String Theory

By Christopher Dow • Photography by Tommy Lavergne

Physics today is buzzing with talk of cosmological strings—elongated regions of densely vibrating energy that might be what create the underlying structure of reality. There is a whole class of strings, however, whose behavior affects millions of people daily but remains largely unexplored by science: the strands that are the heart of a violin, harp, or guitar.

Most people probably think the behavior of musical instrument strings has long been understood, but that’s not the case. Even though the mathematical equations that describe the behavior of musical strings have been around since the 1750s, and it’s one of the examples that most engineering and science students see when they’re introduced to theories of wave propagation, an in-depth look at the subject seems to have fallen through the cracks. “The traditional mathematical model of string behavior is so clean and simple that almost anyone with a year of calculus can follow the derivation,” says Steven Cox, professor of computational and applied mathematics. “The trouble is, real strings don’t behave the way these idealized equations present them.”

Cox and his colleague, Mark Embree, assistant professor of computational and applied mathematics, are interested in the behavior of strings and the oddities and paradoxes that arise from that behavior. “There are fascinating mathematical questions and fascinating physical questions,” Cox says, “and those lead to a need to actually test hypotheses on a bench by building real strings and by modeling imaginary strings and watching how those evolve.”

Both researchers have long-standing interests in the physics of strings, but the current project was sparked by Sean Hardesty, a graduate student who did his undergraduate work at the California Institute of Technology and worked in the musical instrument lab there. The summer before he came to Rice, Hardesty interned with D’Addario & Company, a string manufacturer located on Long Island, New York. There, he worked with another CalTech graduate who was leading the way toward rational design of musical instrument strings—the design of strings based on physical principles and measurements instead of just what sounds good to the ear. When Hardesty came to Rice, he brought an enthusiasm for the subject with him, and it resonated immediately with Cox and Embree.

Although stringed instruments have been around for thousands of years and some modern string manufacturers have hundreds of years of experience in creating their product, the process of string development continues to be through trial and error. “String manufacturers clearly know where to begin looking to achieve something a musician has asked them to reproduce,” Cox says. “And they do fairly sophisticated signal processing after the fact. But they’re finding that the old trial and error process is somewhat limited. We’d like to build a predictive model to help them eliminate the guesswork.”

Cox and Embree are looking into a subset within mathematics called functional analysis. “Functional analysis largely got its birth from a mathematical understanding of vibrations,” Cox explains. “The theory of vibration led to a classical model that predicts the frequencies at which strings should vibrate, and the fit from the theory to the experiment is just gorgeous. But what’s missing is a theory that predicts how long those tones stick around. That’s a lot harder to come up with.” The rate at which tones vanish is called decay. The problem with the classical string model is that it describes what are termed “conservative systems,” which are systems without decay or dissipation—as if, once the string is plucked, the tone is everlasting. Such systems have received fairly complete mathematical treatment, but real-world situations do contain decay or dissipation, and the mathematics for that is considerably more difficult.

“General theory no longer holds for nonconservative systems,” Embree says. “The question is, how much insight can we take from the classical theory as we look at more specific cases? Intellectually, that’s the great puzzle—to see how robust the old theory was and whether we can generate new tools from it that say something about energy decay in strings and, perhaps, other systems.”

When a string vibrates, its movement isn’t just up and down or back and forth but is a three-dimensional whirling. People listening to a string vibrating on an instrument hear a single note or tone, but in fact, a string produces a wide range of sounds. What lends it the semblance of a single tone is that the size of the instrument’s resonating chamber permits only a portion of the spectrum of sounds produced by the string to escape while dampening unwanted tones. In essence, the chamber acts as a selective amplifier. This is true of all instruments with resonating chambers, including drums and wind instruments.

