# Solitary waves

Stories from Physics for 11-14 14-16

In 1834, John Scott Russell, a Scottish engineer and scientist, noticed an unusual type of wave whilst conducting experiments to increase the efficiency of steamboats:

“I was observing the motion of a boat which was rapidly drawn along a narrow channel by a pair of horses, when the boat suddenly stopped — not so the mass of water in the channel which it had put in motion; it accumulated round the prow of the vessel in a state of violent agitation, then suddenly leaving it behind, rolled forward with great velocity, assuming the form of a large solitary elevation, a rounded, smooth and well-defined heap of water, which continued its course along the channel apparently without change of form or diminution of speed. I followed it on horseback, and overtook it still rolling on at a rate of some eight or nine miles an hour, preserving its original figure some thirty feet long and a foot to a foot and a half in height. Its height gradually diminished, and after a chase of one or two miles. I lost it in the windings of the channel. Such, in the month of August 1834, was my first chance interview with that singular and beautiful phenomenon which I have called the Wave of Translation.”

Other researchers investigated Scott Russell’s waves of translation in tanks and found that the waves had a curious property — unlike other waves, a wave of translation retained its bell-shape and velocity as it travelled and the waves would not merge if a small wave was overtaken by a larger wave.

Waves of translation are now called *solitary* waves. They were not well understood in the 19th century, but turned out to have interesting applications in other fields. Sixty years later, Enrico Fermi, John Pasta and Stanislaw Ulam, working at Los Alamos, were trying to understand the surprising result that the predicted thermal conductivity of atomic lattices was infinite when the lattices were modelled as a series of onedimensional springs.

The apparent paradox was solved by Norman Zabusky and Joseph Kruskal in a paper published after Fermi’s death. They proposed localised excitations called solitons, which turned out to be related to the curious wave Scott Russell had observed.

The calculations for Fermi, Pasta and Ulan’s work were carried out on MANIAC, one of the earliest computers built in 1952 for the Manhattan Project’s work on the hydrogen bomb. The programming for the paper was done by the mathematician Mary Tsingou who is credited, along with Pasta, with creating one of the first computer graphics displays — a visualisation of an explosion problem on an oscilloscope.

### References

A. Warn-Varnas, J. Hawkins, K. G. Lamb, S. Piacsek, S. Chin-Bing, D. King, G. & Burgos, Solitary wave generation dynamics at Luzon Strait. *Ocean Modelling*, vol. 31, no. 1, 2010, pp. 9-27.

S. N. Ward, Tsunami in In H. Gupta (Ed.) *Encyclopedia of Solid Earth Geophysics*, Dordrecht, Springer, 2011, 1473-1493, p. 1484.

J. S. Russell, *Report on Waves: Made to the Meetings of the British Association in 1842-4*3, London, Richard and John, E. Taylor, 1845, p.13.

M. Remoissenet, *Waves Called Solitons: Concepts and Experiments*, Berlin, Springer, 1996, p.4.

A. C. Newell, *Solitons in Mathematics and Physic*s, Philadelphia, PA, Society for Industrial and Applied Mathematics, 1985, p.3.

T. Dauxois, M. Peyrard, & S. Ruffo, The Fermi–Pasta–Ulam ‘numerical experiment’: history and pedagogical perspectives.* European Journal of Physics*, vol. 26, no. 5, 2005, pp. 3-11, p.5.

T. Dauxois, Fermi, Pasta, Ulam, and a mysterious lady. *Physics Today*, vol. 61, no. 1, 2008, pp. 55-57.