On the Oval Shapes of Beach Stones

Abstract

This article introduces a new stochastic non-isotropic frictional abrasion model, in the form of a single short partial integro-differential equation, to show how frictional abrasion alone of a stone on a planar beach might 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 the likelihood the stone is 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 the potential energy of the stone. No closed-form or even asymptotic solution is known for the basic equation, which is both non-linear and non-local. On the other hand, preliminary numerical experiments 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.