Branched Flow of Waves and Rays

A rogue wave is a terrifying thought for a sailor, a wave 50 times as tall as normal waves. The branched flow of waves and rays may seem like an esoteric topic in theoretical physics. However, this discovery has applications in many diverse fields ranging from oceanography to acoustics. On September 27, Professor Eric Heller of Harvard University presented a talk at the Physics Colloquium on the capacity of waves to describe the motion of practically anything, spanning from sound to all the way to water.

This phenomenon was first described by Eric Heller et al. in an experiment on the electron flow across a quantum point contact, or a constriction between two electrically charged regions, in which the width is comparable to the de Broglie electronic wavelength. The picture of electronic flux, given the standard picture of waves emanating from a small slit, was extremely anomalous. However, after constructing a classical trajectory of paths through a bumpy potential function, the picture of branching webs arises from the initial conditions, creating a complex web of different paths.

However, this is not restricted to a two-dimensional electron gas; this is really a product of any set of waves traveling over a non-uniform potential. That’s quite abstract, so rather than thinking of a potential field, consider a lens. Normally, if we have a perfect lens, we would expect the light to converge onto a single point. However, if we instead consider an imperfect lens, the light converges into a cone-like shape instead. If we then consider the collection of velocities in the y-direction at some slice of the light rays, we notice that before the lens, all y-velocities are zero, whereas after the lens, the y-velocities are shifted. The light rays within the collection are then off-phase from one another, creating interference. What this means is that most lens will group the lights into collections of light in the same phase, creating branching patterns.

A physical example of this branching is seen in the SOFAR channel, used by both the USSR and USA in the Cold War. Essentially, submarines realized that if they dive to a depth of one kilometer, sound waves can travel across the ocean while maintaining coherency. In that way, submarines may communicate thousands of miles apart with simply a speaker and receiver. Because of how sound waves travels though pressure and temperature gradients, the sound waves diffract and propagate recursively until the waves gather into large branches that tend to stay together within this SOFAR channel. Other familiar phenomena include the intricate ribbon-like patterns of light on a pool bottom and the twinkling of starlight due to interstellar clouds. Each of these involves a set of diffractions and propagations organized in such a way that the amount of light observed depends heavily on how the light branches out after traveling through the medium. A final example is rogue waves, which happen far more often than simple probability suggests. When we consider the nature of the ocean though, with its eddies and currents, we recall that understanding the path of a wave in the ocean is far more complex than just considering the ocean as a set of waves coming from all directions. It turns out that if we do describe waves as such and then give them a spread of initial angles, we would see what physicists and statisticians call 5-σ events – events that have a one in two million probability, when, in reality, such events happen with much greater frequency.

So then, where does that leave us? We have a working theory of how oceanic waves, sound waves, and light waves, but where else can we apply this phenomenon? Professor Kristina Lynch pondered whether this could theoretically be used to describe propagation in politics or economics, areas in which we can describe certain factors in the market or model information as some sort of wave traveling through people. This could theoretically be used in optics and quantum computing to manipulate and possibly transmit quantum bits through circuitries. The possibilities are limited only to our imaginations, all because someone decided to construct a set of trajectories for electrons.

A copy of Eric Heller’s book, Why You Hear What You Hear is available online at www.whyyouhearwhatyouhear.com. This site contains a bounty of information on sound and sound waves, as well as a pdf copy of the book itself.