Resonance

Type of physical science: Astronomy; Astrophysics; Atomic physics; Classical physics; Condensed matter physics; Elementary particle (high-energy) physics; Nuclear physics; Relativity

When the natural oscillatory motion of an object is greatly amplified by an impressed vibration of the same frequency, the object is said to be resonating. Resonance can occur in mechanical, structural, and acoustical systems, as well as in alternating current electric circuits. Resonance phenomena are also prevalent in atomic and nuclear physics, where, for example, electromagnetic waves of certain frequencies excite the orbital electrons of atoms to higher energy levels.

Overview

A dog's wagging tail, the swinging pendulum of a grandfather clock, and a sounding violin are all examples of vibration--a periodic motion that repeats itself over and over again in equal time intervals. The rate at which this motion repeats (termed the frequency of vibration) depends upon the mass of the object and the force that returns the displaced object to equilibrium. In general, objects having a small mass vibrate rapidly. Larger, more massive objects oscillate more slowly. The wings of a mosquito vibrate rapidly enough to produce a high-pitched hum, whereas an earthquake can jolt the earth, causing very slow undulations taking longer than one hour to repeat. In general, a simple vibrating object that is displaced from its equilibrium, or rest, position vibrates with a frequency that is directly proportional to the force attempting to restore the object to equilibrium (the restoring force) and inversely proportional to the mass of the object. Thus, massive objects generally will have a lower natural frequency, while objects having a greater restoring force will vibrate at a higher frequency.

The frequency with which a simple system oscillates when perturbed is called its natural frequency. For most free vibrators, the energy supplied by an initial impulse dies out very rapidly. Energy must be supplied continuously by an outside driving force to maintain the vibration. During free vibration, the characteristic frequency of the object remains constant, but the amplitude continually decreases. During forced vibration, however, when a vibrator is driven by an oscillating driving force, the vibrator oscillates at a frequency that is the same as that of the driving force, but generally with a smaller amplitude. For example, when a sounding tuning fork is held over the end of an open bottle, a faint sound having the same pitch as the tuning fork can be heard emanating from the bottle. Blowing across the mouth of the bottle produces a much louder sound of different pitch, the pitch being determined by the dimensions of the air cavity.

One can examine a simple vibrating system driven by a periodic force of variable frequency, which starts well below the natural frequency of the system. As the frequency of the driving force is slowly increased to a point well above the natural frequency, it is found that the amplitude of vibration increases gradually to a maximum value at the natural frequency. This is followed by a slow decrease in amplitude as the frequency continues to increase. The condition of large amplitude vibration, which occurs when a simple vibrator is driven at a frequency equal or near to its natural frequency, is termed resonance. A common example of resonance is the process used by a bell ringer who wishes to sound a large church bell. Since the bell is too heavy to be set ringing by a single pull on the rope, the ringer pulls the rope very hard and then releases it. Although the bell does not sound, it begins to swing at its natural frequency of oscillation.

After the bell has executed a complete swing, the rope is back to the original position and moving downward. If the ringer now pulls the rope in the same direction that it is already moving, the amplitude of the bell's swing will be increased with the exertion of only a slight additional force. By repeating this process, the amplitude of the swing increases until the bell loudly sounds. Physically, the periodic force exerted by the ringer has the same frequency as the free vibration frequency of the bell; thus the system oscillates with a large amplitude at its resonant frequency.

Not all mechanically vibrating systems exhibit resonances as narrow as the swinging church bell. Blowing across the mouth of an empty bottle produces a sound at the natural frequency of the bottle's air cavity. When a tuning fork having the same frequency is held over the bottle, resonance causes a considerably louder sound to be heard. If the frequency of the tuning fork is changed slightly by attaching a small lump of clay to one tine, there is very little change in the loudness of the resonant sound. In fact, as the frequency of the fork is varied slowly from the resonant frequency of the bottle, the response of the bottle varies only slightly. A fairly large change in frequency must occur before the bottle's response decreases appreciably because the bottle, unlike a tuning fork, has a broad resonance (it responds over a broad range of frequencies).

The amount of resistance, or damping, which removes energy from a vibrating system, determines how broad the resonance will be. An increase in damping increases how rapidly energy is removed from the system, how quickly the amplitude of the free vibrator decreases, and causes the resonance to broaden. Although damped systems respond to a broader range of frequencies, they do so less energetically. In general, the greater the resistance, the less the amplitude of vibration at resonance. Therefore, large damping causes free vibrations to decay rapidly but results in broad response curves of lower amplitude for forced vibration. Small damping results in a long-lasting free vibration, but narrow resonance curves of higher amplitude for forced vibration.

