UCL Mathematical & Physical Sciences


Researchers derive an equation to describe how stones skim across water

6 January 2023

A new mathematical model that predicts how a tossed stone will skim across the surface of water has potential applications in aircraft design, finds a study involving UCL researchers.

Skipping stone

The mathematical model, published in Proceedings of the Royal Society A, factors together the possible shapes and weights of a stone, the different speeds and directions of a throw and the momentum and pressure of the water as the stone impacts. It predicts how long, how far and how high a stone can skip off the water.

The team from UCL and the University of Bristol found that flatter stones are better at skimming farther, but larger “fatter” stones can elicit a “super-elastic” response from the surface of the water and bounce higher.

Lead author Dr Ryan Palmer of the University of Bristol said: "Our research shows the role of a skimmer’s mass and shape in determining whether it successful skips out of the water or sinks into it. It also reveals a relationship between these two characteristics, with a more curved underside enabling heavier bodies to skim that would otherwise sink if they were flatter."

The findings have multiple applications, including for engineers building the landing gear for planes that land on bodies of water, who need to be able to accurately predict how their designs will behave when touching down.

Study co-author Professor Frank Smith (UCL Mathematics) said: “There is strictly no optimal shape: it depends on what you are after and in addition there are so many factors such as weight, inclination, throw speed, wind and water response in reality. However, curved bodies do usually perform better and weight does matter a lot. Traditionally in stone skipping flatter round ones are preferred but heavier curvier ones can do a great job.”

The team found that there is an important relationship between the mass of a tossed stone and the curvature of its underside hitting the water that affects how high and far it will rebound into the air. When a stone first descends into the water’s surface, pressure builds up underneath as the water resists the stone’s entry. If the pressure is high enough and at the proper angle, the stone will skip out, but if not, the stone sinks below the surface.

As a tossed object gets larger and heavier, it needs a curvier bottom to build up enough pressure to leave the water. However, even large stones with a rounded enough underside can induce a significant bounce when thrown at the right speed and angle.

Other potential applications include ensuring aircraft can shed ice particles when flying through clouds to prevent build-up on their exteriors, which the model can help predict.

Professor Smith added: “The research is important for air, land and sea vehicles. Up in the clouds for instance huge numbers of ice particles impact upon meltwater on the aircraft surface and then skim downstream. The skim transfers heat. This and all the accompanying ice formation affect the aerodynamics in a risky and sometimes disastrous way.

“The sweet application to stone skipping just comes out as a by-product of the serious stuff. Much of the same physics is involved throughout. There is also the natural curiosity, enjoyment and popularity to consider of course.”

The research was funded by the Engineering and Physical Sciences Research Council and AeroTex UK.



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Mike Lucibella

  • E: m.lucibella [at] ucl.ac.uk