# Active Tunable Elastic Metasurface for Abnormal Flexural Wave Transmission
## 1 Introduction  
Metasurfaces [1, 2, 3, 4, 5] are artificially constructed materials with subwavelength thicknesses in the direction of wave propagation, exhibiting an excellent ability to manipulate wavefronts, as well as having a more compact physical space and simplicity compared to conventional metamaterials [6, 7, 8, 9, 10]. Metasurfaces have been widely used in various fields, such as medical imaging, holographic imaging, and signal processing, following studies on their extraordinary beam modulation ability in the fields of electromagnetics [11, 12, 13] and acoustics [14, 15, 16, 17, 18, 19]. Recently, the research on metasurfaces has been expanded to the field of elastic waves [20, 21, 22, 23, 24, 25, 26, 27, 28]. However, elastic waves possess more degrees of freedom during solid propagation [29, 30, 31], which leads to a great challenge for researchers. Determining a method to extract and control the propagation of specific modes of guided waves from multimodal guided waves in a solid has become a hot research topic. Researchers have designed elastic wave metasurfaces based on the generalized Snell's law (GSL) for wavefront modulation behaviors such as abnormal refraction, wave focusing, and Bessel waves. [PERSON] et al. [21] first proposed metasurfaces for the abnormal deflection of A\({}_{0}\) Lamb waves. [PERSON] et al. [26] utilized the relationship between the effective mass and stiffness to achieve the modulation of transmitted traveling waves. Classical zigzag topology was used to alter the propagation path of flexural waves, taking advantage of the different propagation paths of the flexural wave to realize the phase shift of the basic unitcell across the full 2\(\pi\)[23]. A series of abnormal modulation behaviors of the flexural wave was realized by adjusting the phase gradient. A structure for the passive modulation of multi-mode guided waves was achieved by adding different lengths of cover layers on the unit element, and the theoretical analysis was verified by finite element simulations and experiments [25]. The above case can be identified as the passive modulation [32; 33; 34; 35; 36] of elastic waves. Passive modulation requires a change in the phase gradient by changing the geometry after fixing the phase distribution, which makes metasurfaces much less practical. Therefore, adaptive elastic metasurfaces have attracted considerable interest. [PERSON] et al. [37] proposed a nuts-and-screws structure where the tunability of the phase shift is induced by the screw-in depth based on the dispersion theory of locally resonant phononic crystals. By adjusting the position of the nut on the screw to achieve the full 2\(\pi\) coverage of the phase shift of the functional element, the abnormal refraction and wave-focusing functions of the broadband plate wave were achieved based on the GSL without re-fabricating the metasurface. There are various ways to use piezoelectric materials for active modulation. Piezoelectric patches can be stacked together to form a unit cell, which can be adjusted via negative capacitance circuits connected to achieve a wide frequency range of elastic longitudinal waves with a variety of modulation functions. It is also possible to attach a piezoelectric sheet to the substrate material and use electromechanical tuning to modulate the propagation of asymmetric mode Lamb waves [38]. [PERSON] et al. [39] presented a novel metasurface that is proposed to simultaneously manipulate multi-mode guided waves in the plate, including shear horizontal waves, symmetric mode, and anti-symmetric mode Lamb waves. Monolithic in-plane and out-of-plane polarized piezoelectric metasurface sheets were staggered on both faces of the substrate, and each piezoelectric patch was individually connected to a negative capacitance circuit. The metasurface can be multifunctionally modulated in each mode by adjusting the shunt's negative capacitance.