Speed Oscillations of Walking Droplets along a Circular Orbit

Walking droplets constitute a macroscopic realization of wave-particle duality, in which a droplet self-propels on the surface of a vertically vibrating fluid bath through a resonant interaction with its own pilot-wave field. A defining feature of this system is 'path memory': at each impact, the droplet responds to the cumulative wave field generated by its prior bounces, with the effective memory increasing as the bath forcing approaches the Faraday threshold. At sufficiently high memory, walkers travelling along straight trajectories may spontaneously develop in-line speed oscillations, a key mechanism behind the emergence of wave-like statistics near topographic impurities. Here, we investigate how such speed oscillations are modified when droplets are constrained to circular paths, forcing continuous interaction with their own curved wave wake. Through simulations and linear stability analysis, we find that the onset of speed oscillations becomes an oscillatory function of the orbital radius. This threshold varies on the scale of the Faraday wavelength, with a structure that depends sensitively on particle inertia and converges to the straight-path limit. We discuss how these behaviors may result from nonlocal wave-mediated forces acting along the circumference of the orbit, and explore the potential for these tangential instabilities to serve as an additional mechanism for orbital quantization in walking droplets.