Scattering theory of walking droplets in the presence of obstacles

Abstract

We aim to describe a droplet bouncing on a vibrating bath using a simple and highly versatile model inspired from quantum mechanics. Close to the Faraday instability, a long-lived surface wave is created at each bounce, which serves as a pilot wave for the droplet. This leads to so called walking droplets or walkers. Since the seminal experiment by Couder et al (2006 Phys. Rev. Lett. https://doi.org/10.1103/physrevlett.97.154101 97 https://doi.org/10.1103/physrevlett.97.154101 ) there have been many attempts to accurately reproduce the experimental results.We propose to describe the trajectories of a walker using a Green function approach. The Green function is related to the Helmholtz equation with Neumann boundary conditions on the obstacle(s) and outgoing boundary conditions at infinity. For a single-slit geometry our model is exactly solvable and reproduces some general features observed experimentally. It stands for a promising candidate to account for the presence of arbitrary boundaries in the walker’s dynamics.

Keywords

• drops
• nonlinear dynamics
• walking droplets
• quantum mechanics
• 47.55.D-
• 03.65.-w