The Willis coupling vectors are recently discovered material properties that enable acoustic fluids with broken time reversal symmetry, reciprocity, and energy conservation. The time modulation of material properties has been identified as another approach to break similar taboos in wave dynamics. Combining both mechanisms promises unparalleled new freedom to manipulate waves, but the significant challenge of physically realizing bulk materials with time-modulated Willis vectors has left the opportunity unexplored. Here, we introduce an active metamaterial of two-dimensional geometry that can be programmed for desired Willis vector magnitude and direction, and modulated very quickly in time. To highlight the newfound possibilities for wave manipulation, we experimentally demonstrate a nonreciprocal directional filter and a steerable beamforming shell. Through time modulation of the Willis vectors, the operating direction of these devices is changed at sonic speeds. The results illustrate a small fraction of the potential applications for time-varying bulk Willis media. The Willis coupling vectors offer a novel approach to manipulating acoustic waves, yet their practical implementation remains challenging. Here, the authors present an active metamaterial capable of rapid time modulation of Willis vectors, demonstrating nonreciprocal filtering and steerable beamforming, thus unlocking new possibilities for dynamic wave control.
Kovacevich et al. (2026) studied this question.