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February 14th, 2013

Solitons, pentagons and other star-shaped waves

Published on February 14th, 2013 | By: Eileen De Guire

Examples of star-shaped waves resulting from vibrations in a tank containing liquid oil. Credit Rajchenbach et al., APS.

Today I am linking to videos mainly because they are cool and not because they have a strong tie to materials science. These short looping videos, which are oddly hypnotic (for me, anyway), are the work of Jean Rajchenbach, Didier Clamond, and Alphonse Leroux, whose paper on “Observation of star-shaped surface gravity waves,” is slated for a future issue of Physical Review Letters (preprint pdf).

The authors work at CNRS, in France, and their paper reports on their observation of new types of standing waves. Nonlinear effects in certain fluids, according to an email from the American Physical Society, “can give rise to remarkable phenomena such as ‘freak,’ ‘horse-shoe,’ and ‘solitary’ waves (solitons), which sometimes are found to have important counterparts in other domains of physics, from optics to cold-atom physics. … The shapes of the waves are entirely determined by gravity, rather than surface tension, and can display a star-like or a polygonal shape, depending on the amplitude and frequency of the driving vibration. The resulting wave patterns, according to the authors’ model, stem from the resonant and nonlinear interactions between multiple waves traversing the liquid.”

This wave near a Maui beach is a soliton—a single peak with no leading or trailing waves—which can appear when conditions are right. Solitons act in some ways like single particles and have been observed in fluids, optics, and Bose-Einstein condensates. Credit: R. Odom/Univ. of Washington.

As it it turns out, the emergence of the idea that solitons actually exist is a great example of early computer modeling and simulation. According to a separate APS story, the 1965 discovery and explanation for solitons was a direct result of the nascent computer technology available in federal lab. The story goes:

In 1955 Enrico Fermi, John Pasta, and Stanislaw Ulam (FPU) came across a puzzling result when using the MANIAC I computer at what was then called the Los Alamos Scientific Laboratory in New Mexico. They wrote a program to follow the motion of up to 64 masses connected by springs in a horizontal line. Each mass could move only in the direction of the line, stretching or compressing the two springs connected to it.

The team started the simulation by displacing each mass from its initial position in a pattern that formed one half of a sine wave, with the end masses having zero displacement and the middle masses having the greatest displacement. The masses would then oscillate, and if the springs were strictly linear—that is, if their force were proportional to the amount of stretch or compression—then a snapshot of the motion at any time in the future would show the masses still in a sine wave pattern. But Fermi and his colleagues added a small degree of nonlinearity to the springs’ force, expecting it to break up the sine wave and cause the oscillation energy to become, in time, equally distributed among all the masses.

That’s not what happened. Although the sine wave indeed evolved into a more complex oscillation, the motion of the masses never became completely disorderly, and in fact it periodically returned to the initial state.

A decade later, Norman Zabusky, then at Bell Labs in Whippany, N.J., collaborated with Martin Kruskal of Princeton University to re-examine the FPU work. They transformed the discrete masses-and-springs equation into one for a continuous system similar to water waves. The team then programmed a computer to calculate the wave motion over a fixed horizontal distance but in such a way that a disturbance passing out of one side of the range reappeared at the other side. Like FPU, Zabusky and Kruskal set the system in motion with an initial sine wave pattern. As the wave rolled along, its leading edge became steadily steeper and then developed smaller-wavelength ripples. These ripples eventually grew into individual waves that moved independently, with a velocity that depended on their height. Remarkably, when these new waves occasionally collided, they passed through each other, emerging almost wholly unscathed from their encounters. In addition, the waves would regularly align to reproduce the initial sine wave, momentarily, before separating again and repeating—not quite perfectly—the same cycle. This phenomenon was similar to the periodic return to the initial state that FPU had observed.

The discovery may have brought a sense of relief to those who had reported (documented sightings go back to the 1830s) seeing single waves in oceans and canals.

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