Theoretical and numerical approach to "magic angle" of stone skipping

We investigate the condition for the bounce of circular disks which obliquely impacts on the fluid surface. An experiment [C. Clanet, F. Hersen, and L. Bocquet, Nature (London)NATUAS0028-0836 427, 29 (2004)10.1038/427029a] revealed that there exists a "magic angle" of 20° between a disk's face and water surface in which the condition of the lowest impact speed necessary for a bounce is minimized. We perform a three-dimensional simulation of the disk-water impact by means of the smoothed particle hydrodynamics. Furthermore, we analyze the impact with a model of the ordinary differential equation (ODE). Our simulation is in good agreement with the experiment. The analysis with the ODE model gives us a theoretical insight into the "magic angle" of stone skipping.