beneficial for both the system dynamics and manufacturing. First proposed by [PERSON] [11] for electromagnetic waves, meta-materials have found, in recent years, a wide application in the acoustic field, since the insertion of artificial structural elements with periodic pattern allows one to control the propagation of wave phenomena [12]. Basically, the periodic geometry induces wave interference with constructive and destructive effects [13], which may lead to an enhancement of sound-absorbing properties with respect to traditional materials, such as porous ones [10]. Several types of meta-materials have been proposed in the relevant literature. For example, [PERSON] et al. [14] introduce a decorated membrane, consisting of a pretensioned elastic membrane with rigid weights adhered on its surface; [PERSON] et al. [15] obtain high sound absorption through a thin decorated membrane, while [PERSON] et al. [16] investigate its upper bound. Moreover, [PERSON] et al. [17; 18] propose a complete analytical model for the absorption behavior of the membrane and confirm the energy absorption limit. Many works are inspired by Helmholtz resonators; for example, [PERSON] et al. [19] combine several low-wavelength Helmholtz resonators in series in order to obtain an ultrasonic resonator; [PERSON] et al. [20] propose an ultrathin open meta-material consisting in a large orifice in each unit cell; [PERSON] et al. [21] experimentally investigate a ventilated tunable acoustic meta-material to address the challenge of acoustical performance and ventilation efficiency in conventional noise barriers; and [PERSON] et al. and [PERSON] et al. [22; 23] demonstrate that Helmholtz resonators can exhibit monopolar resonances, in contrast to single decorated membranes, in which only dipole resonances occur. However, [PERSON] et al. [24] show that a pair of coupled decorated membranes can provide monopolar resonances, too. The application of a shift cell technique, which is a reformulation of classical Floquet-Bloch conditions, is proposed by [PERSON] et al. [25], which then consolidate the research concerning the acoustic performances of vibroacoustic treatments based on periodic patterns by embedding inclusions into a foam layer [26; 27; 28]. Nowadays, the use of meta-materials is suitable for different applications, such as energy, civil, and transportation (aerospace, automotive, and railway) engineering fields, where space, weight, and acoustic comfort still represent key aspects. For example, foam-based meta-materials find a wide range of applications in fuselage acoustic optimization, as reported by [PERSON] et al. [29], which propose and investigate a meta-core solution made of a foam with cylindrical inclusions.