## 1 Introduction  
The first definitions for the term metamaterial were given by researchers in the electromagnetic (EM) field. In 1999, [PERSON] and [PERSON] [1] defined a metamaterial as a \"three-dimensional, periodic cellular architecture, designed to produce an optimized combination not available in nature of two or more responses to a specific excitation\". This definition was completed and extended by [PERSON], [PERSON] and [PERSON] [2] in 2010 who stated that a metamaterial is \"a macroscopic composite of periodic or non-periodic structures whose function is due to both its cellular architecture and chemical composition\". From these principal definitions, there emerges the possibility to achieve properties that are hard or impossible to find in nature by conventionally engineered materials. Metamaterials derive their properties from their designed structures and geometries more than from their chemical composition, and the response of properly defined unit cells can be translated into averaged effective parameters, e.g., an effective density and effective bulk modulus. Research on metamaterials started in the electromagnetic field. [PERSON] [3] first demonstrated analytically that a negative refraction index is attainable if a medium exhibits both negative permeability and negative permittivity simultaneously, and this preliminary concept was experimentally realized by [PERSON] [4] in 2000, when a superlens was proposed, exhibiting a negative refraction index in the frequency bandwidth of visible light or microwaves [4]. The results of this left handed behavior of negatively refracting metamaterials obtained in the EM field sparked the interest of the scientific community in other fields, such as (but not only) acoustics. This was the case for the early work by [PERSON] et al. [5], who proposed an acoustic metamaterial that attenuates noise in the mid-frequency range (40 to 1100 Hz), which is composed of periodically-arranged, high density, spherical cores (made of lead), uniformly coated with a layer of soft and elastic material (silicone rubber). In 2004, [PERSON] and [PERSON] [6] mathematically demonstrated, for the first time, the possibility of designing a metamaterial to achieve a negative effective bulk modulus and density, acoustically introducing slow inclusions in a faster medium. In a frequency range such that the wavelength inside one inclusion is comparable to the dimension of the inclusion itself, but still much larger than the distance between the inclusions, the effective density and bulk modulus of the metamaterial appear to be negative [7; 8]. Physically, a double negative metamaterial shows counterintuitive behaviors--it expands when compressed and reacts with an acceleration in phase opposition with respect to the force that induces it. This consideration better explains the meaning of negative density, which reads asthe negative dynamic mass density term in [PERSON]'s second law [9; 10]. When both mass density (\(\rho\)) and the bulk modulus (\(\mathcal{K}\)) are negative, an acoustic wave is allowed to propagate inside the medium, thus exhibiting a negative refractive index. This causes energy to flow in the opposite direction with respect to the wave, i.e., the wave vector and the Poynting vector (describing the energy flow associated with the acoustic propagation) point in opposite directions. A metamaterial exposing the negative refraction index can be exploited for applications like acoustic superlenses [4; 11], i.e., a device capable of focusing an incident acoustic wave (with a sub-wavelength spacial resolution), even beyond the diffraction limit, in a sub-wavelength spot, and, in general, for bending waves and shaping the acoustic field almost arbitrarily. Near-total reflection can be obtained when only one of the two material parameters, \(\rho\) and \(\mathcal{K}\), becomes negative. This can be achieved using resonant elements to let \(\rho\) turn negative [5], or with a membrane decorated with a resonant mass(es) [12; 13; 14; 15], even without [16; 17].