Scattering theory of walking droplets in the presence of obstacles

TL;DR

This paper presents a Green function-based model to describe the trajectories of walking droplets near obstacles, capturing experimental features and accounting for boundary effects in a quantum-inspired framework.

Contribution

It introduces an exactly solvable Green function approach to model walking droplets with obstacles, advancing the understanding of boundary effects in pilot-wave hydrodynamics.

Findings

Model reproduces experimental features of walking droplets near obstacles

Green function approach accounts for boundary conditions effectively

Single-slit geometry solution demonstrates model's accuracy

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 {\it Couder et al} [Phys. Rev. Lett. {\bf 97}, 154101 (2006)] 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.