Waves On Strings

Type of physical science: Classical physics

Field of study: Mechanics

A study of the mechanical motion of pulses and standing waves in stretched strings is essential to understanding the physical behavior of the musical instruments of the string family, each of whose musical sounds derives from the bowing, plucking, or striking of strings under tension.

Overview

Waves and pulses are distinctive in that, once started in elastic substances, they propagate away from their points of origin entirely by themselves. A stone dropped into water, for example, creates a disturbance in which the water at the point of impact piles up slightly, and this local displacement is immediately transmitted to neighboring points on the surface, which in turn influence points still farther out, and so on. As a result, a localized group of waves propagate spontaneously outward. A cork floating on the surface notes the transit of the wave by bobbing up and down in the direction at right angles to that in which the wave is moving. This oscillatory behavior in the medium through which the wave is traveling is a characteristic of all wave motions. Locally, as a wave goes by a given point, the medium simply oscillates back and forth; it does not move bodily in the direction in which the wave disturbance is traveling. When the oscillatory motion is at right angles to the direction of the wave's travel, as in the case of waves on water, the wave is said to be transverse. Waves on strings are transverse. If the oscillatory motion is parallel to the direction of travel, as is the case for the motion of sound waves in air, the wave is said to be longitudinal.

The speed at which any wave moves forward depends on two properties of the medium. If the medium is very dense (a heavy rope, for example), its inertia will be high and it will oscillate relatively slowly, and so a traveling wave will also propagate slowly through it. The second factor has to do with the medium's elasticity. A material such as steel is highly elastic in the sense that very large forces come into play when it is deformed, while wood, on the other hand, is relatively inelastic. Hence, a sound wave traveling in steel will move more rapidly than in wood because the higher elasticity of steel more than outweighs the effect of its greater density. In a perfectly general way, the speed of transmission of any wave or pulse increases according to the ratio of the medium's elasticity to its density.

For waves on a stretched string, the speed of propagation can be shown to be given mathematically by the square root of the ratio of the tension in it (the relevant elastic factor, measured in newtons) to its mass density (the inertia factor in kilograms per meter of length).

Imagine that there is a string under tension, tied at one end to a fixed support on a wall while the other end is held in the hand. One could initiate a pulse in the string by moving the free end rapidly up and down. Because a string can be thought of as a system of connected mass particles, the initial upward motion of the hand starts the particle next to it moving up, too. Since the first is attached to the number-two particle in line, it also experiences an upward pull and begins to move up, though its motion is slightly delayed in time with respect to that of the first. The upward displacement of successive particles in the string is identical, but delayed in time, and so the leading edge of the pulse is created in this way. Having reached its maximum displacement, the hand moves back toward its starting position. Again, this downward motion is transmitted successively to particles farther down the string, with a time delay, so that when the hand again arrives at its original position, a positive pulse will have been created that moves away at a speed determined by the tension and mass per length of the string. An identical, but downward, motion of the hand will then create the second half of what is recognized as a wavy configuration in the string: The whole shape would occupy what is termed one "wavelength" in the string. (If it appears exactly like the mathematical sine function, the wave is a sinusoid.) The maximum displacement of the hand would have fixed the maximum sideways displacement of the string, which is called the wave "amplitude." A wave transports energy, and it is not difficult to show that the energy it carries depends on the square of the amplitude.

Suppose that the hand is moved steadily up and down at a fixed rate--for example, three times each second: This is the source "frequency." If each full wavelength occupies a space of 2 meters, then the wave disturbance will have moved a distance 3 x 2 = 6 meters in that one second. That is, the speed of the wave will be 6 meters per second. In general, the speed at which a wave travels is given by the product of its frequency and wavelength. This speed is often referred to as the "phase velocity." Since the phase velocity is governed by the properties of the medium itself, increasing the frequency of the source (the hand, in the example above) will mean that the wavelength must decrease: It will do so in such a way that the speed, the product of frequency and wavelength, stays exactly constant. This means that all waves in the string, regardless of frequency, will travel at exactly the same speed. Such media are "nondispersive."

There are media, however, in which the velocity of propagation does depend on frequency; in those media, two waves that differ in frequency get farther and farther apart as they move away from their source.

