In 2002 an American called Kurt Steiner set a new world record when he threw a stone across a river in Pennsylvania and made it bounce 40 times. Readers of Physics World may not have been quite as successful as Steiner, but many will be familiar with the principle of stone skipping – to throw a flattish stone across the surface of a body of water so that it bounces as many times as possible (figure 1). Although the phenomenon is well known, it does raise a number of questions, such as why does a stone skip at all, how many skips can it perform, and how can the number of skips be maximized?

Stone skipping is just one of many other everyday but intriguing phenomena that can be understood using a number of tools and concepts from physics, such as hydrodynamics, elasticity and capillarity. Solving the mysteries of these phenomena lies in bringing these separate ingredients together correctly, a process that can sometimes involve extraordinary complexity. Much of the basic physics of stone skipping has now been cracked, thanks in part to the use of specially designed laboratory equipment to skip stones and record their motion.

Balancing forces

The origin of the force that causes a skipping stone to bounce is easy to identify. The conservation of momentum dictates that as the stone enters the water and pushes some of it downwards, the stone is, in turn, forced upwards. This force is equal to the hydrodynamic pressure on the stone multiplied by its area, which, using dimensional analysis, can be shown to scales as ρU²S, where ρ is the density of the water, U is the stone’s velocity and S is its cross-sectional area. Assuming that this force is balanced against the weight of the stone Mg, where M is its mass and g is the acceleration due to gravity, there exists a minimum velocity – of the order of a few kilometres per hour – above which the stone will bounce.

These basic assumptions have been supported by experimental results gathered by us and our colleagues (2005 J. Fluid Mech. 543 137). To carry out our experiments, we have developed a catapult device that can throw aluminium disks at fixed translational and rotational velocities, and can then record the “splashes” using high-speed video recording (figures 1 and 2). Our results show that a stone does indeed need to have a minimum velocity in order to bounce. If its velocity is less than this value, the stone “surfs” on the water for a short distance and then sinks.

However, the measured threshold velocity is actually much higher than that predicted by the naïve argument above. This is because the assumptions made thus far do not account for the inertia of the stone, i.e. its resistance to changes in its motion. Purely in terms of dimensions, inertia is equal to mass × length × time^-2, and in the case of a stone is estimated to be MR/τ², where R is the radius of the stone and τ is the time over which the collision with the water takes place. Balancing this against the lift force, ρU²S, leads to the following formula for the collision time: τ = (MR/ρS)^-1/2/U.

We have verified this simple formula experimentally by varying the velocity of the stone and using stones with different mass, radius and thickness, and we have been able to do so even though our photographs show that the collision involves highly complex hydrodynamical processes. Indeed, a slightly more detailed version of the same equation can accurately predict all of our experimental results, even when the stone is thrown with a range of velocities and in a variety of different directions.

Our experiments have also shown that a bouncing stone must spin with a certain minimum rotational velocity if it is to be stable, i.e. if the angle between the plane of the stone and the water surface is to remain constant (as is the case in figure 2). This stabilization is known as the “gyroscopic effect”, which is used in many applications from spinning tops to the gyroscopes found on spacecraft. To remain stable, a stone typically needs to rotate at least once during its collision time, which means that it must have a minimum spin velocity roughly equal to the inverse of the collision time. If this rotation does not take place, the stone’s collision becomes quite complex and a second bounce becomes much less likely. This is something that stone skippers realize intuitively, rotating the stone with a flick from the finger.

The need for speed

Given these minimum requirements, how can skippers then maximize the number of bounces? Unsurprisingly, the answer is to throw the stone faster. We have found experimentally that the number of skips is more or less proportional to the throwing speed, given the minimum velocity discussed above. However, throwing the stone at high speeds while controlling the velocity and direction of the throw can quickly become a technical challenge. Our catapult was able to throw a 15 g stone up to 10 m s^-1, resulting in 20 skips, but could not throw any faster. This was some way short of the 20 m s^-1 that would have been needed to break Steiner’s world record. Too bad.

To demonstrate this result mathematically we need to assess what causes the stone to eventually stop skipping. Surprisingly, the stone does not slow down (as can be observed in figure 1, where the stone travels equal distances between successive snapshots, which are equally spaced in time). Instead, the stone’s trajectory “flattens” with time – in other words the vertical component of its velocity decreases while its horizontal one stays constant. This is because the angle with which the stone moves relative to the surface of the water dictates that the stone displaces more water when it moves down than when it rises (as revealed by the shape of the impact cavity in figure 2); this results in a smaller transfer of momentum in the latter stage of each skip and therefore in reduced lift. With successive skips, the stone’s vertical velocity continually decreases and its energy diminishes. When the stone no longer has the energy to jump, it simply surfs over the water before finally sinking. Translating this reasoning into mathematics provides an excellent prediction for the number of skips as a function of velocity.

The number of skips is also determined by the type of stone used and the angle at which it is thrown. We have not investigated these factors experimentally, but we estimate that the optimum angle between the plane of the stone and the surface of the water is about 10-20 °. And as all stone skippers know, the flatter the stone, the better. This is because, for a given mass, flat stones are able to displace more water than rounder stones.

A fruitful pastime

The physics of stone skipping does have its applications, such as the famous bouncing bomb designed by Barnes Wallis and used by the “Dambusters” in the Second World War. More recently, groups at Tohoku University and the Tokyo Institute of Technology in Japan have used our experimental results as a benchmark in order to model the impact of solid objects on a liquid surface. Such research could, for example, be used to improve the design of certain torpedoes that spend part of the time above water in order to reduce their journey times.

However, any such applications are in a sense accidental. The main reason for investigating stone skipping is simply to satisfy one’s curiosity.

• A video of Kurt Steiner’s record-breaking stone-skipping throw can be found at pastoneskipping.com/steiner.htm