A Trajectory Equation for Walking Droplets: Hydrodynamic Pilot-Wave Theory

Anand U. Oza

Courant Institute of Mathematical Sciences

New York University




Yves Couder and coworkers have demonstrated that millimetric droplets walking on a vibrating fluid bath exhibit several features previously thought to be peculiar to the microscopic quantum realm, including single-particle diffraction, tunneling, quantized orbits, and wave-like statistics in a corral. We here develop an integro-differential trajectory equation for these walking droplets with a view to gaining insight into their subtle dynamics. We then rationalize the emergence of orbital quantization in a rotating frame by assessing the stability of the orbital solutions. The stability analysis also predicts the existence of wobbling orbital states reported in recent experiments, and the absence of stable orbits in the limit of large vibrational forcing. In this limit, the complex walker dynamics gives rise to a coherent statistical behavior with wave-like features.