Focus: One-Way Waves

Phys. Rev. Focus 22, 21
With an array of water-activated levers, researchers demonstrate the first mechanical medium where waves travel in only one direction.
J. Lindner/College of Wooster
Tipsy-turvy. Using water hoses, an array of levers propagates waves in only one direction, leading to unusual wave behaviors. (See video below.)

In most cases, waves travel backward just as easily as they do forward. But a team reporting in the December Physical Review E has built the first mechanical medium where waves go only one-way. Their Rube-Goldberg-like contraption contains an array of levers coupled by jets of water. It demonstrates that with a one-way medium, waves can propagate continuously without an external driving force, as others have suggested based on electronic circuits and computer simulations. The set-up also shows other unusual wave behaviors not predicted in simulations and may serve as a laboratory to study waves in so-called anisotropic media–which are important in geology, optics, and materials science.

Imagine a string of pearls. You can start a wave by wiggling the first pearl or the last; the waves can travel either way because each pearl is coupled equally to both neighbors. But researchers have lately become interested in “unidirectional” coupling, in which the force between neighbors only allows waves to move in one direction. This can be seen as an extreme example of anisotropic media, in which the wave speed depends on the direction. Computer simulations have shown how waves will propagate through unidirectional arrays, and researchers have built electronic circuits that exhibit unidirectional coupling [1]. But these circuits had only three “pearls” in the array–too small to see all of the wave propagation effects predicted in the simulations.

Video courtesy of John Lindner, College of Wooster, Ohio.
Waves travel continuously down the array of coupled levers without any conventional driving force. This is a phenomenon unique to a “frustrated,” one-way wave medium.

So John Lindner of the College of Wooster in Ohio, Barbara Breen of the University of Portland in Oregon, and their colleagues, built a mechanical device that could plainly demonstrate one-way waves. Drawing from several prototypes assembled with construction toys, the current machine–costing only a few hundred dollars–is a linear array of levers that tip to one side or the other inside a trough. Each lever is bolted to a tray that is hit with water streaming down from a vertical hose. The lever-tray assembly tips whichever way the water directs it, and the hose position is controlled by the previous lever in the array. A mechanical connection causes each hose to move in the direction opposite from its controlling lever. If, for example, the first lever is tilted to the right, its hose will be shifted to the left, which will push the second lever to the left (if it wasn’t already there). The third lever will then be pushed to the right, and so on. To make the system cyclic, a set of gears and an axle allows the last lever to control a hose that drops water on the first lever’s tray.

If the levers were two-way coupled (say, with springs attached between them), the system would quickly settle into a static configuration, but with one-way coupling , the levers either never reach equilibrium or do so only after a long time. When the researchers used an odd number of levers (15), it was impossible for the system to stabilize into an alternating left-right equilibrium. Instead, a wave spontaneously traveled through the “frustrated” system. A driving force akin to shaking one end of the string of pearls wasn’t needed to excite this continuous motion.

With an even number of levers placed initially in random positions, two waves formed and chased each other around the array, eventually “annihilating” when one caught up with the other and left the system motionless. Although these wave phenomena were expected, no one had demonstrated them in a mechanical system before, and it allowed the team to verify and improve a number of quantitative predictions from simulations, such as the distribution of annihilation times for wave pairs.

The work is “a classic example of an elegant experiment being put together to demonstrate a new idea and predict behavior that can now be further explored by theory,” says Adi Bulsara of the Space and Naval Warfare Systems Center in San Diego. One-way coupling may help researchers understand wave propagation in anisotropic media such as birefringent crystals, in which the speed of light depends on its polarization, Breen says. The team is now developing a two-dimensional array with unidirectional coupling and looking at the transition from an isotropic to an anisotropic medium.

–Michael Schirber

Michael Schirber is a Corresponding Editor for Physics based in Lyon, France.


  1. V. In et al., “Coupling-Induced Oscillations in Overdamped Bistable Systems,” Phys. Rev. E 68, 045102(R) (2003)

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