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Abstract In solid state physics, a bandgap (BG) refers to a range of energies where no electronic states can exist. This concept was extended to classical waves, spawning the entire fields of photonic and phononic crystals where BGs are frequency (or wavelength) intervals where wave propagation is prohibited. For elastic waves, BGs are found in periodically alternating mechanical properties (i.e., stiffness and density). This gives birth to phononic crystals and later elastic metamaterials that have enabled unprecedented functionalities for a wide range of applications. Planar metamaterials are built for vibration shielding, while a myriad of works focus on integrating phononic crystals in microsystems for filtering, waveguiding, and dynamical strain energy confinement in optomechanical systems. Furthermore, the past decade has witnessed the rise of topological insulators, which leads to the creation of elastodynamic analogs of topological insulators for robust manipulation of mechanical waves. Meanwhile, additive manufacturing has enabled the realization of 3D architected elastic metamaterials, which extends their functionalities. This review aims to comprehensively delineate the rich physical background and the state‐of‐the art in elastic metamaterials and phononic crystals that possess engineered BGs for different functionalities and applications, and to provide a roadmap for future directions of these manmade materials. 

Abstract Acoustic holograms have promising applications in sound‐field reconstruction, particle manipulation, ultrasonic haptics, and therapy. This study reports on the theoretical, numerical, and experimental investigation of multiplexed acoustic holograms at both audio and ultrasonic frequencies via a rationally designed transmission‐type acoustic metamaterial. The proposed metahologram is composed of two Fabry–Pérot resonant channels per unit cell, which enables the simultaneous modulation of the transmitted amplitude and phase at two desired frequencies. In contrast to conventional acoustic metamaterial‐based holograms, the design strategy proposed here provides a new degree of freedom (frequency) that can actively tailor holograms that are otherwise completely passive and significantly enhances the information encoded in acoustic metamaterials. To demonstrate the multiplexed acoustic metamaterial, the projection of two different high‐quality metaholograms is first shown at 14 and 17 kHz, with the patterns of the letters N and S. Then, two‐channel ultrasound focusing and annular beams generation for the incident ultrasonic frequencies of 35 and 42.5 kHz are demonstrated. These multiplexed acoustic metaholograms offer a technical advance to tackle the rising challenges in the fields of acoustic metamaterials, architectural acoustics, and medical ultrasound.