Surfaces and Essences: Analogy as the Fuel and Fire of Thinking (52 page)

On a more mundane scale, flags wave in the wind. And we wave to our friends, saying hello or good-bye. That is to say, not unlike flags in the wind, our arms flap back and forth in a roughly periodic fashion for a little while. And if we are teachers, we are inclined at times to succumb to the temptation to give hand-waving explanations of things that are very hard to grasp.

On a far huger scale, America was built by legendary “waves of immigration” washing one after another across her shores. Even if one didn’t grow up near the ocean, one can almost see a long series of these mighty swells crossing the Atlantic or the Pacific Ocean, one by one approaching the shoreline, and finally crashing onto
terra firma
, possibly making a huge roar and breaking into spray.

These are the kinds of phenomena that go into the everyday conception of what waves are, and they are also the kinds of concrete imagery upon which early thinkers built as they started to put together a picture of the undulatory nature of certain fundamental and universal phenomena. However, despite their frequent presence in our lives, water waves were not the best sources of inspiration — not by a long shot, as it turns out. There are too many diverse and complex phenomena involved in water waves; thus, there are surface waves (such as the delicate ripples made by water skates), which have to do with surface tension, and there are also tsunamis, which have nothing to do with surface tension, but which instead involve gravitation pulling the water down (think of water sloshing back and forth in a bathtub; when one end goes up, the other is pulled down, and vice versa). And adding to their complexity, water waves of different wavelengths propagate at different speeds, which (as it turns out) is a profoundly complicating factor when one tries to understand waves through mathematics.

From antiquity, the first physicists were inspired by water waves, and as a result they formulated the most basic ideas about them, such as that of
wavelength
, whose very definition betrays its watery origins: “the distance between successive crests” (or equivalently, between successive troughs). Likewise, the
period
of a wave is the time between arrivals of successive crests (or troughs), and the
frequency
of a wave is the reciprocal of the period. Note that these notions don’t apply to an isolated breaker, but only to a long series of swells coming in toward a beach. For a physicist, a wave is first and foremost seen as a periodically repeating phenomenon (although, in the end, that requirement, too, can go by the boards…).

Starting out with these basic notions, as well as the extra notion of a wave’s
velocity
, which is its wavelength divided by its period, physicists were able, already several centuries ago, to analyze many phenomena involving water waves, such as
reflection
(everyone has seen ripples bounce off the edge of a swimming pool),
refraction
(the slight shift in direction that takes place when a wave crosses over from one medium to another — for instance, ripples moving from one liquid to another, or from a shallow basin to a deep basin), and
interference
(what happens when waves from different sources crisscross). The discovery of these water-wave phenomena, and later of the mathematical laws governing them, was in itself a significant accomplishment, but it was merely a forerunner of far greater achievements in understanding other important natural phenomena.

Already in roughly 240 B.C., the Greek philosopher Chrysippus had speculated that sound was a kind of wave, and some 200 years later, his ideas were developed more fully by the Roman architect Vitruvius, who explicitly likened the spreading of sound waves from a source to the circular spreading of ripples on water. What Vitruvius did is in fact extremely typical of the thinking style of all physicists: taking a familiar, visible, everyday phenomenon and seeing it, in one’s mind’s eye, as taking place in another medium, sometimes at a vastly different spatial or temporal scale, so that it is inaccessible to one’s senses (recall the invisible “shadows” in the cathode-ray tubes). In this case, the familiar phenomenon is ripples, whose wavelength and frequency are extremely apparent, and the new medium is of course air. The wavelengths and frequencies of sound waves are not perceptible and in fact their frequencies are very different from those of ripples or ocean waves. Because of these major differences, it was a very bold act of Vitruvius to apply the same word to two phenomena of which one was very well known and the other was hardly known at all (much like Galileo’s daring extension of the word “Moon” to infinitesimal dots that moved, when he observed them through his telescope). It took many centuries more, however, before the theory of sound waves was further advanced, thanks to the work of such insightful scientists as Galileo, Marin Mersenne, Robert Boyle, Isaac Newton, and Leonhard Euler.

Not surprisingly, there were some significant discrepancies between sound waves in air and waves on water. Among the most important is the fact that, unlike water waves, sound waves are
longitudinal
, meaning that they involve motions of air molecules along the direction of propagation of the noise. It’s perhaps easiest to explain this by another
analogy. When a line of cars moves down a road that has a series of traffic lights, the distance between neighboring cars diminishes each time they must come to a stop, and it increases when they start up again. This is sometimes called a
compression wave
, since the traffic is getting more and then less compressed. Compression waves always are longitudinal, in that the distances grow and shrink along the direction that the wave itself is traveling in. Similarly, the density of air molecules oscillates rapidly as a sound wave passes down a corridor, and the molecules, like the cars on a crowded road, get alternately closer and farther from each other, their relative motion being along the same direction as the sound itself is traveling (once again, that’s the meaning behind the term “longitudinal”).

Comparing sound waves with water waves seems easy, but there are hidden subtleties. It’s obvious to a casual observer that as a ripple passes by, the water at and near the surface moves up and down, and this up-and-down motion is perpendicular to the ripple’s direction of travel (which is horizontal). This is called a
transverse
wave. However, it happens that ripples are not that simple. The water actually does another dance at the same time: it also oscillates forwards and backwards, where “forwards” means “in the direction the ripple is going”. That constitutes a
longitudinal
oscillation (much as in the case of sound, and also like the cars on the highway). These two motions, transverse and longitudinal, take place simultaneously, at and below the water’s surface, and to confuse matters more, the up-and-down motion is not in phase with the back-and-forth motion, but they are, as physicists would say, “90 degrees out of phase”. As a result, as a ripple passes by, the dancing water molecules at and below the surface move in perfect
vertical circles
, aligned with the ripples’ motion (in the same way as a bicycle wheel is a vertical circle aligned with the bicycle’s motion). One last elegant feature of these surface-tension waves is that the circles’ radii grow smaller and smaller the further below the surface one descends. As this shows, water waves certainly are not the simplest waves of all.