Resonance is also exhibited in oscillating electric circuits. When a simple circuit, consisting of a resistor, a capacitor, and an inductor in series, is driven by an alternating current source of variable frequency, the circuit will have a resonant frequency where the current rises to a maximum value.

The vast majority of objects visible to humans do not emit light, but rather reflect a part of the incident radiation to the eyes. The colors of light to which the eyes respond vary from 400 nanometers (violet) to 700 nanometers (red). Thus, a red rose appears red because long-wavelength radiation is reflected while the shorter waves are absorbed. For example, one can think of the atoms of a solid object as being tiny three-dimensional oscillators that vibrate in response to a driving force imposed by the periodic electric field contained within the incident electromagnetic wave. These electron oscillators have a natural frequency that is different for different atoms. When light passes over an atom, the electron cloud vibrates under the influence of the light at the same frequency but with low amplitude. Once oscillating, the atoms radiate light in all directions, which yields (after interference effects) refracted and reflected waves with frequencies equal to the atomic oscillation frequency. When the frequency of the incident light is a poor match to the resonant frequency of the electron cloud, the vibration amplitude is small and much of the incident energy is absorbed and transformed into heat. The refracted portion of the wave may be transmitted through the substance or it may be absorbed. The reflected portion of the wave, which originates in a layer within about one-half a wavelength of the surface, consists of the reradiated energy that has not been absorbed. When all the incident energy is reradiated, the object appears white (all colors are radiated equally). Most objects absorb some light and reradiate the remainder. The reradiated energy reaches the eyes as reflected light and is perceived as the color of the object.

According to quantum theory, light is emitted and absorbed as small bundles of energy called photons. Atoms can be characterized as having well-defined energy levels, which may or may not be occupied by electrons. If an incident photon has an energy equal to the difference of two atomic energy levels, the upper level being unoccupied, the photon will be absorbed and the electron will jump from the lower to the upper level. For this situation, the light is said to be in "resonance" with the atom. When white light passes through a gas, the photons causing such atomic resonances are absorbed, leaving dark regions superimposed on the continuous spectrum produced when the light is passed through a prism. Absorption spectra such as this is characteristic of the light emitted by stars.

Applications

Acoustic resonance is an important aspect of nearly all sound-producing systems, including musical instruments, the human vocal apparatus, and even high-fidelity loudspeakers.

The sound board on a piano has a very broad resonance; thus it amplifies the many different frequencies occurring over its seven-octave playing range. Brass instruments, on the other hand, utilize a fairly narrow set of resonances that enable the performer to produce a harmonic series of notes by buzzing the lips at one of these frequencies. The resonances must be neither too broad nor too narrow. Resonances that are too narrow do not allow the player to miss the exact frequency and still get a good response. If the resonances are too broad, however, the note will not be able to be held on pitch. Well-designed violins use both the resonance of the enclosed air cavity and the resonant vibration of the wood to enhance the weak sounds of the strings. One of the important differences between a high-quality violin and an inferior instrument is that the quality violin has its important resonances occurring at the same frequencies as the open strings.

The human vocal apparatus uses a variable resonator (the vocal tract) to transform the buzzy sound of the vocal cords into different vowel sounds.

For mechanical structures, resonance is to be avoided. Roman soldiers never marched across a bridge, because if the periodic driving force of all the warriors' feet striking the bridge in unison happened to correspond to a resonant frequency of the structure, the resulting large amplitude vibrations could cause the bridge to collapse. Engineers who design large structures must always consider the possible detrimental effects of resonance. In November, 1940, only months after opening, the Tacoma Narrows Bridge (in Washington State) was subjected to gale-force winds of about 65 kilometers per hour. The wind produced a fluctuating driving force that excited one of the natural resonant frequencies of the bridge. The resultant large amplitude vibrations destroyed the bridge within hours.

A loudspeaker diaphragm, having mass and stiffness, also has a resonant frequency.

Since the goal of high-fidelity systems is to reproduce an original sound as accurately as possible, it is important that speakers be designed to allow their resonance to help maintain as uniform a frequency response curve as possible. Woofers (bass speakers) are designed accordingly, with the resonance of the diaphragm occurring in the lowest range of frequencies to which the speaker can respond. This boosts the output at the low-frequency end. The enclosure surrounding a speaker diaphragm also has a resonant frequency that may be used advantageously to help augment the bass, that is, to lower the frequency response of the speaker. The open-back cabinet, commonly used in console phonographs and television sets, typically has a resonance between 100 and 200 hertz. This technique boosts an otherwise deficient bass caused by undersized speakers. Although the response is often characterized as being "boomy," it is an inexpensive alternative to an otherwise lackluster bass response.

