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Energetics of pilot-wave hydrodynamics: Nonresonant effects

Tino Damiani1, Matthew Durey2, Bauyrzhan K. Primkulov3, and John W. M. Bush1,*

  • *Contact author: bush@math.mit.edu
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Phys. Rev. Fluids 11, 084002 – Published 12 August, 2026

DOI: https://doi.org/10.1103/fpg5-myb3

Abstract

A millimetric droplet can bounce indefinitely on a vertically vibrating bath and generate a Faraday pilot-wave field that propels it forward. This hydrodynamic pilot-wave system, in which a walking droplet is accompanied by its quasi-monochromatic wave field, exhibits several features previously thought to be exclusive to the quantum realm. For steady dynamical states, the energy input from the vibrational forcing and output from viscous dissipation must balance, so one can consider the partitioning between the droplet and wave energy. The energetics of the hydrodynamic pilot-wave system have been considered only in the special case where there is resonance between the bouncing drop and its pilot wave, so one may neglect the drop's vertical dynamics. We here deepen our understanding of pilot-wave energetics by simulating these vertical dynamics, allowing us to assess the energetics of both resonant and nonresonant walking states. Specifically, we characterize the dependence of the system energetics on the droplet's bouncing or walking mode, and assess whether resonance has a significant influence on the partitioning between drop and wave energy. In all instances, the horizontal kinetic energy of the drop is negligible relative to its gravitational potential energy. At high memory, where the pilot waves are most persistent, the pilot-wave energy dominates the total system energy, exceeding the droplet energy by an order of magnitude. When the resonance between drop and wave is broken, the wave energy is slightly diminished due to wave interference, but remains dominant.

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Phys. Rev. Fluids 11, 084002– Published 12 August, 2026
  • Received 16 December 2025
  • Accepted 20 July 2026
Article part of CHORUS
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