  We present and analyze a theoretical model for the dynamics and interactions
of "capillary surfers," which are millimetric objects that self-propel while
floating at the interface of a vibrating fluid bath. In our companion paper
[1], we reported the results of an experimental investigation of the surfer
system, which showed that surfer pairs may lock into one of seven bound states,
and that larger collectives of surfers self-organize into coherent flocking
states. Our theoretical model for the surfers' positional and orientational
dynamics approximates a surfer as a pair of vertically oscillating point
sources of weakly viscous gravity-capillary waves. We derive an analytical
solution for the associated interfacial deformation and thus the hydrodynamic
force exerted by one surfer on another. Our model recovers the bound states
found in experiments and exhibits good quantitative agreement with experimental
data. Moreover, a linear stability analysis shows that the bound states are
quantized on the capillary wavelength, with stable branches of equilibria
separated by unstable ones. Generally, our work shows that self-propelling
objects coupled by interfacial flows constitute a promising platform for
studying active matter systems in which both inertial and viscous effects are
relevant.
