The revealed secrets of stone skipping
Christophe Clanet , Fabien Hersen , and Lydéric Bocquet
IRPHE, UMR 6594, 49 rue F. Joliot-Curie, BP 146, 13384 Marseille, France
Ecole Polytechnique, 91128 Palaiseau, France and
Laboratoire PMCN, UMR CNRS 5586, Université Lyon-I,
43 Bd du 11 Novembre 1918, 69622 Villeurbanne Cedex, France
Abstract
Skipping stones on water appears to have been a continuous human activity for
thousand of years. Although not written, the rules of this international game have
not changed since the ancient Greeks [1]: a stone or a shell is thrown over a water
surface and the maximum number of bounces distinguishes the winner [2]. However
beyond the obvious context of the game, only very few attempts have tried to understand the mechanisms behind this ancestral human activity [3–5]. We focus here on
the crucial moment of the stone skipping : the bouncing of the stone on the water
surface. We report the first exhaustive experimental study of the collision of a circular
spinning stone with water. Unexpectedly the energy dissipated during a collision is
found to exhibit a minimum for a specific ”magical angle” (around 20 degrees) of the
stone relative to the water surface. This angle provides the optimal throw conditions,
yielding the maximum possible number of bounces.
1
z
ω
t=0
α
β
U
1
x
2
3
4
t=τ
5
6
7
8
FIG. 1: Chrono-photography of a skipping stone, obtained with an aluminium disc of radius R =
2.5 cm, thickness e = 2.75 mm, translation velocity U = 3.5 m.s−1 , angular velocity ω = 65 rot.s−1 ,
attack angle α = 20◦ , trajectory angle β = 20◦ . Time increases from left to right and from top to
bottom with the time step ∆t = 6.5 ms.
A stone skipping throw involves a quite specific gesture, probably optimized generation
after generation : a sweeping movement of the arm gives a velocity to the stone, while a
spin velocity is provided by a small kick with the finger, making the stone to rotate. A good
throw is generally close to the water surface, with both a grazing translational velocity,
and a small angle between the stone and the water surface. All these different ingredients
actually define the control parameters of a water-stone collision, as shown on Fig. 1. Four
parameters characterize the collision : U and ω are the translational and spin velocity, while
α is the “attack angle” of the stone relative to the water surface, and β is the impact angle
of the translational velocity. We have developed an experimental set up, designed to control
independently each one of the four previous parameters, as shown on Fig. 2. This allows us
to study one stone-water collision, which is the key element of the stone skipping sequence.
A typical collision sequence is shown on Fig. 1. Many indications, both qualitative and
quantitative, can be extracted from such movies. The first obvious information is whether
the stone successfully bounces or not. But beyond this binary information, the collision
time, the change of orientation of the stone, the shape of the liquid cavity, etc. can be
2
U
α +β
β
ω
FIG. 2: Schematic description of the experimental setup. A spin velocity ω is provided to the stone
by a rotating wheel. The contact between these two elements is maintained by an air depression.
This whole block can be translated on a rail and is given a velocity U . Both the angles of the
rotating wheel (α) and translation rail (β) can be varied. At launched time, the air depression
between the stone and the rotating wheel is released via a contactor, and the stone pursues its
trajectory towards the impact with water. This experimental setup therefore allows to control all
four collision parameters, U , ω, α and β. The collision of the stone with water is then recorded
using a high speed video camera (Kodak HS4540, with 2250 frames per second). All the experiments are conducted with water and with an aluminium stone, corresponding to a stone (s) to
water (w) density ratio : ρs /ρw ≈ 2.7. In all the figures except 5-b, the reported experiments have
been obtained using a disc of mass m = 15 g, with a radius R = 2.5 cm and thickness e = 2.75 mm.
collected from these pictures. As an example, the collision on Fig. 1 extends from image 2
to image 7, yielding τ ≈ 32 ms. All these quantitative informations provide much detailed
insight into the mechanisms at play during the collision.
