The role of bouncing-phase variation for walking droplets

Experimental and theoretical studies of droplets walking on a vibrating bath show that the droplet’s impact phase depends on the driving acceleration. Experiments also show that this phase may change in the presence of boundaries or other walkers, indicating a dependence of phase on local wave amplitude. One expects that this phase variation may alter the stability of various dynamical states. We here introduce an integro-differential model for a walker's horizontal motion that accounts for the variability of impact phase, and use it to predict the stability of rectilinear walking and orbital solutions. Our model predictions are compared with those of previous constant-phase models and related to experiments whenever possible.