One interesting aspect the researchers have noticed is that a tone’s decay is not necessarily uniform but can experience rapid drops or short periods of sustain. Factors that can alter both the pitch and decay are the way the string is held down at the two ends—the bridge and the nut—and the interplay between the player’s fingers and the bow or plectrum in relation to the bridge and the nut. Even the exact qualities and pressure of the musician’s finger pads have an effect. These are the kinds of real-world contingencies that aren’t well accounted for in the classical model. The dampening effect on the string caused by a player’s finger pad is quite complex mathematically, and Cox admits he and Embree still don’t understand all the physics. To help them model the impact of a finger pad on a string, they’ve approached mechanical engineers researching haptics, which is the study of touch, who are teaching robotic hands to manipulate objects without crushing or dropping them. They also have experimented with noncontact ways of dampening. One method they’ve used with metal strings is magnetic braking. “You can brake the string by passing it through a magnetic field,” Embree explains. “Using magnets, we can knock out one mode and make it decay very rapidly.”

Clockwise from top left: Steven Cox, Mark Embree, Jeffrey Hokanson, and Sean Hardesty

Another important factor relating to pitch and decay is the exact composition of the string and the materials used to construct it. Most instrument strings have three layers. The first is an elastic wire core. Around that is some type of insulating material, and the insulating layer is then wrapped with an outer wire winding. It is this multilayer composition that creates mathematical complexity with regard to tone. “If you know the number of windings per centimeter, and you know the elastic properties of the insulation between the core and the windings,” Cox says, “you probably can begin to rough out a crude model. But the process for doing that is not well understood.”

Making the process of understanding even more difficult is that some modern strings are not uniformly constructed along their entire length. In other words, a string might not be one kind of wire for the core, one kind of insulation, and one kind of wire for the winding. Instead, any of the three might be composites made of different alloys or materials in different parts of the string. This can give musicians a much wider palette of sound than is possible from a monolithically constructed string, but it plays havoc with the math.

Questions about the physics of a musician’s finger pads on a string also relates to the researchers’ interest in harmonics. A harmonic is a flute-like or bell-like tone produced on a stringed instrument by lightly touching a vibrating string at a nodal point, a point on a vibrating string that is relatively free of vibration. Such points occur at each end of a string, where it is held tightly in place, and at all points in between where the sound wave is midway between its peak and its trough. The base wave is called the fundamental, and each successive harmonic wave is half the length of the preceding harmonic. To get an eighth harmonic on an instrument string, for instance, a musician has to press at an eighth of the length of the string.

Musicians, through training, know by ear how hard to press and when to let go to get the bell-like tone instead of a screech or a thud. What they’re actually doing in eliciting that pure tone is using precise finger pressure to eliminate nonharmonic tones by making them decay rapidly, leaving only the desired tone. “To get that high tone on the fingerboard, a bass player has to press farther down than the fingerboard actually extends,” Cox says. “So to elicit a tone three or four octaves above where the player is fingering the freeboard is quite difficult. I think there’s even some kind of informal competition among bass players to achieve the highest harmonic possible. There are people out there who actually can accurately press at an eighteenth the length of a string.” A mathematical model should be able to pinpoint the correct touch at a given node to cause unwanted tones to decay as quickly as possible.

Cox and Embree’s course, the Physics of Strings, is funded by Rice’s Vertical Integration of Research and Education (VIGRE) program. In its third year, the program involves faculty, postdoctoral instructors, graduate students, and undergraduates from the departments of mathematics, statistics, and computational and applied mathematics. Funded with a five-year, $2.3 million grant from the National Science Foundation (NSF), VIGRE is intended to open up new dimensions of the mathematical sciences to Rice students.

“We’re really pleased with how diverse the group of students is,” Embree says. “We have students from the usual VIGRE departments and from mechanical engineering, physics, and even chemical engineering. The project is a model of interdisciplinary work. And interestingly enough, a majority of the students involved in the project are musicians, too.”

The largest draw so far has been students in engineering and physics. “I guess that’s because, in this early stage, we are building things,” Cox says, “and they can see tangible evidence of their work. But in the process, they’re learning a lot of mathematical models and a lot of computer modeling. The more advanced of them also are learning mathematical analysis.”

