Crowds disperse, fumes disperse -- in either case, they just spread out and dissipate until they are unrecognizable. What does it mean for waves to disperse?
Imagine listening to a symphony orchestra or an excellent home sound system. You depend on the sound arriving at your ear in pretty much the same form that it left the sound system. If sound dispersed like fumes, as they spread out around the room, what would be the point of paying a lot of money for tickets, or a sound system with less than 0.02% total harmonic distortion?
For sound waves, things tend to work out pretty well as long as waves of different frequency all travel at exactly the same speed. But imagine waves for which high frequencies travelled faster than low ones. If the orchestra all hit the first note together, you'd hear the piccolo start before the base viol did. But it's worse than that: you know from Fourier analysis that arbitrary sounds can be decomposed into constituent waves, with a spectrum of frequencies. That would mean that a sound leaving your home sound system speakers would decompose as it propagated towards you -- the consituent frequencies would be teased apart, in the sound, with the high ones arriving first, and low ones coming along later. Seats closer to the sound would be much better, because the sounds would virtually fall apart by the time they got to the cheap seats in the back.
Fortunately, this doesn't usually happen with sound. But it can, actually, for sound in a sewer pipe, say, or other sort of waveguide structure -- satisfying boundary conditions in space can change the relationship of the waves to themselves, and make wavefronts of one frequency travel at a different speed than wavefronts of another frequency. In free space, sound waves are nondispersive, but in a waveguide, or for sound waves in materials other than air, for transverse waves or for extreme frequencies, waves can be dispersive.
In a waveguide there are different ways (modes) in which waves can propagate. Each mode of propagation can have a different relationship between frequency and wavefront speed. The same is true for optical waves. In telecommunications in fiber-optic waveguides (optical fibers), light waves can be dispersive, both because of the glass they travel in (material dispersion) and also because of the waveguide effect (modal dispersion). It may even be possible for the two kinds of dispersion to behave oppositely, and partially cancel out.
This is of tremendous practical importance. Imagine sending digital information in which the bits were ultrafast optical pulses. What's the information capacity of such a system? Well, if the pulses are dispersive, and each optical pulse spreads out, with high frequency components within each pulse running ahead of the low-frequency components, eventually the adjacent bits will spread out until they overlap each other. If this means that you can't tell the bits apart, this will imply a serious limit for how much information can be passed per second. Or, equally, how far a fiber optic link can reach before the bits must be detected and generated again afresh.
So these experiments will introduce you to dispersion, beginning with an acoustic waveguide in which you can identify modes, measure modal dispersion relations, and discover the effects of dispersion for wavepackets of sound.
The fiber-optic experiments that follow, in this lab, will help you study different modes in multimode optical fibers, and learn how to couple light into fibers and out again.
Finally, the ultrafast Er fiber laser will let you measure material dispersion and modal dispersion in optical fibers, as well as a number of linear and nonlinear optical effects that make such a laser possible.
A really large acoustic waveguideYou may find it useful to get an illustration of what this is all about, before starting. Here's a video and sound clip of acoustic wave propagation inside a long cylindrical acoustic waveguide. A very sharp sound can have a very brief duration -- this rapidity requires quite high frequency components in its Fouier spectrum. The wide range of frequencies involved show up the effects of dispersion more readily. Thus a sharp clap of the hands comes back sounding more like a ricocheting bullet.
Our experiment here uses a much shorter tube, so the effect isn't so pronounced, but the computer equipment provided will help you to record, measure, and analyze the same effects.
EM waves in a magnetized plasmaDuring the first World War, short-wave radio operators noticed curious noises in addition to the talk they were listening for. These were descending tones, like a slide whistle, in the audio signal. Consequently, they were termed 'whistlers'.
This was eventually discovered to be the result of lightning strokes from storms in the southern hemisphere, making a burst of radio noise. These powerful EM pulses would radiate to the ionosphere, where they would propagate as special electromagnetic waves in the ionized plasma, following the earth's magnetic field to the opposite hemisphere, where they were detected. The energy associated with higher-frequency oscillations travels faster than that associated with low-frequency oscillations, and so the high tones arrive first, and lower tones with some delay. For certain frequency ranges, dispersion is strong and these pulses are teased apart conspicuously; this is dispersion. The shape of these dispersed pulses, in frequency vs. time, details the physics of the process of dispersion, for these waves.
More information, and sound clips, are available on websites dedicated to astrophysics and astronomy, planetary physics, and ionospheric plasma physics. Among these are:
- The INSPIRE Project -- INSPIRE is a non-profit scientific, educational corporation whose objective is to bring the excitement of observing natural and manmade radio waves in the audio region to high school students. The whistler wave data above is from there, and a variety of different sorts of VLF radio noise can be heard on that website.
- The Lion Roars provides similar listings
- Both have been provided through the Marshall Space Flight Center's Space Plasma Physics Branch.
The Experiment
Primer on dispersion in waveguides
Some referencespaper on dispersion in acoustic waveguides (Meykens et al., Am. J. Phys. 67, 5, (1999)) 373 kB
Instructor's
links
Last revised: 12 March 2003 -- rsm
