Impacts of Solid Objects on Fluid Surface  -- Physics of Skipping Stones

 Shin-ichiro Nagahiro, May 2005
  At a lakeside, we sometimes see time-wasters throwing a stone on water and counting the number of bounces the stone was able to make. Our intuition gives us some rules for successful bounces: a stone must be flat and thin in shape, and one should throw it with small incident angle with water's surface [1]. Could we explain these empirical rules with the help of classical mechanics and hydrodynamics? The answer is, of course, "yes" but it is very difficult. A stone splashes water and largely deforms its surface. To solve the Navier-Stokes equation with such boundary condition is almost impossible. Hence, the studies on "sold-fluid impacts" have been relied on experiments for long years.

  Recently, C. Clanet, H. Hersen and L. Bocquet performed an experiment of stone skipping using an aluminium disk as a model stone, and revealed a secret for a successful bounce: throwers should tilt the stone about 20 degrees to water's surface [2]. The "magic angle" 20 [deg.] minimizes the required speed necessary for a bounce (see figure 1). 

  To better understand their experiments, we numerically and theoretically studied disk-water impact. For numerical simulation, we used the technique of Smoothed Particle Hydrodynamics (SPH). SPH describes flow with a set of fluid particles which moves in accordance with the Navier-Stokes equation. Because the method does not require a grid for computation, and thus has an advantage to treat free surface motion (see a mpeg video of our SPH simulation).
  Neglecting the effect of viscosity, surface tension and deformation of water's surface, we considered a model of ordinary differential equation (ODE) for a further analysis [3].
  Both approaches successfully agreed with the experiment and our numerical simulation showed that the exact value of the "magic angle" varies slightly with the precise geometry of the impact. But, in most cases of stone-skipping, it remains around 20 [deg.].
  Even though our ODE model was seemingly too simple to reproduce the experiment, the model could give a  qualitative explanation for the disk-water impact. Hence we expect that impact of stone on water could be understood as a very simple dynamic process.

[1] F. Jerdone Coleman "The Secrets of Stone Skipping", Stone Age Sports Publications (1996)
C. Clanet, H. Hersen and L. Bocquet, Nature (London) 427, 29 (2004)
[3] S. Nagahiro and Y. Hayakawa, Phys. Rev. Lett. 94, 174501 (2005)