Hokusai’s Great Wave and Roguish Behaviour

Hokusai’s woodcut “The Great Wave off Kanagawa”.

“The Great Wave off Kanagawa”, one of the most iconic works of Japanese art, shows a huge breaking wave with foam thrusting forward at its crest, towering over three fishing boats, with Mt Fuji in the background [TM158 or search for “thatsmaths” at irishtimes.com].

This woodcut, produced by artist Hokusai around 1830, is full of dynamism and dramatic tension. The wave, with height estimated at more than ten meters, has inspired many artists and it appeared on the original score of Claude Debussy’s orchestral work, La Mer.

In recent decades, many enormous sea waves have been observed, removing all doubt about the existence of rogue waves. These waves have heights more than double the surrounding waves. In January 2014, the height of a wave off Killard Point in Co. Clare was measured at almost 30 meters. Although they are quite rare, rogue waves are part of the normal behaviour of the oceans.

Rogue waves have caused controversy, and consensus on their origins and behaviour is only beginning to emerge. It seems likely that several physical mechanisms are involved in the unusual amplification of these waves. Rogue waves are of limited extent, being localized in both the transverse and longitudinal directions. They are unrelated to tsunamis, which are extended in space and owe their existence to quite different causes.

Nonlinear Modelling

Earlier mathematical studies of waves used linear theory, in which two waves can reinforce each other, but which cannot simulate rogue waves realistically. In nonlinear models, wave interactions can trigger instabilities that produce extreme peaks in wave height. The resulting rogue waves appear suddenly and disappear again just as suddenly.

Photograph of a rogue wave, taken from the French ship Astrolabe

A photograph of a rogue wave, taken from the French ship Astrolabe in the Southern Ocean between Hobart and the Antarctic, shows it to be localized and on the point of breaking. Its structural similarity to Hokusai’s wave is striking.

Recent research by Prof Frédéric Dias of UCD and co-workers examines the process of directional focusing in crossing seas, where wave trains with different directions can constructively interfere to produce extreme amplitudes at isolated locations. They support their theory with computer simulations having solutions very like the observed rogue waves.

Rogue Waves in a Tank

Recently, rogue waves have been generated under laboratory conditions. We thus have corroborative evidence from mathematical theory, computer simulations, wave-tank experiments and observations, putting the essential characteristics and behaviour of rogue waves beyond doubt.

Rogue wave in a laboratory wave-tank
[image from McAllister, et al.]

Ongoing research aims to determine the critical factors for the formation of rogue waves. Mariners’ lives depend on their ability to avoid them, and new theoretical descriptions may enable us to anticipate their likely occurrence. Despite progress, many questions about rogue waves remain unanswered and research continues. The pay-offs include greater accuracy of wave predictions and saving of money and of lives.


Dudley, J. M., Sarano, V., and Dias, F.: On Hokusai’s Great wave off Kanagawa: localization, linearity and a rogue wave in sub-Antarctic waters, Notes and Records of the Royal Society of London, 67, 159–164, 2013. doi: 10.1098/rsnr.2012.0066.

M L McAllister, M L, S Draycott, T A A Adcock, P H Taylor and T S van den Bremer, 2019: Laboratory recreation of the Draupner wave and the role of breaking in crossing seas. J. Fluid Mech., 860, 767—786.

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