Oza, A. U., Harris, D. M., Rosales, R. R., & Bush, J. W. (2014). Pilot-wave dynamics in a rotating frame: on the emergence of orbital quantization. Journal of Fluid Mechanics744, 404-429.

We present the results of a theoretical investigation of droplets walking on a
rotating vibrating fluid bath. The droplet’s trajectory is described in terms of an
integro-differential equation that incorporates the influence of its propulsive wave
force. Predictions for the dependence of the orbital radius on the bath’s rotation
rate compare favourably with experimental data and capture the progression from
continuous to quantized orbits as the vibrational acceleration is increased. The orbital
quantization is rationalized by assessing the stability of the orbital solutions, and may
be understood as resulting directly from the dynamic constraint imposed on the drop
by its monochromatic guiding wave. The stability analysis also predicts the existence
of wobbling orbital states reported in recent experiments, and the absence of stable
orbits in the limit of large vibrational forcing