Surfaces and Essences

by Douglas Hofstadter

  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 i
n 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.

  In the early twentieth century, radio waves (really just long-wavelength light waves) were used as a host medium for carrying sound waves. In other words, sound waves hitch a ride on the much faster medium of electromagnetic waves. Amplitude modulation (AM) is a kind of transverse way of letting the hitchhiking sound waves locally distort the medium through which they are traveling, whereas frequency modulation (FM) is essentially a longitudinal way of letting the hitchhiking sound waves hop aboard the very fast host waves. FM, in short, is a very abstract sort of compression wave. We can’t enter into the details here, but the brilliant though tricky idea of waves riding on waves gradually grew into an ever more common leitmotiv in physics.

  Later during the twentieth century, temperature waves were discovered, in which the value of the temperature of a substance is what oscillates “up and down”. In other words, what is moving “up and down” is the number of degrees (Fahrenheit or Celsius) at each point in the substance, like millions of imaginary columns of mercury moving physically up and down in so many imaginary thermometers all through space (and of course the thermometers need not be in phase with each other).

  Also discovered were spin waves, where the direction of spin of electrons (think of a room filled with millions of tiny spinning tops, some pointing up and some pointing down) can “ripple” across the medium, with spins periodically flipping from up to down and then back to up. And then there are gravitational waves, where the amount of gravitational pull at some point in space oscillates periodically in time, as if an invisible ripple were silently shimmering through space and, by its local size, telling thousands of little pebbles floating in space how strongly, and in what direction, they are being pulled by a rapidly shifting, totally invisible force.

  Last but not least, among the very most important kinds of waves in all of physics are quantum-mechanical waves, sometimes called matter waves or probability waves. Roughly speaking, at every point in space such a wave has a value that changes over time, and when that value is squared, it tells how likely one is to find a particle in the given spot at the given instant.

  We could list dozens of other types of abstract waves, but this modest sampler will suffice. As we have seen, the notion of wave in physics has reached an enormous degree of abstraction and sophistication today, and yet all of the latest and most abstract forms of waves are tied by analogy and by heritage to the earliest kinds of extremely concrete, tangible, palpable waves in bodies of water and in fields of amber grain — waves that we can see with our eyes and feel with our bodies.

  Of Sandwiches

  Oh, the poor fourth Earl of Sandwich! How often is his name abused these days! As is well known (or at least rumored), it was the august Earl who concocted the clever idea, around 1750, of putting some meat (or perhaps some cheese) between two slices of bread. Sandwich’s original “bread–meat–bread” pattern spurred an enormous number of copycat variations in the gastronomic world, which we don’t need to spell out here.

  It might be amusing, however, to mention that one of the authors of this book has been known, to his great gustatory delight, to savor the act of devouring a handful of macadamia nuts placed on a square of American cheese that is then folded back on itself, thus becoming both an upper and a lower layer at once. The analogy is tremendously obvious, and yet there is pleasure in spelling it out: the two layers of surrounding cheese “are” the two layers of bread, and the macadamia nuts “are�
� the meat. The only question left, then, is whether we also need quotation marks around the word “is” if we assert that this is a sandwich. Of course it is like a sandwich, but what would the Earl himself say? Or is there any reason to presume that the fourth Earl of Sandwich would be the ultimate authority concerning membership in the category bearing his title’s name? Could there in fact be any ultimate authority on the topic? Would it not be a fine thing to create an elite Sandwich Memorial Board that would officially make all such rulings?

  Let’s move on to sandwiches lying beyond the realm of the edible. Our first variation on the theme — our first inedible extension of this category — is not too far removed from that realm, however. For at least a couple of hundred years, restaurants in large cities have had the practice of hiring hungry souls to serve as walking advertisements, having them stroll the sidewalks wearing wooden or cardboard posters on front and back. Such people are usually called “sandwich men” (although of course a sandwich man need not be a man1). Needless to say, there is many a discrepancy between a slice of bread and a piece of wood with slogans painted on it, and likewise there is many a discrepancy between a slab of meat and a living human being; these two facts already cast some doubt on the sandwichhood of the described item. But in addition, this kind of “sandwich” is not meant to be consumed, except in a very abstract sense — namely, by a visual system. And perhaps more subtly, a human sandwich of this sort challenges the great U.S.A. (that is, the Unspoken Sandwich Axiom), which is the tacit idea that a sandwich must always be horizontal. In sum, there are numerous reasons to wonder about the membership of this kind of entity in the category sandwich. The question arises as to the limits of the concept, and whether there are any limits at all to it. How far out does sandwichhood stretch, and in what directions? This is a most provocative question.