Let us first concentrate on the influence of the rotation of the stone. The rotation is expected
to stabilize the stone thanks to a gyroscopic effect [3]. This is indeed what is observed, as
shown on Fig. 3, where a low spinning stone does not bounce but dives. This stabilizing
behaviour is highlighted more quantitatively on Fig. 4-a : the collision time τ is found to
diverge at small spin velocity ω, while it converges asymptotically to a constant value at high
ω. For example, under the conditions of Fig. 4-a, a spin velocity of -at least- ω ≥ 40 rot.s−1
is necessary to stabilize the stone.
3
1
2
3
4
5
6
7
8
FIG. 3: Chrono-photography of a diving stone, obtained with an aluminium disc of radius
R = 2.5 cm, thickness e = 2.75 mm, translation velocity U = 3.5 m.s−1 , attack angle α = 35◦ ,
trajectory angle β = 20◦ . angular velocity ω = 5 rot.s−1 . The time step ∆t between each image is
∆t = 8.9 ms.
In the following, we focus more specifically on this high spin velocity limit. With the three
remaining control parameters, {U, α, β}, a dynamical phase diagram can be constructed,
highlighting the conditions for a successful bounce (the ”skipping stone” domain). Cross
sections in the {U, α} and {α, β} variables are shown on Figs. 4-b and 4-c. Unexpectedly
this phase diagram points out the specific role played by the value α 20◦ : the lowest
velocity for a rebound, Umin , reaches a minimum for α 20◦ , while the maximal successful
domain in β is also achieved for this specific value of α.
Looking more into details of the chrono-photography yields further insight into the collision
mechanism. One observation is the building up of a dissymetric air cavity during the impact
of the stone with water (image 8 in figure 1). More unexpected is the fact that after a
transient period, a liquid film is observed ahead of the stone on the chrono-photography.
This shows that the whole surface of the stone is quickly wetted by water during the collision,
and not only the immersed area, as was assumed in Ref. [3].
Now pursuing with a more quantitative analysis of the collision, we report on Fig 5-a the
experimental measurements for the collision time as a function of the attack angle α. The
4
(a)
(b)
τ (ms)
U min (m.s -1 )
60
6
50
5
(c)
60
50
skipping stone
40
4
40
30
3
20
2
20
10
1
10
β
30
skipping stone
0
0
0
20
40
60
80
100
0
0
10
-1
20
30
40
α
ω (rot.s )
50
60
0
10
20
30
40
α
FIG. 4: Definition of the skipping stone domain: (a) Evolution of the collision time τ with the
spinning velocity ω for R = 2.5 cm, e = 2.75 mm, U = 3.5 m.s−1 , α = 20◦ , β = 20◦ ; (b) Domain
of the skipping stone in the {α, Umin } plane with R = 2.5 cm, e = 2.75 mm, ω = 65 rot.s−1 ,
β = 20◦ ; (c) Domain of the skipping stone in the {α, β} plane with R = 2.5 cm, e = 2.75 mm,
ω = 65 rot.s−1 , U = 3.5 m.s−1 . We quote that the domain β < 15◦ was not reachable with our
experimental set-up.
main feature on this plot is the existence of a minimal value of the collision time τmin
again obtained for α ≈ αmin = 20◦ . In order to understand more specifically the physical
mechanisms at play, we studied the evolution of this minimal collision time τmin as a function
of velocity U for different stones diameters and thickness. As indicated on figure 5-b, the
√
minimal contact time is found to obey a very simple scaling, τmin ∝ eR/U (for fixed α ≈
20◦ and β = 20◦ ). This scaling is actually suggested by a simple dimensional analysis. Since
the lift force Flif t is the key point in the rebound process, a collision time can be constructed
as τ ∼ mR/Flif t with R the radius of the stone and m its mass. Now for the velocities
under consideration, the Reynolds number is quite large (Re = UR/ν ∼ 105 , with ν the
kinematic viscosity of water) and the lift force is expected to scale as Flif t ∼ ρw Swetted U 2 ,
where ρw is the mass density of water and the wetted area scales as Swetted ∼ πR2 [7, 8].
