SOME BELLS RING WITH MUSICAL quality; others only clang. What determines the quality of a bell and the frequencies it emits? Strange to relate, the note that is perceived by a listener may not be produced by the bell. This month I shall examine the acoustics of bells in some experiments designed by G. Theodore Wood, a colleague at Cleveland State University. We concentrated on a ship's bell but also examined several handbells lent to us by Katherine Marshall of Cleveland.

We were aided by three references. In the l9th century Lord Rayleigh examined the eight bells at his parish church in Terling, laying the groundwork for all later study of bells. Recently Stephen David Kelby and Robin Paul Middleton, n undergraduate students at the University of Birmingham, did experimental work on handbells. The source of the illusory note from a bell has been investigated by Arthur H. Benade of Case Western Reserve University.

The key to musical sounds of any kind lies in the study of acoustic resonance. When a bell is struck by its clapper or some other object, it is made to vibrate. What one hears from the bell are the sound waves generated by the vibration. Only certain of the vibrations generate sound waves loud enough to be heard. This special set of vibrations forms what are called resonance modes.

The study of resonance is usually introduced with a simple system in which a string is made taut between parallel supports. When the string is plucked, waves race from the point at which it is plucked to the support points, are reflected and travel back along the string. For the sake of simplicity let us assume that the waves have the sinusoidal shape shown in Figure 2. Each wave has a length that is the distance along the string required for the wave pattern to repeat. It also has a frequency that is the number of times per second (designated in hertz) a crest passes through a point on the string.

Suppose two identical waves are sent along the string in opposite directions. Reaching the support points, they are reflected and pass through each other, maintaining their initial amplitudes. (I shall ignore their loss of energy as they pull on the string and beat against the air.) The shape of the vibrating string at any instant is established by the interference of the waves at that instant. For example, if the waves are in step, the string is pulled into a sinusoidal pattern. Where the waves have crests, the string has crests that are twice as high; where the waves have valleys, the string has valleys that are twice as deep. Such interference is said to be constructive. At other instants the crests of one wave are in the valleys of the other wave. Then the string is flat because any pull upward is exactly countered by a pull downward. Such interference is said to be destructive. At instants that are intermediate the shape of the string is established by some intermediate amount of interference.

I can control the type of waves I send along the string by controlling the frequency generated by the initial vibration of the string. For nearly all choices of frequency the interference of the waves leaves the string with virtually no vibration. For a special set of frequencies- the resonance frequencies-the interference gives rise to a repeating pattern on the string that beats vigorously enough against the air to be heard.

When I pluck the string, a great many waves are generated simultaneously. Most of them have the wrong frequencies, result in no repeated pattern and are not heard. The few waves with a resonance frequency force the string into vigorous, repeated vibrations.

The simplest of the resonance patterns, called the fundamental or the first harmonic, always results from a pluck. The ends of the string do not vibrate because they are tied in place. The center of the string vibrates the most. Point on the string displaying no vibration are called nodes; those displaying the maxi mum vibration are antinodes. Hence the pattern for the fundamental has an anti node at the center and a node at each end. To create this pattern the waves must have a length that is twice the length of the string. The interference at the center of the string varies smoothly between a large crest, a flat string and a deep valley. The vigorous beating against the air sends out sound waves that have the same frequency as the string waves.

Other patterns of interference yield higher frequencies of sound. The next simple pattern, the second harmonic, forms if the lengths of the string waves are equal to the string's length. This pattern has three nodes (at the end points and at the center) and two antinodes. Although waves are traveling over the string, they always interfere destructively at the center and therefore alternate between destructive and constructive interference at the antinodes. The frequency of the sound generated by the string is the same as that of the waves on the string, which is twice the frequency of the fundamental.

One can find other resonances for the string by considering shorter wavelengths (and higher frequencies) for the waves on the string. The values for the frequencies fall into the neat mathematical sequence known as a harmonic series: each frequency is an integral multiple of the frequency of the fundamental. For example, the third harmonic (three antinodes and four nodes) has a frequency that is three times the fundamental's. The fourth harmonic (four antinodes and five nodes) has a frequency that is four times the fundamental's. When a string is plucked, the fundamental and some of the higher harmonics are generated because the plucking generates string waves with the appropriate wavelengths. It also generates waves with inappropriate wavelengths, which fail to move the string significantly and therefore are not heard.

Usually only the lower harmonics are loud enough to be heard, but which of them are generated depends partly on where the string is plucked. For example, suppose I pluck a string at its midpoint. The fundamental is sure to be generated because it has an antinode there. The second harmonic cannot be generated, however, because it demands a node at the very place I force to vibrate with the pluck. Any other harmonic calling for a node at the center is also absent. The other harmonics will be generated, some (like the fundamental) strongly and others weakly.

