EVEN A BAD SINGER SOUNDS GOOD in the shower, at least to himself. I certainly qualify as unskilled in singing, but in the shower my voice is a thing of beauty, making it easy for me to overrate my ability. The shower stall is so kind to musical sounds that a friend of mine practiced her violin lessons in one.

The secret of the shower stall as an aid to musical performance lies in its acoustic resonance, which greatly increases the strength of the sound. Like any other sound, a sound generated by the vocal apparatus travels through the air as a wave of variations in pressure. If the voice could be held steady at one frequency (or pitch), the wave could be represented by a sinusoidal curve. The curve depicts the regular variation in the pressure of the air along the path of the sound wave from relatively high (above the ambient pressure) to relatively low.

The variation results from slight oscillations of the air molecules parallel to the path of the sound wave. The oscillations generate a fairly high density of air (a high pressure) at some places and a fairly low density (a low pressure) at others. The entire pattern of high and low pressure travels away from the mouth at the speed of sound, which at room temperature is about 340 meters per second.

An associated factor is the frequency of the sound, which is measured by the number of times per second a high-pressure section of the wave passes an imaginary point along the path. If the number is 500, the frequency is 500 hertz (cycles per second). The range of hearing for a young human adult is roughly from 20 hertz to 20,000.

A wave moving from one place to another is called a traveling wave. Another type of wave, the standing wave, is important to singing and acoustic resonance. Consider a cylindrical pipe open at both ends. In the simplest approximation its diameter makes no difference with respect to resonance, but its length is important. A traveling sound wave of a single frequency is fed continuously into the pipe. Imagine following one high-pressure segment of the wave along the pipe. At the far end of the pipe part of the sound is reflected backward, even though the pipe is open. The rest of the sound continues on outward. The reflected part now overlaps e waves still moving toward the far end of the pipe and interferes with them. M a given instant a particular section of the air has a tendency to rise in pressure because of the wave going one way and to fall in pressure because of the wave going the other way. The result is a constantly shifting, complex pattern of pressure variations. The sound coming out of the tube is no louder than the sound that went into it.

At certain frequencies a much simpler pattern of pressure variations develops. The pattern is the harmonic frequencies for the pipe. These frequencies are integral multiples of the lowest frequency: the fundamental. The pattern is a standing wave. When such a wave is created, the sound coming out from the pipe is much louder than the sound that went into it. This condition is called acoustic resonance.

Suppose the fundamental frequency is directed into the pipe. The interference of the waves traveling in opposite directions ensures that at the ends of the pipe (where the waves reflect) the pressure never varies. Such a place is called a node. The pressure changes most in the middle of the pipe, varying smoothly from high to low (high when the oppositely traveling waves interact to increase the pressure, low when they interact to decrease it). Such a place is an antinode. The pattern of pressure variations along the pipe is called a standing wave because the nodes and antinodes stand still.

A standard graphical representation of standing waves in a pipe appears in the illustration below. The points where the curves cross are nodes and the points where the curves have the greatest separation are antinodes. (One can be misled by the standard representation into thinking that the molecules oscillate up and down in the pipe. Actually the direction of their oscillation is along the length of the pipe.)

Things get more complicated as the higher frequencies of a harmonic series are directed into the pipe. With the second harmonic (twice the frequency of the fundamental) the standing wave shows three nodes, one node at each end and one in the middle. Antinodes lie between the nodes. The third harmonic (three times the frequency of the fundamental) has four nodes and three antinodes.

Acoustic resonance is significant because it creates a large standing wave of sound, which is louder than the sound directed into the pipe. The pipe is therefore an amplifier. In my simple model the values of the harmonic series in a pipe depend only on the speed of sound and the length of the pipe. Heating the air, which makes the sound travel faster, shifts the harmonic frequencies upward. Shortening the pipe has the same effect. That is why the fundamental is higher in a short organ pipe than it is in a long one.