√
Using m = ρs eπR2 with ρs the mass density of the stone, one gets eventually τ ∝ eR/U,
as is indeed measured experimentally.
This simple argument, although informative, does not help in understanding the presence of
5
50
60
(a)
(b)
(c)
τ (ms)
τ min (ms)
z (mm)
80
60
0
50
60
-5
40
40
30
τ min
20
-10
20
10
0
0
0
15
α min
30
45
60
-15
0
1
2
3
eR
(ms)
U
α (°)
4
0
5
10
15
20
25
30
t (ms)
FIG. 5: (a) Evolution of the collision time τ as a function of the attack angle α for R = 2.5 cm,
e = 2.75 mm and different conditions: U = 3.5 m.s−1 , β = 20◦ , U = 3.5 m.s−1 , β = 30◦ ,
• U = 5 m.s−1 , β = 20◦ , the solid lines are parabolic fits which provide guides for the eye. (b)
√
Evolution of the minimal contact time τmin with the characteristic time scale eR/U with β = 20◦
and α ≈ 20◦ and different stones: R = 2.5 cm, e = 2.75 mm, R = 5 cm, e = 2.55 mm, •
R = 2.5 cm, e = 5.55 mm, ◦ R = 5 cm, e = 5.55 mm, the solid line present the results obtained
numerically through the integration of equation (1); (c) time evolution of the vertical location of
the stone for U = 3.0 m.s−1 , α = 20◦ ,β = 20◦ : experimental data, the solid line results from the
numerical integration of equation (1).
a minimum collision time. A more detailed description is thus required, as the one proposed
e.g. in Ref. [3]. However the latter description suffers from various drawbacks since it is
e.g. unable to account neither for the existence of a minimum in Umin as a function of α,
nor for the role of αmin ∼ 20◦ .
We now propose a model to account for these behaviours. First, the presence of the water
film ahead of the stone suggests to use the analogy with the classical problem of the oblique
impact of a jet on a surface to compute the potential force acting normally to the stone
surface : Fp ≈ Cp ρw Swetted U 2 sin (α + β) with Swetted is the wetted area and Cp is a dimensionless coefficient [6]. Neglecting gravity, the equation of motion of the stone along the z
6
35
40
direction thus writes :
m
d2 z
= Cp ρw U 2 Swetted sin (α + β) cos α
2
dt
(1)
However the stone is not immediately fully wetted and one should account for the time
dependent raise of the water film over the stone surface. The wetted area thus depends on
the water film level on the stone, which is roughly given ≈ Ut for a stone with velocity U.
For a circular stone, the wetted area can be roughly estimated as Swetted ≈ π2 R. Assuming
U, α and β to be constant, the previous equation, Eq. (1) can be integrated. Together
with the initial conditions z = 0, dz/dt = −U sin β one gets the following evaluation of the
contact time of the stone with water :
τ2 =
12 ρs eR
f (α, β)
Cp ρw U 2
where f (α, β) ≡ sin β/ [sin (α + β) cos α]. The scaling τ ∼
(2)
√
eR/U is retrieved as well
as the existence of a minimum value of the collision time for a finite α. Now including
gravity, g, in the model provides an estimate for the minimum value to bounce, Umin , as
Umin ≡ mg/Cp ρw πR2 sin (α + β) cos α. For α = β = 20◦ , this gives Umin ∼ 0.4 m.s−1 ,
which clearly too small compared to the experimental values Umin ∼ 3 m.s−1 .
Thus even if this simple model account for most of the physics of the rebound, some improvements are necessary : (i) The model only holds for time smaller than the complete wetting
time τwetting ≡ 2R/U. For times larger than τwetting , the surface of contact remains constant
and equal to πR2 ; (ii) Even if the attack angle α remains constant during the impact, due
to the stabilizing rotation effect, the trajectory angle β changes as well as the velocity U.
