Velocity Oscillations of a Walking Droplet*

Couder \emph{et al.} (2005) demonstrated that a droplet bouncing on the surface of a vertically oscillating bath may destabilize to a walking state characterized by rectilinear motion across the bath surface at a constant speed. When a walking droplet is perturbed, there is experimental (Wind-Willassen \emph{et al.} (2013)) and theoretical (Bacot \emph{et al.} (2019)) evidence suggesting that its velocity may return to its free walking speed via over- or underdamped oscillations. By revisiting the stroboscopic pilot-wave model of Oza \emph{et al.} (2013), we demonstrate that linear stability analysis of the walking state predicts velocity oscillations over a lengthscale that becomes commensurate with the Faraday wavelength as the vibrational acceleration approaches the Faraday threshold. Furthermore, we demonstrate that this model predicts that the walking state destabilizes via a subcritical Hopf bifurcation, where the associated unstable limit cycle consists of periodic velocity oscillations, also on a scale comparable to the Faraday wavelength. This behavior conditions the droplet's histogram to develop a coherent structure on the scale of the Faraday wavelength, and so provides new insight into the emergence of quantum-like statistics from pilot-wave hydrodynamics.