## 1 Introduction  
Periodic arrays of solid scatterers embedded in a solid host, known as phononic crystals, have been exploited in the last decades for acoustic wave control, giving rise to several applications ranging from the radiofrequency regime to seismic waves [1; 2]. These systems have a particular dispersion relation with the presence of band gaps [3]. Recently, with the advent of 3D printing, more complex and efficient designs are allowed [4; 5; 6; 7]. In the particular case of rigid scatterers embedded in a fluid host, the system is known as sonic crystal because they are particularly suited for sound waves [8]. Characterization of their dispersion properties has been performed using different methods, for both ideal, i.e., infinite structures [3; 9] and realistic, i.e., finite ones [10], see reference [11] for a review on band-structure calculation methods. Combining the dispersion and the bandgaps, perhaps the most developed application of sonic crystals is design of acoustic barriers for the sound mitigation of traffic noise [12; 13]. Recently, a proof of concept for ultrasonic waves in water has shown that the diffusive properties of sonic crystals can be used to efficiently spread the acoustic waves when used as a cover of a cavity [14]. This system has been scaled in order to evaluate these wave diffusion properties in the case of the noise produced during the launch of a rocket. In this situation the cavity covered by the sonic crystal plays the role of the channel for exhaust gases. The results show a sound pressure level reduction in the area where the rocket is located close to 2 dB in a frequency band covering two octaves (equivalent to a decrease of 37% of the acoustic energy). Moreover, one of the main interests of using sonic crystals in this situation is that they present low flow resistivity [15] allowing the flow of both air and water through the structure. Therefore, a noise reduction solution based on sonic crystals can be added to the usual mitigation methods used during the launch of a rocket consisting in the injection of a hot supersonic pressurized water flow all around the pad [16], producing a level reduction of 3-5 dB. The band gap in sonic crystals as well as their strong dispersion properties appear around the Bragg frequency, which is inversely proportional to the distance between the rigid scatterers in the array [8]. Therefore, if low frequency sound must be mitigated, then bulky structures are needed. In order to avoid this problem, resonant inclusions have been proposed as building blocks of such periodic arrays giving rise to the well known locally resonant materials or metamaterials [17; 18]. Due to the resonance effect, new hybridized band gaps appear close to the resonance frequency of the local resonators, which can be tuned at much lower frequencies than the Bragg frequency. Some works have shown the filtering effect of such kind of locally resonant systems as well as the ultra-band gap created once the resonance of the resonant building block and the Bragg frequency are tuned [19; 20; 21]. Ultra-thin meta-diffusers have also been designed showing very good performance at low frequencies [22; 23]. Locally resonant materials have been also used for the mitigation of sound for traffic and railway noise showing their efficiency at low frequencies [24; 25; 26; 27; 28; 29; 30; 31; 32]. In addition to the effects of periodicity and local resonances, the presence of viscothermal losses in a realistic system must be taken into account [33; 34]. Viscothermal losses limit the strong dispersive effects in sonic crystals [35] and in acoustic metamaterials [21]. However, recently the presence of losses has been exploited to design perfect acoustic absorbers in reflection [36; 37; 38] and in transmission problems [39; 40]. The local resonators in the acoustic metamaterials are open, meaning that they radiate energy to the host medium, and lossy, as they present viscothermal dissipation [41].