My model of string resonance is an ideal one. With a real string the higher resonance frequencies are not exactly integral multiples of the fundamental frequency. The term harmonic is reserved for frequencies with an exact fit to the mathematical sequence. The term partial is employed to describe resonances that fail to fit the sequence. Therefore when a string is plucked, the fundamental and several partials are excited. Part of the richness of a stringed instrument can depend on how much the partials differ from the ideal harmonics.

The striking of a bell is similar to the plucking of a string under tension. Vibrational waves race throughout the bell, interfering with one another. Most of the waves have wavelengths that are inappropriate for creating any strong, repeating pattern of vibration. Those waves do not contribute to the sound of the bell. The few waves with a resonance length interfere to force the bell into repeating vibrational patterns. The bell, beating against the air, generates sound waves having the same frequencies as those vibrational waves.

The resonance frequencies do not form a harmonic series and therefore are partials. Still, according to Benade and others, an observer might think the bell has emitted a harmonic series. If a listener is presented with a purely harmonic series from any generator of sound, he perceives only the fundamental frequency. In effect the brain examines the harmonics, picks out the fundamental and brings that frequency to consciousness. The assignment of frequency is easy when a full set of, say, the first five harmonics is present. Suppose, however, one of them is missing. The assignment is still made, even if the missing harmonic is the fundamental itself. Somehow the brain compares the sounds, realizes they are part of a harmonic series and then still perceives the fundamental.

Suppose the sounds presented to a listener do not form any harmonic series. Then the listener might perceive them in several ways. If the brain concludes that they approximately fit a harmonic series, the listener is forced to perceive the fundamental of the series even if that frequency is absent in the sounds. Instead the brain might utilize most of the sounds to achieve a best fit to a harmonic series and leave one or more of them separate. Then the listener might be forced to simultaneously perceive the absent fundamental and the partials left out of the best fit. Judging the frequency of the perceived fundamental introduces a problem. Commonly an error of one octave is made. For example, middle C (C₄) on a piano has a frequency of 261.6 hertz, but a listener might interpret the sound as being an octave higher, that is, C₅, which has twice the frequency of middle C.

Such analysis and interpretation apply to the sounds from a bell. The nonharmonic sounds generated by the bell may be forced by the brain into a best fit for a harmonic series. The perceived fundamental of that series may actually be absent from the sound. Moreover, the assignment of a frequency value to the perceived fundamental may be in error by an octave (or a factor of two). If some of the partials from the bell are not forced into a best fit, the listener might hear them along with the perceived fundamental.

A tuned bell is one designed to generate the frequencies and vibrational patterns displayed in Figure 3. The lowest frequency (the persistent one) is called the hum note. The next-higher frequency, called either the fundamental or the prime, is twice the hum note. (Since the term fundamental here can be confused with the term applied to string resonance, I shall call this frequency the prime.) The next several partials are referenced to the prime. The third note is a musical minor third up in frequency from the prime, that is, its frequency is 1.2 times the prime one. The next note is up from the prime frequency by a musical fifth (a factor of 1.5). The note called an octave is twice the prime frequency. The note called an upper third is 2.5 times the prime frequency. Also shown in the illustration are two other partials of a tuned bell.

When a tuned bell is rung, the hum note and several of the other partials are excited. Each partial has associated with it a certain vibrational pattern of nodes and antinodes. I shall label a line of nodal points (no vibration) according to whether it runs up and down the bell (a longitudinal line) or around it (a latitudinal line). The hum note has two longitudinal nodal lines along which the bell does not vibrate. (The pattern is also depicted in the illustration by a circle representing an overhead view of the bell.) Midway between the nodal lines are the antinodes of the vibration. Since striking the bell forces it to vibrate at that point, one of the antinodal lines for the hum note must pass through the same point. The ensuing waves of vibration racing around the bell set up the rest of the pattern. For example, nodal lines appear at 45 degrees longitude on each side of the strike point.

The vibrational pattern for the prime frequency is similar except that it has a latitudinal nodal line. Hence when the prime is excited, there are two nodal lines running up and down the bell and there is also a nodal line running around it. The position of this last line depends on the distribution of mass in the bell. Figure 3 shows the vibrational patterns for the other partials of the bell. In each case I have located the latitudinal nodal lines only approximately.

These patterns can be labeled according to the number of longitudinal and latitudinal nodal lines they have. For example, the pattern for the hum note is called a (2,0) mode to represent the two longitudinal nodal lines and the absence of any latitudinal nodal lines. Both the minor third and the fifth are labeled (3,1). Although these patterns are identical in their types of nodal lines, they differ in their latitudinal nodal lines and in the frequencies of the waves responsible for the patterns. A bell cannot vibrate in the (1,0) mode or any mode that lacks longitudinal nodal lines. Such a pattern would require that the circumference of the bell's mouth oscillate in size. The bell is too strong to allow such stretching and contraction of the rim.

