On the formation of elliptic stones due to periodic water waves

On the beaches of the oceans may be found amongst many stones with irregular shapes also stones that are nearly perfect ellipsoids. In this work, two grinding processes in the surf for stones of ellipsoidal shape are described mathematically on the basis of simple physical models. The first process is the abrasion of material perpendicular to the shortest axis of the stone sliding on the sand of the beach. The model explains the experimental fact that oblong stones with small b/a-ratios as a rule are slim stones with a small c/a-ratio. The second process is connected with a rolling motion around the axis of the largest moment of inertia, which leads to an effective grinding on a surface strip perpendicular to the ab-plane. The corresponding calculation yields a criterion for the form stability of the ellipsoid. The obtained result is virtually identical with the experimentally observed maximum of the error distribution function for the b/a-ratio. Copyright EDP Sciences, SIF, Springer-Verlag Berlin Heidelberg 2013