The timeless childhood pursuit of skipping stones across a still pond involves some pretty tricky physics—and now researchers have developed a mathematical model that explains much of it. The model could help engineers design airplane wing surfaces better able to shake off icy buildup and boat hulls that can glide more smoothly through the water.
Scientists have been trying to unravel the physics of skipping for years. They know what happens when a stone first hits the water—or when tiny ice crystals strike the moisture-coated wings of an aircraft—but things get murky after that.
Mathematicians Peter Hicks and Frank Smith of University College London had been working on formulas to describe collisions between particles blowing in the wind. Eventually, they realized that many of the same aspects of those collisions could be applied to bodies skipping over water.
Based on their earlier research, Hicks and Smith constructed a mathematical formula that predicts the effect of objects of various masses and shapes hitting and then rebounding out of water at different velocities and angles. In particular, the model can calculate the amount of pressure water will exert on an impacting object depending on its shape and mass as well as its velocity and angle of entry.
Reporting  online this month in the Proceedings of the Royal Society A, Hicks and Smith detail the several formulas they combined to produce the model. The formulas also incorporate factors such as the length between the object's leading and trailing edges and the area of contact between the leading edge and the water—a factor called the "wetted portion."
Regardless of the variables, pressure remains the critical factor, Hicks explains. Enough pressure applied against the leading edge of the impacting object and the stone or ice crystal will rebound back into the air. The model "allows us to look at the motion of the body through the liquid and predict [whether] the body can rebound," Hicks says. "There are individual interactions you see with each bounce when you try and skip a stone across a lake," he adds, "although in my case the stone invariably sinks without a trace."
The model can also help boat designers, because the same principles applied to skipping can be used to improve the lift and stability of vessels cruising across the water, as well as the comfort of their passengers, says applied mathematician Alexander Korobkin of the University of East Anglia in Norwich, U.K. But understanding the physics becomes critical in the case of ditching an aircraft in water, he says, when a pilot must apply the proper angle of attack to maximize the aircraft's survival chances. The paper, Korobkin says, has "shed some important light into this complicated and not well-understood phenomenon."