The frequency relations I have described are true only for a tuned bell. Such a bell is musically pleasing because of those relations. When a listener hears the sounds from the bell, he might force them into a harmonic series consisting of the prime, the octave and the partials that are at three and four times the frequencies of the prime. Then the listener will be likely to perceive the prime as the fundamental of the series. The hum note and the other frequencies left out of the series might be perceived separately from the prime.

Alternatively the listener may unconsciously choose a harmonic series of the hum note and the partials that are integral multiples of its frequency. Then the bell will seem to ring at the hum note and perhaps also at the minor third (which is left out of the series). If the listener is asked to estimate the frequency of the bell, he might give the frequency of the hum note or make the octave error and give the prime frequency. Although what is heard and what frequency is assigned to the sound are the result of an unconscious process, predicting the outcome is difficult.

When a bell has not been tuned, prediction is even more difficult. Wood and I examined an apparently untuned ship's bell with a mouth 25 centimeters in diameter. It was a heavy brass bell suspended from a horizontal support that was clamped onto table stands. The clapper had been removed. To ring the bell we tapped it with hammers that had a head of hard plastic, hard rubber or steel. The softer blow with the rubber head generated mostly the lower frequencies; the harder blow with the steel head generated the higher frequencies as well. A microphone under the mouth of the bell was connected to a computer Wood had programmed to separate the frequencies in the sound and display them on a monitor along with their relative amplitudes. The computer could also produce a graph of the frequencies. The analysis was limited to frequencies lower than 5,000 hertz.

A tap about halfway up the bell produced a hum note with a distinct beat, or warble. The computer revealed that the hum note was split into two frequencies at 811 and 819.9 hertz. We were hearing those frequencies and their interference with each other. Periodically the two sounds arrived at the ear so as to reinforce each other. At intermediate times they arrived so as to cancel (or almost cancel) each other. We heard the true frequencies and also an extra sound (at the average of the two hum notes) that varied in loudness at a rate equal to the difference between the true frequencies. Thus the beat in the hum note was at about nine hertz.

Lord Rayleigh had explained such beats in the output of bells in terms of the distribution of mass around the bell. Suppose the bell is perfectly symmetrical in its mass distribution. Then striking the bell anywhere should excite the hum note. The strike point is automatically an antinode and the rest of the pattern is determined by its position. A different choice of strike point results in the same pattern but shifted so that again the strike point is at an antinode. Regardless of the strike position, the same frequency is generated for the hum note.

If the bell is asymmetric in its mass distribution, the symmetry of the vibrational pattern for the hum note is broken. There now can be two vibrational patterns for the hum note, one of them shifted in longitude by 45 degrees from the other. Hence the nodal lines for one pattern fall along the antinodal lines for the other. These patterns vibrate with different frequencies. Provided the frequencies are close enough, they can beat against each other when both patterns are excited.

On an asymmetric bell the nodal and antinodal lines are fixed in place. Thus a tap at one strike point on the bell might excite only one of the patterns. A tap at another strike point might excite the other pattern. Taps at still other points might excite both patterns. If only one pattern is excited, only its frequency is emitted. When both patterns are excited equally, both frequencies are emitted with equal loudness and the beats resulting from their interference are most noticeable.

Wood and I set out to find the nodal and antinodal lines for each hum-note pattern. Halfway up the bell we tapped point by point along a latitude line. At one point we heard only a single hum note and no beats. We marked this point as being on the antinodal line for the first hum-note pattern. Since it was on the nodal line for the second pattern, only the first pattern was being excited. The computer determined the frequency to be 819.9 hertz.

As we explored farther along the latitude line beats developed, indicating we were exciting both patterns. Taps farther along led to less beating and finally a pure tone for the hum note again. This time the frequency was 811 hertz. A tap at this point on the bell excited only the second pattern because it was on an antinodal line for that pattern and on a nodal line for the other pattern. We continued around the bell, marking points of pure tone and maximum beating. We verified that the two patterns for the hum notes were shifted from each other by 45 degrees.

Wood and I reasoned we could alter the frequencies of the hum notes by adding mass to the bell. We stuck Apiezon sealing compound (modeling clay would serve the same purpose) on the rim of the bell at a point on an antinodal line of the second pattern. We figured that when we tapped to excite only the second pattern, the added mass would retard the vibration, decreasing its frequency. The frequency did fall from 811 hertz to 808.4. Since the mass was on a nodal line of the first pattern, it should not alter the frequency of that pattern. Indeed, when we tapped at the point that would excite only the first pattern, the frequency was approximately the same.