Taking these points into account, we have integrated numerically the equation (1), together
with the corresponding equation for x. Most of the experimental feature are then found to be
(nearly quantitatively) reproduced by this approach : The numerical results for the contact
time compares qualitatively and quantitatively with the experimental values, as seen on
figure 5-b (the best agreement with the experimental results providing the value Cp = 0.3);
The asymmetric shape of the trajectory of the stone inside the water is nearly quantitatively
reproduced as can be seen on Fig. 5-c; the minimum velocity Umin is found to exhibit a
minimum around α ∼ 20◦ , although experimental values are still slightly underestimated
(typically for α = β = 20◦ , the model predicts Umin ∼ 2m.s−1 , while Umin ∼ 3m.s−1 in the
experiment); For a given U a minimum in the collision time is always found for values of α
ranging between 15◦ and 30◦ for β varying between 10◦ to 40◦ (data not shown).
7
These results show that in spite of the complex nature of the water-stone collision,
quantitative predictions might be obtained within a simple hydrodynamic description. One
unexpected result which emerges from the measurements is the presence of a minimum in
the stone-water collision time for a given “magical” angle α ∼ 20◦ . It is then easy to show
that this angle does also maximize the number of bounces, Nmax . Let us estimate this number. Starting from a throw velocity U0 , Nmax is reached when the velocity U has decreased
to its minimum value Umin . Now, for each collision, the variation ∆U of the velocity can
be estimated on the basis of the averaged force parallel to the water surface, Fx , and the
collision time τ , as ∆U ∼ Fx τ /m. Using the previous estimates for the hydrodynamic
forces and the collision time, Eq. (2) , one gets ∆U/U ∼ f (α, β) for each rebound. This
predicts a maximum number of bounces Nmax scaling as Nmax ∼ 1/f (α, β) log(U0 /Umin ),
with U0 the initial throw velocity. The dependence of Nmax on U0 is much weaker than the
one predicted from the more naive approach in Ref. [3] a result which is in much better
agreement with Everyone’s experience. Moreover, as can be seen immediately, Nmax is also
maximized for a value of the angle α which minimize the collision time. In contrast to what
one would have believed, the best throw strategy is not a vanishing attack angle α : a finite
angle α helps a lot in increasing the number of bounces. There is still to be learned from
science for stone skipping... Back to your stones !
Acknowledgements:
The conception of the catapult has been conducted at the Ecole Polytechnique as a Scientific Project
by Vincent BALLARIN, Antoine MAITROT, Alexandre MIRAILLES, Lionel ROSELLINI, and Vincent
ROTIVAL. Its contruction has been performed at IRPHE by Franck DUTERTRE and Jacky MINELLI.
The idea of the depression to sustain the stone is due to Patrice LE GAL and part of the funding to
Emmanuel VILLERMAUX.
Nicolas WITKOWSKI has found and provided probably the first scientific article on the subject [9].
May all of them find here the expression of our recognition.
We also thank M.-L. Bocquet for a careful reading of the manuscript
[1] D’Arcy Thomson, Alliage 44, 77 (2000).
8
[2] The actual world record appears to be 38 rebounds (by J. Coleman-McGhee). See, for example,
¡http://www.stoneskipping.com¿ for more information on stone skipping competitions.
[3] L. Bocquet, The Physics of Stone Skipping, American Journal of Physics, 71, 150-155 (2003).
[4] Some pictures of the bouncing process of a circular stone on water and sand can be found in
C. L. Stong, The Amateur Scientist, Sci. Amer. 219 (2), 112–118 (1968).
[5] H. R. Crane, “How things work: What can a dimple do for skipping stones?,” Phys. Teach. 26
(5) 300–301 (1988).
[6] E. Guyon, J.-P. Hulin, L. Petit, Hydrodynamique Physique, Inter Editions et Editions du CNRS,
1991 (p. 201).
[7] D. J. Tritton, Physical Fluid Dynamics (Oxford University Press, 1988), 2nd ed., pp. 97–105.
[8] L. D. Landau and E. M. Lifshitz, Fluid Mechanics (Pergamon Press, 1959), pp. 168–175.
[9] Encyclopoedia Diderot-d’Alembert, Paris, 1751 (article “Ricochet” from d’Alembert), or
“Traité des fluides” from d’Alembert, Paris, 1744 (p. 221).
9