  This article introduces a new geophysical theory, in the form of a single
simple partial integro-differential equation, to explain how frictional
abrasion alone of a stone on a planar beach can lead to the oval shapes
observed empirically. The underlying idea in this theory is the intuitive
observation that the rate of ablation at a point on the surface of the stone is
proportional to the product of the curvature of the stone at that point and how
often the stone is likely to be in contact with the beach at that point.
Specifically, key roles in this new model are played by both the random wave
process and the global (non-local) shape of the stone, i.e., its shape away
from the point of contact with the beach. The underlying physical mechanism for
this process is the conversion of energy from the wave process into potential
energy of the stone. No closed-form or even asymptotic solution is known for
the basic equation, even in a 2-dimensional setting, but basic numerical
solutions are presented in both the deterministic continuous-time setting using
standard curve-shortening algorithms, and a stochastic discrete-time
polyhedral-slicing setting using Monte Carlo simulation.
