Mathematicians develop new models to describe ripples

Math is a language, and like a language, mathematics can be used to describe related phenomena. Recently, researchers at the University of Buffalo developed new math models to describe the ripple-like wave phenomena known as undular bore.

Undular bore involves the propagation of rapid oscillations. The ripple-like disturbance can occur in water or plasmas, on large or small scales. Undular bore is often observed in the atmosphere.

Mathematicians at Buffalo wanted to describe the spreading of oscillations along two axes.

"You see these effects in water, in plasmas, in the atmosphere, so the equations that describe these waves come up in a bunch of different fields," Gino Biondini, a professor of mathematics, said in a news release. "We like to say that the math is universal -- the same mathematics allows you to describe many different scenarios."

Mathematician Gerald B. Whitham developed a formula for describing undular bore in the 1960s, but his equations could only describe the wave propagations along a single access -- a ripple moving down a narrow channel.

The new models, detailed this week in the journal Proceedings of the Royal Society A, describe the propagation of oscillations in two directions, along two axes. So far, researchers have used their formulas to analyze undular bore with wave height variation along just one of the two axes. But Biondini and his partner Mark J. Ablowitz, a professor of applied mathematics at the University of Colorado, hope to soon measure wave height variation in two dimensions.

"The equations we formulated mark a step forward for describing these interesting phenomena," Biondini says. "Also, the methods we used can be applied to study a variety of related physical problems, so we hope that our results will open up a long series of works on these kinds of topics."

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