It can be shown mathematically that a well-localized, single pulse can be synthesized in a medium by simply adding together a large group of pure waves (sinusoids), each component of which differs slightly in frequency and wavelength from the next. If the medium is nondispersive, each component wave will travel at the same speed, and so the pulse will maintain its original shape, traveling at the same phase speed as each component. If, on the other hand, the medium were such that the component waves in the pulse had speeds that depended on their frequencies, then the faster components would get ahead of the slower ones, and the pulse shape itself would become more spread out, less well-localized. A medium of that kind is termed "dispersive." The speed at which the center of the pulse travels, which is a measure of energy flow, is not the same as that of any single component wave's phase velocity, or even of the average. It is termed the "group velocity" and is generally smaller than the average phase velocity of the components making up the group.

One might wonder what will happen in a medium if two independent waves arrive at the same place at the same time. Clearly, the medium can do only one thing. A person in a small boat, for example, might be subjected simultaneously to waves from two independent sources. At a given moment, one might have caused the boat to rise by three feet, while the other wave acting alone would have lowered it by an equal distance. It is found that the water simply sums algebraically the two individual displacements: +3 feet + (-3 feet) = 0. That is, at that particular instant, the boat suffers no displacement at all. This concept for the straightforward algebraic addition of individual displacements is called the "superposition principle."

Returning to the earlier example of a single, positive pulse traveling toward the fixed end of a string, one can see what happens when it arrives there. Until that moment, the various segments of string, being free to move, have simply responded to the forces imposed upon them during the passage of the pulse. Arriving at the fixed end, the leading edge attempts to pull up the wall at the point of attachment. While the rope is unable to cause any motion of the wall, its upward pull does generate a reaction force from the wall in the downward sense (Sir Isaac Newton's third law of motion--action/reaction), and so the leading edge of the pulse is pulled down, rather than up. The pulse itself is thus turned down and reflected back toward the hand.

This mechanism explains why it is that, upon reflection from a fixed support, the crests of a continuous train of waves are returned as troughs, and the troughs as crests--or, one could say that a phase change had occurred upon reflection. Moreover, if there is no loss of wave energy, the incident and reflected waves will have identical amplitudes. Hence, when a wave travels down a string and is returned from a fixed end, the reflected wave has the same wavelength, frequency, amplitude, and speed.

The superposition principle allows one to determine the overall situation when two identical but oppositely moving waves exist simultaneously in the string. Exact analysis shows that, provided the tension is just right, the string divides into vibrating segments that are bounded by stationary points called nodal points, or nodes. Between the nodes are regions, antinodes, in which vigorous side-to-side motion is seen. A variety of patterns can be generated by making appropriate adjustments to the tension in the string, thus controlling the velocity--and therefore the wavelength at a given frequency--of the component waves. Since the pattern of nodes and antinodes remains stationary on the string, a wave of this type is called a "standing wave." In a standing wave, the distance between any two adjacent nodes is exactly one-half the wavelength of the component waves. The fixed ends of the string are, of necessity, themselves nodes.

Since the product of wavelength and frequency gives the wave speed, an increase in frequency means a decrease in wavelength, and vice versa. Thus, the standing wave mode of longest wavelength is the one in which only a single antinode lies between the end nodes, and it is generated at the lowest source frequency. It is called the "fundamental." When the string is vibrating in its fundamental mode, one can easily find the wavelength of the component waves by multiplying the string length by two; then the speed of a traveling wave can be determined by finding the product of wavelength and frequency. The next higher mode shows one wavelength standing in the string, the next one up three-halves, then four-halves, and so on. Since for a given medium (here, a string under fixed tension), the wave speed is fixed, and since each higher mode above the fundamental has progressively decreasing wavelength, the frequency of each mode increases correspondingly as one moves up in the series.

For an ideal, perfectly flexible string, the modes form a harmonic series in which the frequency of each member is related as the ratios of the integers 1:2:3: etc. These are called the normal modes of oscillation.