To test different models and study the physical properties of real-world strings, the researchers obtained an NSF grant and set up a lab where they can create strings with different parameters and measure their vibrations under a variety of conditions. “We’re really indebted to the mechanical engineering shop masters, especially Joe Gesenhues,” Embree says. “They’ve been of enormous help to us in designing and fabricating equipment, some of which has tolerances of something like 1/5,000th of an inch.”

The mechanism consists of a pair of specialized clamps called collets that hold the string fixed at both ends, a crank to put tension on the string, and a force transducer to accurately read the amount of tension on the string and turn that reading into a voltage that can be recorded by a computer. “It helps students visualize what is happening,” Embree says. “They can put a string on our apparatus and see how it performs. Sometimes real data gives you interesting wrinkles that you didn’t expect.”

But the researchers do not simply stretch various sorts of strings in the device and study the waveforms produced by the vibrations. Instead, they alter the dynamics of strings by placing metal beads at different locations along their length.

“The subject developed historically from a study of beads,” Embree says. “It’s easier to assume you have a string with no mass except at a few points—the beads—and then to take measurements as you get more and more massive points until the string has a continuous mass. The key is that a string with beads affixed to it vibrates in a limited number of ways rather than an infinite number of ways. This makes it simpler to tackle the problem of how small changes in design affect the string’s tone.”

The researchers put the beads on the string and “forget” where they put them, then see if they can pluck the string and determine mathematically where the beads are and how much they weigh. “It’s the beginning of a model of a nonuniform string,” Cox says. “String manufacturers experiment with different materials and different approaches to get the desired sound, and for us, beads are a way to change the mass distribution of the string. We’re not expecting people to play beaded strings, but they’re easy to make and a good way to introduce numerical questions.”

Now that the lab is operational, a lot of the research has become student-driven. “The students are always coming up with questions or paradoxes that need to be explained, leading us down new avenues,” Cox says. “It’s been a great give and take.” Embree is equally enthusiastic. “It’s a fun project,” he says. “As much fun for us as for the students.”

In addition to teaching the academic course, Cox has given seminars and lectures to high school science teachers and in a course for the Glasscock School of Continuing Studies. The continuing studies lecture was accompanied by a musical demonstration of the principles Cox described performed by Shepherd School of Music alum Shawn Conley ’05, a double bassist who played a piece consisting almost entirely of harmonics.

The research Cox and Embree are doing is of great potential benefit for musicians. The researchers foresee developing a computational tool in about five years that will enable musicians to adjust a set of tone controls until they have the sound they want from a string, which the manufacturer will then be able to construct. “We’ve established a lot of the theoretical groundwork,” Embree says. “The work we’re doing at the present is a big step toward a practical technique we could propose to manufacturers.”

But there are a number of other potential nonmusical applications for the work, too. “The title of our NSF grant is ‘Design and Identification of Dissipative Bodies,’” Cox says. “Strings are the easiest example that one can experiment with in a lab, but the work is a prototype for solutions of a broader class of very hard problems that involve optimal dampening treatments to remove undesired vibrations.” Likely applications include noise reduction in cars and airplane cabins and decreasing vibrations in machinery and structures. “There is a host of problems in fields such as biology,” Embree says. “How do spiders detect vibration in the web, for example? Many areas have this kind of dynamic behavior, and maybe we can give a little insight into some of their specific problems.”

For musicians, the research opens new realms of possibility. Normally, a string is tied down at the bridge and at the nut and the musician fingers or bows it, but instrument designers might soon leave the single-stringed instrument behind. In fact, a couple of Canadian mathematicians recently designed a tritar, which has three necks, one fingerboard, and several pickups. Instead of hitting a bridge at the end of the neck, the string forks down two additional necks.

“It’s just a small step from tritars to instruments that are networks of strings,” Embree says. “Some of our students have visions of developing instruments like that.”

The Connexions course, Music, Waves, Physics, by Nelson Lee, at cnx.org/content/col10341/latest/, has a fun interactive wave generator that allows you to adjust the amplitude, frequency, and other wave characteristics to see how waves work. The animation is at cnx.org/content/m13513/latest/.