The art and science of skipping stones explained

Skimming stones. (Photo: ChatGPT)

By Don Pinnock

3 Sep 2024

On encountering a pebbly shore beside placid water, at some time in your life you are almost guaranteed to have selected a flat stone and skimmed it. However, there’s more to this than you might imagine.

“It is very well known that stones thrown slantwise upon water skip, and this is a favourite amusement of boys playing on the banks of rivers.” This is not taken from a modern pamphlet on parenting, but from a scientific paper written in the 1750s by the Italian scientist Lazzaro Spallanzani, who was fascinated by hydrodynamics.

He was better known for demonstrating that life could not arise out of thin air (“spontaneous generation” was the current theory at the time and Darwin was long in the future), that sperm was essential for fertilisation and that bats could navigate using their ears. His work on skipping stones was small cheese compared to that stuff, but some trawling turned up “De lapidibus ab aqua resilientibus” written in Latin. ChatGPT offered to translate all 55 pages but, helpfully, offered a summary.

Lazzaro Spallanzani. (Photo: Supplied)

The thesis seems to have been a response to the assertion by the Jesuit mathematician Jacques Bellogarde that stones bounced because of the elasticity of water. From his work on hydrodynamics, Spallanzani disagreed, but then had to explain why stones bounced when thrown on water. Was it air trapped under the stone or some physical reaction between stone and water?

Through no doubt countless repetitions and a large pile of stones, Spallanzani’s eagle eye noticed what happened to the water when the stone struck it. A true scientist, he started with the obvious: if you dropped a flattish stone into the water, thin edge first, it “sought the bottom”.

But with the right angle of approach (not straight down) the stone created a dent in the water which pushed up water ahead. The stone, he wrote, “seemed to descend by one plane and ascend the other, then to spring off”. This was assisted by the angle of the stone, which tended to tip slightly backwards, the following edge hitting the water before the front, a perfect angle for takeoff.

The energy of the throw propelled the stone skywards up the slope out of the dent, defying gravity until it descended to bounce again and again until it ran out of energy and dived. The lower the angle of the throw, the more likely it was to skip.

He tested stone shapes and sizes and found that stones with flat, smooth surfaces and a specific angle of impact would bounce more frequently.

There the theory rested, unchallenged, for around two centuries until 1957, when an article on the subject appeared in Scientific American magazine written by a retired professor of English, Ernest Hunter Wright, who confused the matter by skimming stones on a wet beach to enable him to see the shape of the indentation.

Skimming stones. (Photo: ChatGPT)

He was startled to find that the second bounce track appeared a mere 10cm after the first, followed by a long skip, then another short one until the stone ran out of momentum. He thought the same thing must happen in water but couldn’t explain why. He asked eminent scientists and got answers that didn’t satisfy him.

One was that the stones flipped over on every impact, creating the two close marks before leaping; another was that it hit first with its rear end and then its front.

There it rested for around 30 years until in 1988 a professor of physics, Richard Crane, explored the role of the spin. Writing in the American Physics Teacher, he calculated that spin was essential. On hitting the water, the spin caused a wobble which increased the tilt and improved the bounce. Or something to that effect.

In 1991 at the Massachusetts Institute of Technology, research took a jump when a student named Kirston Koths employed high-speed cameras, purpose-designed sandstone discs and a strobe light firing every millionth of a second. “About as long as it takes an automobile moving at 60 miles an hour to travel the thickness of its paint,” he explained.

On skipping stones on sand, he found, Wright was correct: they tumbled then jumped. In water, Koths found that the stone tilts slightly backwards when hitting the water, causing a wave to build up under its leading edge. Because of its forward momentum, it launches itself up the wave and into the air again. In other words, Spallanzani was correct.

In 2002 a French scientist, Lydéric Bocquet, applied complex mathematics to the subject and laid the need for further research to rest. Spallanzani had nailed it from the start. It has nothing to do with the elasticity of water.

The last word is an almost unintelligible (to us mortals) disquisition of stone skipping that appeared in Nature in 2004 written by French researchers Christophe Clanet, Fabien Hersen and Bocquet.

“The minimal collision time is found to obey a simple scaling when the velocity, U, radius, R, and thickness, e, of the stone are varied: namely, τmin∝√(eR)/U (for fixed α and β). This scaling is inferred from a simple dimensional analysis. As the lift force, Flift, is the key point in the rebounding process, a collision time can be constructed from the dynamical law as τ∝√(mR/Flift), where m is the mass of the stone,” they found.

Um… yes, indeed.

To keep it simple, here’s some useful advice for your next competition beside the water, though it may seem obvious. Select a stone that’s not too heavy and not too light; then the lower the angle, the harder the throw, the flatter the stone and the greater the spin, the greater will be the number of hops.

The world record for the number of bounces of a stone skipped on water is an astounding 88. This was set by Kurt Steiner on September 6, 2013 at Red Bridge in Pennsylvania, US. Steiner has held the record since that date. If you want to know his technique, see this video. DM

https://www.youtube.com/watch?v=6GWL8Gt-BsQ

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