Anomalous interference drives oscillatory dynamics in wave-dressed active particles

A recent surge of discoveries has sparked significant interest in active systems where a particle moves autonomously in resonance with its self-generated wave field, leading to notable wave-mediated effects including new propulsion mechanisms, self-sustained speed modulations, and quantum-like phenomena. Drawing from walking droplets, an archetypical model of wave-dressed active particles, we identify a wave-mediated non-local force driving their oscillatory dynamics, arising from the particle's path memory and an unconventional form of wave interference near jerking points, locations where the particle's velocity changes rapidly. In contrast to the typical case of constructive interference at points of stationary phase, waves excited by the particle near jerking points avoid cancellation through rapid changes in frequency. Through an asymptotic analysis, we derive the wave force from jerking points, revealing it as an exponentially small yet crucial remnant of the particle's past motion. This previously unrecognized force underlies a range of phenomena that have thus far been treated independently, including in-line speed oscillations, wave-like statistics in potential wells, and non-specular reflections, thereby unifying them within a common framework rooted in generic wave superposition principles.