Dynamics of an elastic ball bouncing on an oscillating plane and the oscillon

The oscillon is a highly localized dynamical phenomena occurring in a thin horizontal layer of granular material, which rests on a rigid metal plate a nd the plate oscillates in the vertical direction. It is axially symmetric and physically resembles a splash of liquid due to a falling drop, except t hat it continually perpetuates itself and does not generate a spreading wav e, as is the case for a liquid splash. If the plate vibrates with amplitude A and period T = 2 pi/omega, then the oscillon moves from "peak" to "crate r" in time T-1 and "crater" to "peak" time T-2, such that the time from "pe ak" to "peak" or "crater" to "crater" is twice the period of the oscillatin g prate namely T-1 + T-2 = 2T. At present the physics of granular phenomena is not properly understood and there is no continuum mechanical theory of granular materials which is widely accepted as accurately describing their behavior. Here we present an elementary analysis of a single elastic ball b ouncing on an oscillating plate, and we demonstrate that under certain circ umstances the ball can perform a "big" bounce followed by a "little" bounce , and then simply repeat the sequence ad infinitum. For a perfectly elastic ball initially at rest on the oscillating plate, the theory with T-1 = T-2 predicts oscillonic behavior with an acceleration amplitude Gamma = A omeg a(2)/g (g is the acceleration due to gravity) of about 4.6, while experimen tally oscillons have been observed to occur for Gamma around 2.5. However, for T-1 not equal T-2 the theory predicts oscillonic behavior for values of Gamma which are well in accord with those observed experimentally. The ele mentary analysis presented here at least provides specific alternative Gamm a values for future experimentation, as well providing some insight into wh at is otherwise a complex physical phenomena. (C) 2000 Elsevier Science Inc . All rights reserved.