October 30, 2007

Skipping stone has been a popular pastime for thousands of years. Nearly everyone have tried it at least once. The rules have remained unchanged through history; one must throw a stone over a water surface so that it makes a maximum number of bounces. The current world record is detained by Russell Byars with 51 skip [1]. The first studies of object bouncing on water date from the beginning of last century. The reasons at this time were not as peaceful as skipping stone can be. Actually the main motivation of these early studies was focused on military application and more particularly on dambusters. They clarified the rebound condition of cylindrical and spherical objects on water. However if the physics of skipping stone is very similar, the geometry of the stone leads to a more complex analysis. Only recently (Stong in 1968, Crane in 1988 and Boquet in 2003 [2]) proposed a theoretical description of the physical mechanisms at play and shortly after an experimental study follows (Clanet et al. 2004 [3]). The aim of this report is to present the model developed and its agreement with this recent experimental result.

Fig. 1: Convention and geometrical properties
of the disk. |

The stone is modeled by a disk of thickness h, radius R and mass M. The water surface is assumed to be perfectly flat. The stone-skipping process involves then four parameters: U the translational speed, Ω the rotational speed, α the "attack" angle between the stone and the water surface and β the "impact" angle between the water surface and the translational speed (see Fig. 1). The typical values of these parameters are useful to determine the hydrodynamic regime involved and then further simplify the model. U is of the order of a few meters per second, Ω is several tens of rotations per second and α, β typically lie between 0 and 30 degrees.

First the influence of the rotational speed can
easily be removed from the equation. In the limit of small Ω, the
stone tumbles at the impact and dives under water. In the limit of very
large rotational speed the attack angle does not change during the
collision. Actually the effect of Ω is to stabilize the stone
through the gyroscopic effect. In the following model the rotational
speed is assumed to be high enough so that α is kept constant
through the bounce. The typical values of the parameters motioned
previously lead to a Reynolds number of 10^{5}. This number is a
ratio of the inertial effect over the viscous effect of water. In this
limits of a large Reynolds number the viscous effect of water can be
neglected at first and may be reintroduced later as a dissipation
term.

The critical physical point in the model is evidently the action of the water on the stone. In the limit of high Reynolds number the action of water on the stone is expected to take the following form:

where S_{wetted} is the area of the disk in
contact with water; C_{l} is called the lift coefficient. The
wetted area can be expressed as:

where s=|z|/sin(α) with z is the high of the lowest point of the disk relative to the surface. The lift force is considered to be zero when the disk is totally immersed. This is not actually true but since no rebound is possible at this point it does not influence the rebound condition. The difficulty of the problem comes from the nonlinearity introduce by expression of the wetted area. If one considers a square stone instead of a round one, an analytic expression can then be derived easily. An approximated description for the function f can be found by studying the action of a stationary water stream incident on a plane surface. Then an estimation of the momentum transferred to the plate gives

This dependence of the lift force has been experimentally verified by [4]. One must keep in mind that this expression of the lift force neglects the complex dynamic of the water flow around the stone. The trajectory of the disk can then been found by integrating the equation of motion

When the stone is not in contact with water, the motion simplify to a ballistic one. In this expression no dissipation term is included. However is easy to see that it still leads to a finite number of bounce. Indeed the action of the water on the disk projected on the x axis is always in opposite direction of the movement in that direction. It causes a constant deceleration. Along the z direction the situation is more subtle. Actually one can see that the lift force is smaller when the disk is going up that when it is going down (when the disk is going up the sign of β becomes negative). As a consequence the translational velocity decreases from one bounce to the other until it becomes to low for the disk to bounce.

Fig. 2: Chronophotography of a skipping stone
from Rosellini et al [4] (R=2.5cm, h=2.75mm,
U=3.5 m/s, Ω=65 s^{-1}, α=20° and
β=20°). |

A quantitative experimental work was done by Clanet
et al [3,4]. The experiment was designed to control independently
Ω, U, α, β and the collision of the stone was recorded
using a high speed video camera. The disk was made of aluminum leading
to a density ratio ρ_{s}/ρ_{w}=2.7. Fig. (2)
shows a chronophotography of a collision sequence. In this example the
collision time is 32 ms.

By varying the three control parameter (U,
α and β) the author constructed some phase diagram that
highlights the "skipping stone" domain (see Fig. (3)). The
author reported that the value α=20° is found to be the
optimal "attack" angle. Indeed it minimizes the required velocity
U_{min} for a rebound for a large range of β. The
maximal "impact" angle β angle is also reached for this value
of α. Moreover this particular value of α is also the
one that minimizes the collision time. One may observe that no
rebound is possible for value of β larger than 45°.

The model shown previously has been compared with these experimental sets of data and despite its simplicity a good agreement has been found. The optimal attack angle was found to be 3° off the measured one. This indicates that the approximated lift force that was used is able to capture the physical mechanism at play.

The collision process has also been investigated using a sophisticated 3D simulation by Nagahiro and Hayakawa [5]. It was done using a smoothed particle hydrodynamic technique (SHP). This method is based on a Lagrangian description of the fluid that is represented by a set of particles that moves according to Navier-Stocks equation. This simulation was able to reproduce some of the complex features of dynamic properties of the water flow around the stone.

© 2007 A. Serpry. The author grants permission to copy, distribute and display this work in unaltered form, with attribution to the author, for noncommercial purposes only. All other rights, including commercial rights, are reserved to the author.

[1] www.skippingstone.com.

[2] L. Bocquet, "The Physics of Stone
Skipping," Am. J. Phys. **71**, 150 (2003).

[3] C. Clanet, F. Hersen and L. Bocquet, "Secrets of
Successful stone-Skipping," Nature **427**, 29 (2004).

[4] L. Rosellini *et al.*, "Skipping Stones," J.
Fluid Mech. **543**, 29 (2005).

[5] S. Nagahiro and Y. Hayakawa, "Theoretical and
Numerical Approach to 'Magic Angle' of Stone Skipping," Phys. Rev. Lett.
**94**, 174501 (2005).