Waves and interaction modes of capillary surfers

Recent experiments on so-called "capillary surfers" have demonstrated that asymmetric millimeter-scale objects may self-propel while floating at the interface of a vibrating fluid bath. The propulsion of these surfers is thought to be driven by interfacial waves, but the details remain elusive. We here present the results of an investigation into the vertical dynamics of floating discs subject to an imposed time-periodic forcing, with a view to elucidating the waves and flows generated by oscillating capillary-scale objects. The axisymmetric and inviscid wavefield is governed by a linear elliptic boundary value problem with mixed boundary conditions, wherein the no-penetration boundary condition is satisfied under the disc while the free surface boundary conditions are enforced away from it. The problem is solved by recasting the system of partial differential equations as a second-kind Fredholm integral equation which is then solved numerically. The solution furnishes a prediction for the dependence of the disc's oscillation amplitude on the forcing frequency, which exhibits excellent agreement with experiments.