So far I have discussed only a pipe with both ends open, but for singing in the shower two other configurations are important. One of them is a pipe with both ends closed. its harmonic series is the same as the series in the open pipe if the pipe is the same length. The standing waves are different but their frequencies are the same. Now an antinode appears at each end of the pipe because the pressure variation is highest at a wall. A picture of the first harmonic (the fundamental) shows an antinode at each end and a node in the middle.

When the pipe is closed at only one end, the harmonic series is subtly modified. A drawing of the first harmonic still follows the rules I have outlined, showing an antinode at the closed end and a node at the open end. The same is true for the next harmonic, but the result is that its frequency is three times the frequency of the first harmonic. Sometimes this standing wave is called the second harmonic because it is the second-lowest frequency possible for resonance in such a pipe. I find this designation confusing. Since the harmonic's frequency is three times that of the fundamental, I shall continue to call it the third harmonic. Only the odd harmonics (first, third, fifth and so on) are possible for resonance in a pipe with one open end. Standing waves for the even harmonics are impossible because of the rules governing pressure variations at the ends of a pipe.

Singing entails creating a resonance in what is essentially a pipe with one open end. The pipe is not the straight, cylindrical one I have been describing, but the model still serves well. The sound originates with the passage of air pushed out of the lungs through the vocal cords (which are not cords but thin membranes). The air emerges from the vocal cords in a series of pulses. The frequency of the pulses depends mainly on the tension of the cords; a higher tension results in more frequent bursts of air and so in higher frequency.

The vocal cords vibrate in a harmonic series of frequencies, yielding a harmonic series of sound waves encompassing the fundamental and all the higher harmonics. The most intense wave is the fundamental.

These waves pass through the vocal tract, consisting of the larynx, the pharynx and the mouth. The vocal tract can be modeled as a pipe with one closed end (at the vocal cords) and one open end (at the mouth). It therefore has a harmonic series of frequencies at which standing waves can be excited. Among the frequencies entering the vocal tract from the vocal cords are some that fall on or near these resonant frequencies. Those frequencies are relatively loud.

A singer generates the loudest sound when the fundamentals of the vocal cords and those of the vocal tract match. A good singer can achieve the match, probably with little conscious effort, in several ways. The tension of the vocal cords can be adjusted somewhat to regulate the fundamental at that point. Once that fundamental is established, however, further matching must be done by varying the shape of the vocal tract.

Here the physics of the acoustic resonance is not as simple as it is in the pipe models, where I ignored the effect of diameter. By appropriate reshaping of the vocal tract a singing adult male can shift the fundamental of the tract to any frequency between 250 and 700 hertz. The third harmonic (bear in mind that only the odd harmonics develop in this kind of pipe) varies between 700 and 2,500 hertz.

I am unskilled in matching the fundamental of my vocal tract to the one being generated in my vocal cords. If I sought to project to an audience a note of, say, 500 hertz, I would have to scream to make it sufficiently audible. A well-trained singer would make the match and so would benefit from the amplification arising from the resonance of the vocal tract. An advantage of singing in the shower is that the unskilled vocalist is aided by the resonances generated between the surfaces of the shower stall.

The stall is in effect a pipe with both ends closed. It differs from the other pipes I have been describing in having three pairs of ends: (1) the floor and the ceiling, (2) one set of parallel walls and (3) the other set of parallel walls. (I treat the shower door or curtain as a solid wall.) A standing wave of sound can be excited between any pair of walls. A depiction of the fundamental for, say, the floor-ceiling pair shows antinodes at the floor and the ceiling and a node halfway between the two. The second harmonic (a pipe with both ends closed can have even harmonics) has three antinodes (floor, ceiling and halfway between) and two nodes (between the antinodes). Similar patterns arise for the other two pairs of walls.

Although the association of singing in the shower and acoustic resonance has long been known, the only study I have found of it is an abstract of a talk given by Daniel W. Haines of the University of South Carolina at the Second Conference on the Teaching of Acoustics and the Physics of Sound and Music in April, 1976. His work revealed several interesting features of the phenomenon For example, suppose the aim is to generate resonance between the floor and the ceiling of the shower stall. If the source of the sound is halfway between the two, there cannot be any resonance because that is the location of a node in the associated standing wave. A node, which is by definition the absence of pressure variations, cannot arise where the sound source itself is creating variations in pressure. To be closer to an antinode of a standing wave one would do better to sing near the floor or the ceiling (although it might be hazardous and might expose one to the ridicule of an onlooker, however friendly).