A radio tuner is merely an adjustable tuned resonant circuit. This circuit selects from myriad possible radio waves the particular radio frequency (station) of interest. The information content, superimposed on this wave, can then be demodulated and transformed into sound. When the frequency of the tuner is adjusted to exactly the same frequency as the incoming electromagnetic wave, the narrow resonance amplifies the signal while excluding nearby stations with slightly different broadcast frequencies.

Most of the colors perceived in the world are the result of preferential absorption of light caused by a substance having an atomic resonance in or near the visible region. Chemical dyes are chain or ring-shaped molecules in which the electrons move freely along the chain or ring. Since the electrons are spread out over a greater distance than in other atoms and molecules, resonances occur at lower energies of the spectrum--that is, the visible rather than the ultraviolet region. This, in and of itself, however, is not a sufficient condition for a molecule to serve as a useful dye. After absorption, the energy of the incident photon must be transformed into heat and not reemitted as light. Next, the resonance must be a broad resonance, since a dye having a narrow resonance would absorb such a limited frequency range that the reflected light would be almost white. Thus, a red dye will absorb light of all colors except red, while green paint absorbs the blue and violet region as well as red and orange.

Context

Although vibrating objects had been observed and studied for centuries, it was Galileo who first described the phenomenon of sympathetic vibration, which is known as resonance. In his DIALOGUE CONCERNING THE TWO CHIEF WORLD SYSTEMS, PTOLEMAIC AND COPERNICAN, published in 1661, after discussing the dependence of frequency on length for a pendulum, he discusses how the vibrations of one object can produce similar vibrations in another distant body. Later, after the development of Sir Isaac Newton's laws of motion and calculus, it became possible to calculate resonance frequencies exactly for simple mechanical systems that are set into vibration.

In the ensuing centuries, it became possible to calculate resonances for more complicated structures and for resonant air columns, but the calculation of resonances for the complicated shapes of orchestral wind instruments seemed impossible. The advent of the computer age, however, has made such calculations routine. Programs now exist that can calculate the resonances of any complicated air column, including open or closed tone holes, from its geometry. Engineers also now routinely use computer modeling techniques to predict, and thus to avoid, detrimental resonances in large structures such as suspension bridges.

Principal terms

ABSORPTION SPECTRUM: a continuous spectrum, such as formed by white light, but containing dark lines caused by the absorption of certain frequencies by a substance through which the radiation has passed

AMPLITUDE: the maximum displacement, from its rest position, of an oscillating substance (a mass for a mechanical vibrator, a pressure for an air wave, or an electric current for a circuit)

FORCED VIBRATOR: an oscillating object to which energy is continuously supplied, thus maintaining the amplitude

FREE VIBRATOR: any oscillating object set into vibration and then allowed to lose its energy; free vibrators have a continously decreasing amplitude

FREQUENCY: the number of complete vibrations in each second; measured in hertz, where 1 hertz is one vibration per second

PHOTON: the basic packet of electromagnetic energy that exhibits particle-like behavior; a photon's energy is proportional to its frequency

RESONANCE: a condition of large amplitude of vibration that occurs when a vibrator is driven at its natural, or free vibrator, frequency; resonances may be either broad (over an extended range of frequencies) or narrow (over a limited range of frequencies)

Bibliography

Asimov, Isaac. UNDERSTANDING PHYSICS. 3 vols. New York: New American Library, 1966. This excellent series includes resonance and its application to sound in volume 1, light and color in volume 2, and basic atomic theory in volume 3.

Hewitt, Paul G. CONCEPTUAL PHYSICS. 6th ed. Scranton, Pa.: HarperCollins, 1989. This nontechnical book includes discussions of resonance and its application to sound and light.

Roederer, Juan G. INTRODUCTION TO THE PHYSICS AND PSYCHOPHYSICS OF MUSIC. New York: Springer-Verlag, 1973. Includes discussions of resonance curves for musical instruments and references to the hearing process. For a wide audience.

Strong, W. J., and George R. Plitnik. MUSIC, SPEECH, AND HIGH FIDELITY. 2d ed. Provo, Utah: Soundprint Press, 1983. A very complete nontechnical treatment of all aspects of acoustics, including resonance and its importance to musical instruments, the human vocal apparatus, and loudspeaker design.

Van Heel, A. C. S., and C. H. F. Velzel. WHAT IS LIGHT? New York: McGraw-Hill, 1968. This book describes the fundamental properties of light, including how light interacts with matter and the photon theory.

Electrons and Atoms

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