Hopping of elliptical pebbles on the beach and laboratory results

Abstract

On sandy ocean beaches, amongst many irregularly shaped pebbles, a surprisingly large number of pebbles that are nearly perfect ellipsoids may also be found. These elliptical stones are characterized by the density rho of the mineral and by the length of their symmetry axes (a, b, c) with (a > b > c). The grinding process which forms the ab-plane of the ellipsoid relates to the rotation around the axis of the greatest moment of inertia (c-axis) in the surf. In a recent paper by fairly simple physical arguments, it was shown that by this rotation around the c-axis the ellipticity varepsilon always increases. This is in agreement with the empirical observation that on sandy beaches of the oceans no spherical or disc-like pebbles can be found. If the pebble velocity during the rotation becomes too large, then hopping of the pebble sets in. This is caused by the fact that the centre-of-mass acceleration during fast rotation exceeds the gravity acceleration g. Since greater forces occur for jumping than for rolling, the hopping can have a strong influence on the b/a-ratio of the elliptical pebble. In this work, it will be shown that the b-axis is worn away more than the a-axis by erosion with jumping. In a laboratory study with a model of an elliptical cylinder, the theoretical results were checked and partially confirmed.