Applications

There is no question that the most familiar practical application of the theory of traveling and standing waves in strings is to the various stringed musical instruments. The instruments of the string family--violin, viola, cello, and double bass--are among the most important members of the modern orchestra; the harp, piano, and guitar also depend on the motion of vibrating strings to create their characteristic sounds. In each of these instruments, a string under tension is either bowed, plucked, or struck. Each technique serves to start waves traveling away from the contact point in the string, and their subsequent reflections from the fixed ends lead to the establishment of complex standing waves. The string vibrations generate sound waves in the air as well as in the body of the instrument itself. The body's interior and outer surfaces serve as resonating members, whose functions are to amplify the sound of the strings and send it out to the listener.

In each of these instruments, the strings are held firmly under a tension that can be adjusted by means of a peg around which the free end of the string is wound, thus setting the frequency and pitch. The overall stress on the body of the instrument can be quite considerable: String tension amounts to some 50 or 60 pounds or so for a violin, while for a modern grand piano, it can reach as high as 20 tons. The strings themselves are usually made of catgut, nylon, or metal. The four basic open-string pitches of a violin (G, D, A, and E of fundamental frequencies 196, 294, 440, and 660 hertz) are clearly insufficient to account for the enormous variety of pitches found within the instrument's range, and the player must therefore make the proper changes in the string lengths, which is done by pressing them down against the fingerboard.

The pitch of a musical sound is directly related to the frequency of the vibrating string.

The string impinges on the surrounding air and sets off a sympathetic vibration at the same frequency. High pitches are most easily generated by light, short strings held under high tension.

Yet, for any given string, there is a whole series of standing waves that it can support. The lowest in frequency is the fundamental. The fundamental pitch is normally the most pronounced and has the greatest amplitude. The other standing wave modes will also be excited to a greater or lesser degree: Surprisingly, an entire group of the string's harmonic series will normally be present simultaneously in the pattern of its motion. The quality, or timbre, of a note depends critically on the relative proportions of the string's harmonics in that overall mixture.

Bowing a string depends upon a phenomenon in which the hairs of the bow first grip the string, thus pushing it to one side, then slip, releasing it, then grip again, and so on. In this way, a steady vibration is set up. The spectrum of harmonics generated, and their relative strengths, depends on the point at which the bow contacts the string. For example, a string bowed near its midpoint will strongly excite the fundamental mode, since it has an antinode near its center. Conversely, the next harmonic will be relatively weakly stimulated, since it has a node at the midpoint. Other even harmonics will be strong for the same reason, while odd ones will tend to be weak. Thus, the position of the bowing or pluck point strongly affects the overall aural quality of the sound. The width of the pluck (its sharpness) also affects the nature of the sound produced by the instrument.

In a piano, there is normally a group of three closely spaced strings assigned to each key. It is found that a richer, more pleasant-sounding tone results when the piano tuner makes minute adjustments of tension in order to ensure that the pitch of each member of the set of three is slightly different. Another factor strongly influences the richness of the piano sound, and it comes about because, unlike the ideal, perfectly flexible string, the wires used in a piano--especially those of the bass notes--have a considerable intrinsic stiffness that tends to decrease their flexibility. For that reason, the ideal string's perfect harmonic series, whose frequencies increase in a strictly integral way, are somewhat compromised, and what are then termed the "partials" of the fundamental are not quite harmonic. This so-called inharmonicity is increasingly important for the higher partials of a given fundamental, and its presence adds a quality of richness to the piano's sound.

Context

The study of waves on strings is one of the oldest known to humankind. The Pythagoreans of the sixth century B.C., whose natural philosophy was founded on the integer numbers 1, 2, 3, and so on, are said to have discovered the pleasing qualities of the sounds emitted by stretched strings whose lengths were proportional to the integers. Nevertheless, a physicist's interest in studying wave motions generally, and standing waves in particular, is by no means restricted to simple, one-dimensional, mechanical systems, such as vibrating strings. They are naturally of greatest interest to musicians. Scientists in many fields have discovered an enormous variety of applications for the general theories of both traveling and standing waves.

Wave motion is, moreover, one of the great unifying concepts of physics. Traveling waves of all kinds are vehicles for the transmission of energy: Mechanical waves are observed as the precursors of often catastrophic seismic events on Earth; ocean waves are now regarded as reservoirs of energy that can be captured and utilized, and sound waves are responsible for the transmission and the perception of speech and music.

