Ponderomotive Effects in a 1D Model of Pilot-Wave Hydrodynamics

Ponderomotive effects in electromagnetics and mechanics occur when a rapidly oscillating force is modulated by a slowly varying envelope, introducing a separation of timescales. These effects can lead to surprising phenomena, such as the dynamical stabilization of unstable equilibria, as exemplified by the Paul trap (Nobel Prize in Physics, 1989). Ponderomotive effects are also known to arise in pilot-wave hydrodynamics, where droplets bouncing on a vibrating bath self-propel – or 'walk' – through a resonant interaction with their wavefield. In particular, they have been identified as playing a critical role in the emergence of the quantum-like statistical behavior in the hydrodynamic analog of the quantum corral. We here investigate the role of ponderomotive effects in a simple theoretical model of one-dimensional walker dynamics in a central force. By exploiting the separation of timescales between the fast vertical bouncing and the slow horizontal drop motion, we characterize the emergence of ponderomotive effects when resonance is lost between drop and bath, specifically when the droplet’s vertical motion is aperiodic with respect to the bath vibration.