# Mathematical equation explains how rogue waves are formed

Posted June 28, 2016

• Mathematical equation explains how waves can combine in unusual circumstances
• Rogue waves last for around 20 second and threaten even large ships

An international team of scientists has developed a new mathematical equation that explains how rogue waves of up to 25 metres high can leap seemingly out of nowhere to sink big ships and engulf oil platforms.

The waves stem from a combination of constructive interference – a known phenomenon of colliding waves – and nonlinear effects specific to the complex dynamics of ocean waves.

An improved understanding of how rogue waves originate could lead to improved techniques for identifying ocean areas likely to spawn them, allowing shipping companies to avoid dangerous seas.

The mathematical equation was developed by Professor Frédéric Dias, UCD School of Mathematics and Statistics, and researchers from Georgia Institute of Technology and Institut FEMTO-ST CNRS-Université de Franche-Comté.

The findings were published in Nature Publishing Group’s Scientific Reports.

The new research is based on an analysis of the 26-metre high Draupner wave that struck an oil rig in 1995 in the North Sea and the Andrea wave that was also recorded on instruments on an oil rig in the North Sea in 2007. The study also analysed the Killard rogue wave that struck in 2014 at a site for marine renewable energy off the coast of Co Clare, Ireland.

##### Pictured right: Professor Frédéric Dias, UCD School of Mathematics and Statistics

Before the Draupner wave struck, waves of this height were widely dismissed as impossible. How they are formed remains a mystery.

The equation combines constructive interference and nonlinear effects. Combined interference happens when energy from two or more waves combines to ‘pile up’ and form a larger single wave.

Though ocean waves have a predominant direction, in the open ocean, waveforms from other directions can arrive. In rare conditions, those waves arrive in an organised way or almost in phase, leading to an unusual case of constructive interference that can double the height of the resulting wave.

But this doubled height still cannot explain the size of the rogue waves observed in the North Sea – and elsewhere. That difference can be accounted for by the nonlinear nature of the ocean waves, which are not sinusoidal – meaning smooth and consistent in both troughs and peaks – but instead have rounded troughs and sharp peaks.

Using both these factors allowed the team to model how waves could combine in unusual circumstances to produce the Draupner, Andrea and the Killard rogue waves.

The simulations they ran based on their model matched the measurements of the three real-world rogue waves almost perfectly.

“We describe the complex energy flow of a wave field by what we call its directional spectra,” said Professor Dias.

“What we have shown is that by combining knowledge of this spectra and using mathematics that accounts for second-order nonlinearities, we can reproduce the measured rogue waves almost exactly.”

The research has been the basis for a new rogue wave model that could be used to identify ocean areas where nonlinear effects could give rise to the waves and to provide new insights into the unsolved problem of wave breaking. The authors hope that their results may eventually be used to improve the warnings given to those onboard ships.

Rogue waves typically last only 20 seconds or so before disappearing – but this is long enough to endanger large ships.

Francesco Fedele, Georgia Institute of Technology, John Dudley, Institut FEMTO-ST CNRS-Université de Franche-Comté, and Sonia Ponce de León and Joesph Brennan, both of UCD School of Mathematics and Statistics, also contributed to the research.

The paper is entitled ‘Real world ocean rogue waves explained without the modulational instability’.

By: Jamie Deasy, digital journalist, UCD University Relations