The wave behavior of light, confirmed experimentally during the early years of the nineteenth century, gradually led to the understanding of light as but one small part of the great electromagnetic spectrum that stretches from very short-wavelength X rays to the much longer-wavelength radio waves. All are fundamentally electric and magnetic in nature and all travel with the speed of light. Yet, as different as they would seem to be from simple waves on strings, they can be described in mathematical terms that are identical, and the concepts that were developed originally for mechanical waves are found to carry over in a very natural way. The startling discovery that matter itself can also exhibit a wavelike behavior led to a postulate by mathematical physicist Erwin Schrodinger (1887-1961) that material particles obey their own characteristic wave equation. This idea not only made possible the interpretation of certain puzzling experiments but also led to the notion that the electrons, which lie in the outer reaches of the atom surrounding the nucleus, could also be thought of as standing electron waves.

The basic idea of the one-dimensional standing wave--as exemplified by mechanical waves on strings--is readily expanded to three dimensions and to systems in which other than purely mechanical motions are involved. Numerous technological applications of these ideas have been exploited in virtually every subfield of physics, from radio transmitters to lasers.

Principal terms

ELASTICITY: the tendency of solid objects to return to their original form after a stress is removed

FREQUENCY: the number of waves passing a fixed point in one unit of time; its common unit is the hertz, which is one per second

INERTIA: the tendency of a moving object to keep on moving

NEWTON: a common unit of force equal to the force required to accelerate a 1-kilogram mass 1 meter per second squared

NODE: a point that remains fixed in a standing wave; nodes separate the regions of maximum motion, the antinodes, in a standing wave

SUPERPOSITION: the addition of two or more waves that are present together in a medium

WAVELENGTH: the distance between adjacent points in a wave having exactly the same displacement and velocity at any given time

Bibliography

Berg, Richard E., and David Stork. THE PHYSICS OF SOUND. Englewood Cliffs, N.J.: Prentice-Hall, 1982. This book contains an excellent discussion of the basic physics of traveling and standing waves and introduces the families of musical instruments, including the strings. Each chapter concludes with an annotated bibliography. Ancillary to this work is a set of videotapes made by the authors that is especially interesting when used in conjunction with the book.

Bolemon, Jay. PHYSICS: AN INTRODUCTION. Englewood Cliffs, N.J.: Prentice-Hall, 1989. This book is a general introductory college physics text written for students not majoring in a science. The discussion is often humorous, though full justice is done to the underlying principles, and the ideas are made accessible. Several chapters are devoted to vibrations, waves, and sound, while later chapters point up the notion of standing waves as applied to atoms.

French, A. P. VIBRATIONS AND WAVES. New York: Norton, 1971. Another standard work that concentrates mainly on mechanical waves. It was written as one unit in the MIT introductory physics series and is both challenging and thorough in its approach. Contains many useful illustrations.

Pierce, John R. ALMOST ALL ABOUT WAVES. Cambridge, Mass.: MIT Press, 1974. This is a classic introductory work on wave motion. Pierce attempts to present the material, some of it highly sophisticated, in reasonably nonmathematical language, although the knowledge of algebra is assumed. The topics extend beyond the normal discussion of simple mechanical waves to include seismic, electromagnetic, and ocean waves, and the discussion of phase and group velocities is unusual.

Rossing, Thomas D. THE SCIENCE OF SOUND. 2d ed. Reading, Mass.: Addison-Wesley, 1989. This is a standard introductory text in the physics of sound and music. It is written in nontechnical language and is intended especially for the student without background in physics. Basic physics, which includes a discussion of traveling and standing waves, is covered in part 1, while the perception and measurement of sound, and the acoustics of instruments, are discussed in parts 2 and 3. The many illustrative photographs and drawings are exceptionally well done.

Van Bergeijk, Willem A., et al. WAVES AND THE EAR. Garden City, N.Y.: Doubleday, 1960. This work, a part of Doubleday's Science Study Series, was conceived as a supplement to the standard introductory physics texts. The primary aim of the series was to capture and stimulate the interest of lay readers and beginning science students alike. Therefore, the discussion, while nonmathematical, is occasionally challenging. The authors, all researchers in the field, present the physical principles of wave motion in the early chapters, then move on to a thorough examination of the physiology and psychoacoustics of hearing.

First five harmonics of a standing wave in a stretched string

Electrons and Atoms

Producing and Detecting Sound