We next removed the sealing compound and explored down a longitudinal line that was coincident with an antinodal line of the first pattern. Starting at the top, we laid out 15 strike points separated by half an inch. We were searching for latitudinal nodal lines of the higher-frequency partials. In particular we wondered if the sequence of frequencies and vibrational patterns fell into the same order as they would for a bell that had been tuned.

As we tapped our way down the side of the bell, the resonance frequency at 1,672 hertz remained strong except at the 12th strike point; there it apparently had a latitudinal nodal line. The resonance frequency at 1,810 hertz had a broad latitudinal nodal line throughout regions of the seventh through the 12th strike points, entirely disappearing at the 11th point. The resonance at 1,870 hertz displayed a wide latitudinal nodal line centered at the sixth strike point. The resonance at 2,573 hertz was strong at the upper strike points, increased to the loudest resonance at the 10th point and then almost disappeared near the rim of the bell. The 3,010-hertz resonange showed a narrow latitudinal nodal line at the seventh strike point. The 4,197-hertz resonance disappeared entirely at the 12th strike point.

With these clues we attempted to connect the resonance frequencies with the vibrational patterns. The hum notes were already established at 811 and 820 hertz. We figured that the prime frequency was at 1,672 hertz, for two reasons. That frequency is approximately twice the hum-note frequency, which would be true for a tuned bell. Moreover, that frequency has a latitudinal node through the 10th strike point, an appropriate position for the prime.

The other resonance frequencies were harder to identify, and none of them fell into the mathematical sequence of a tuned bell. There are two conditions for identifying such a pattern. First, the resonance frequency must be within a few hundred hertz of the predicted (tuned) values, as referenced to the prime. Second, the pattern must have an appropriate latitudinal node. With these conditions the patterns for the resonances at 1,870, 2,573, 3,010 and 4,197 hertz resemble the patterns called respectively the minor third, the fifth, the octave and the upper third.

Further clues could be found by tapping around a latitudinal line and counting the number of nodes and antinodes for each resonance frequency. The asymmetry in the mass of the bell should split each resonance frequency, but the resolution of the computer analysis was apparently too coarse to find

the splits. Nevertheless, the asymmetry fixes in place the nodal patterns for each resonance, so that one can search for the patterns along a latitude line as we did for the hum note. We did not search for these extra clues, being content to find the bell was substantially untuned.

Wood attempted to identify the overall note of the bell in two ways. In one experiment he matched the note of the bell with an electrically generated musical scale. The best match was slightly above G5. (The piano-key note is at 784 hertz.) Next he matched the bell with the output of a signal generator. Here the best match was at 819 hertz. Apparently what he (and I) heard from the bell was close to its hum note. Perhaps the resonances at 820, 1,672, 2,573, 3,010 and 4,197 hertz fit closely enough to a harmonic series for us to make the identification. If that is the case, the fundamental of the series would be near the humnote frequency. The higher frequencies might disappear so quickly, however, that one can only hear and recognize the hum note.

Wood and I also examined the frequencies (up to 5,000 hertz) emitted by four of Marshall's handbells. None of the bells was thought to be tuned, although their hum notes were noticeable One bell, a lead-glass one blown in Portugal, had a pleasing sound but a perplexing number of frequencies. The second bell was a Swiss cowbell that was quite unmusical. The third was a sterling-silver bell with a great many frequencies. The last was a Buddhist prayer bell with high frequencies. Although we could determine the humnote frequency of this bell, we failed to identify the higher frequencies with any vibrational pattern.

If you would like to examine the resonance of bells, I suggest you use a home computer to analyze the frequency spectrum. Programs for such an analysis are sold for many popular makes of computer. A more difficult approach would be the one taken by the makers of bells They search for resonance frequencies by listening for beats between the sound of the bell and a tone they produce either by singing or by playing a tuned instrument.

The task is easier if you slowly sweep a signal generator through a range of frequencies. Listen to the sound from a speaker connected to the generator as you are ringing the bell. When the generator's frequency is near a bell frequency, you can hear beats. You can then eliminate some of the resonance frequencies by striking the bell along its latitudinal nodal lines. If the bell is asymmetric in its mass distribution, you can also eliminate some of the frequencies by striking along longitudinal nodal lines. In this way you can unravel what frequencies and what vibrational patterns are characteristic of the bell.

Bibliography

VIBRATIONS OF PLATES. John William Strutt, Baron Rayleigh, in The Theory of Sound: Vol. 1. Dover Publications, Inc., 1945.

THE VIBRATIONS OF HAND BELLS. S. D. Kelby and R. P. Middleton in Physics Education, Vol. 15, pages 320-323; 1980.

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