Yet another major discrepancy between sound waves in air and ripples on water is that whereas water waves travel at different velocities depending on their wavelength (for which they are said to be “dispersive”), sound waves are much simpler: no matter what their wavelength is, all sound waves propagate at the same velocity in a given medium (which entitles them to the label “nondispersive”). This is lucky for us speaking creatures, since otherwise the different waves constituting our voices would all disperse and we couldn’t understand a thing anyone said (unless we possessed a far more sophisticated auditory system than the one we have, which works only for nondispersive waves).

In a sense, the leap from visibly undulating water to invisibly undulating air, though humble in a way, was also the greatest leap in the story of the development of the
wave
concept, because it opened up people’s minds to the idea of making other daring leaps along similar lines. One success led to another, each new analogical extension making it easier to make the next one. The next big leap — from sound to light — was of course a bold step, but the way had already been paved by the leap from water to sound. To put it differently, the sound-to-light leap was facilitated by a
meta-analogy
,
even if it wasn’t spelled out explicitly — namely, the idea that one analogical leap (from water to sound) had already worked, and so why shouldn’t the
analogous
analogical leap (from sound to light) also work?

Such meta-analogies have permeated the thinking of physicists in the last few centuries: an idea understood well in one domain is tentatively tried out in some new domain, and if it is found to work there, physicists hasten to try to export the old idea once again to even more exotic domains, with each daring new attempt at exportation being analogous to previous exportations. Over the past hundred years or so, making bold analogical extensions in physics has become so standard, so par for the course, that today, the game of doing theoretical physics is largely one of knowing when to jump on the analogy bandwagon, and especially of being able to guess which of many competing analogy bandwagons is the most promising (and this subtle selection is made by making analogies to previous bandwagons, of course!). This highly cerebral game might be called “playing analogy leapfrog”.
Chapter 8
will consider these ideas in greater detail.

But back to waves. One major difference between trying to prove the existence of sound waves and trying to do so for light waves was that whereas in the seventeenth century, it was relatively simple to devise experiments to determine the wavelengths and frequencies of typical sound waves, no such measurements were feasible for light at that time (the wavelength of visible light is microscopic, and its frequency is enormous — hundreds of trillions of “crests” and “troughs” pass by each second). On the other hand, the
sound ⇒ light
leap possessed the happy precedent of the prior
water ⇒ sound
leap, which, as we have said, inspired confidence, by analogy.

The first guesses about light as a wave phenomenon were somewhat wrong, as they were based on an overly simplistic analogy with sound; it was assumed that light, just like sound, was a compression wave that propagated in an elastic medium, such as air. Thus light waves were originally conceived of as
longitudinal
, just like sound waves. (In
Chapter 7
, this kind of assumption will be dubbed a “naïve analogy”, and the nature of such assumptions will be scrutinized in detail.) It took careful experiments in the early 1800s by Thomas Young and Augustin-Jean Fresnel to get beyond this naïveté and to reveal that light was not longitudinal but
transverse
, meaning that whatever was oscillating was doing so
perpendicularly
to the direction of motion of the wave, a finding that was very disconcerting, because no one could give a physical explanation for why such a wave would exist. (In a sense, water waves provided a precedent for this finding, since they had an obvious transverse quality, but it was clear that this quality was due to the existence of a special spatial direction defined by gravity, and light moved through space where there was no gravity and hence no special direction, so this removed any promise that the analogy might have seemed to hold out.)

It was only in about 1860 that James Clerk Maxwell came to the astonishing revelation that light waves did not involve the motion of any material substrate at all, but instead were periodic fluctuations, at each point of the three-dimensional space in which we live, of the magnitudes and directions of certain abstract entities called
electric and magnetic fields.
It was as if the medium that conducts light waves consisted of a gigantic collection of immaterial arrows, one located at every point of empty space
(actually, two — one magnetic and the other electric), and whose numerical values simply grew and shrank, grew and shrank, periodically oscillating. This was certainly extremely different from the visible, tangible motion of water on a lake’s surface or of waving wheat in a field, and some physicists couldn’t relate at all to this kind of highly abstract intangibility, but it was too late to go back and undo it. The concept of
wave
was inexorably growing more and more abstract, spreading relentlessly outwards from its original “city center”, as is the wont of concepts, always moving out to the suburbs.

It didn’t take physicists too long before they started realizing how immensely fertile this concept of
wave
truly was, in the explanation of natural phenomena, ranging from the most ubiquitous, such as sound and light, to all sorts of exotic cases. Any time space was filled with any kind of substance (or with an abstraction that could be likened to a substance), it seemed that local disturbances in that “substance” would naturally propagate to neighboring spots, and so forth, and thus waves would radiate outwards from a source. The disturbance, however, could be very different from ordinary vibration — it could be highly abstract, like the shrinking and growing of invisible abstract arrows. Nonetheless, all the standard old concepts associated with earlier waves could be investigated —
wavelength, period, speed, transverse
or
longitudinal, interference, reflection, refraction, diffraction
, and so on, and many of the same equations carried over beautifully from one medium to another.

For instance,
moonlet waves.
That’s not the standard term for them, but curiously enough, James Clerk Maxwell’s first discovery in physics was the fact that the rings of Saturn are made of billions of tiny “moons”, and he proved his theory by showing that if there were compression waves sloshing back and forth inside the rings around the planet, their calculated behavior would perfectly match the data observed by astronomers.