## 1 Introduction  
Metamaterials are rationally designed artificial structures with periodically or non-periodically arranged subwavelength elements [1; 2; 3; 4], which exhibit unusual properties beyond natural occurring media [5]. The metamaterials have been widely investigated for guiding and controlling the flow of electromagnetic waves and acoustic waves in a supernatural manner since its discovery [6; 7; 8; 9; 10; 11; 12; 13; 14]. Novel wave manipulations such as the negative index, reversed Cherenkov radiation [15], and inverse Doppler effect [16] have been demonstrated experimentally. With such novel physical characteristics, the subwavelength metaatom-based metamaterial gives birth to the perfect lens, invisibility cloaks, and other new types of functional devices. As such, it provides new physical mechanisms in manipulating classical waves in comparison with traditional materials and devices. The features of metamaterials are determined to a great extent by the rationally designed metaatoms, instead of the intrinsic properties of natural materials [17; 18; 19; 20]. Since the geometric parameters of metamaterials can be specifically optimized, desirable properties can be acquired by changing the topology and sizes [21; 22; 23; 24; 25]. Due to their ability to control wave-matter interactions [26; 27; 28; 29; 30], metamaterials have been proven to be an important concept for the development of functional materials [31; 32; 33; 34; 35; 36; 37; 38; 39; 40]. Electromagnetic (EM) metamaterials have been intensively studied since the birth of metamaterials, and the research on EM metamaterials has achieved great progress [41; 42; 43; 44; 45; 46]. As early as the 1960s, the concept of negative refraction was proposed. Sir [PERSON] proposed a kind of composite made of periodic metal rods for realizing negative permittivity at a microwave frequency band in 1996 [1], and then another kind of structure constructed of resonant split rings was proposed to achieve the negative permeability in 1999 [2]. Based on these theoretical considerations, [PERSON] et al. exquisitely designed and fabricated hybrid structures with periodic metal rods and resonant split rings. The existence of a negative index was directly proved through the observation of negative refraction [3; 47]. Since then, metamaterials have attracted intensive studies, and various optical, terahertz, and microwave metamaterials have already been developed in the past decades [48; 49]. Acoustic waves, which play important roles in our modern daily life, are similar to EM waves in their master equations. Since sound waves and electromagnetic waves have a large amount of analogical characteristics, the supernatural phenomena such as negative refraction [50], invisible cloak [51; 52], and wavefront manipulation [53] also occur through specific designed acoustic composite structures, i.e., acoustic metamaterials (AMMs). AMMs are a new type of man-made structure that regulate acoustic behavior, resulting in negative effective mass densities and negative effective bulk modulus based on localized resonance mechanisms as well as extraordinary scattering properties [54]. For traditional acoustic materials, the insulation and absorption of low-frequency sound is confronted with great challenges due to the limitation of mass density law (which states that the acoustic transmission loss through a wall is proportional to the product of wall thickness, the mass density, and the sound frequency) [55; 56]. Recent reports show that AMMs demonstrate an excellent performance of acoustic attenuation surpassing the traditional matter, especially at low frequency. In the study of AMMs, numerous attempts are concentrated on adjusting the density or the bulk modulus of the structures to manipulate the scattering of acoustic waves [57; 58; 59; 60]. AMMs are considered as those complex material structures, rather than the sum of component parts, that are proposed and manufactured to modulate the propagation of acoustic waves in gases, liquids, and solids [61; 62].