The second harmonic and other even harmonics are possible if the source of sound is halfway between the floor and the ceiling of the shower stall. Any other height for the source might give rise to many harmonics, some more intense than others, depending on how close the source is to an antinode. The only impossible harmonics are those where the source and a node coincide. (My explanation here may be faulty. If the source is quite small and radiates sound throughout the stall, its location is probably unimportant. I think, however, that the human head is large enough for its location to matter.)

Haines also noted that the singer can sound good or bad to himself according to the position of his ears. If they are at a node, the vocalist cannot hear the standing wave (since hearing is actuated by pressure waves on the eardrums). If they are at an antinode, the impact on the eardrums is at a maximum.

The first eight harmonics between the floor and the ceiling are shown in the illustration in Figure 2. The mouth and ears

of the singer are approximately a fourth of the distance from the ceiling to the floor. In theory he could both excite and hear the fundamental because his mouth and ears are not at a node of the associated standing wave. Since they are also not at an antinode, he cannot excite or hear a strong fundamental. The second harmonic cannot be excited or heard because his head is at one of its nodes. Among the standing waves depicted in the illustration the third, fourth, seventh and eighth harmonics would come out best.

The singer is somewhat more mobile when it comes to the standing waves between a pair of walls. By moving toward or away from a wall one can modify the spectrum of harmonics that are excited and heard. At the center of the stall the fundamental from each pair of walls would be absent, since both fundamentals have nodes there. The second harmonics have antinodes at that position and would come into play. By moving closer to a wall the singer would eliminate the second harmonic but would excite the third.

The harmonics of the walls are complicated by the horizontal separation between the mouth and the ears. For some harmonics the mouth might be in the right place to excite a standing wave while the ears were at a node. (To study this problem adequately one must also consider the distortion of the standing wave by the head.)

A further complication arises because the harmonic frequencies are not precisely as they are shown in the illustration above. The numbers there depend on the speed of sound, which may change as the air in the stall gets warmer from the shower. Even without any change in the speed of sound the frequencies are not sharp but extend over a range of about 10 hertz. A standing wave is excited most efficiently when the singer's voice is pitched at a frequency listed in the table. the wave is weaker if he is slightly off the theoretical value.

Another complication results from the fact that the singer's body occupies space in the stall, reflecting some of the sound. It is best to assume that this complication is secondary to the principal mechanism of creating standing waves. Modeling the resonances as if the stall is a pipe with closed ends at least makes possible an approximation of the resonant frequencies.

The frequencies tabulated in the illustration above are for my shower stall, which has flat tiled walls, floor and ceiling. One wall is a glass door. I calculated these harmonic frequencies (1,000 hertz or less) assuming the speed of sound to be 346 meters per second in the stall. The frequency of the fundamental is then the speed divided by twice the distance between a pair of reflecting surfaces.

Since the floor and the ceiling are farther apart than the walls, the floor-ceiling fundamental is lower in frequency than the fundamentals for the walls. The higher harmonics are calculated by multiplying the fundamental frequency by an integer. The second harmonic is twice the fundamental, the third three times the fundamental and so on.

Now, suppose I begin to sing in the shower, somehow adjusting my vocal cords so that their fundamental is 330 hertz but failing to match the fundamental of my vocal tract. The vocal cords deliver frequencies at integral multiples of the fundamental, namely 330, 660, 990 hertz and so on. Suppose the fundamental of my vocal tract is set at 450 hertz. Its next resonance is at 1,350 hertz (the third harmonic). Outside the shower stall my singing would sound weak, since none of the frequencies from the vocal cords resonate in the vocal tract. Inside the stall, however, the frequencies at 330 and 660 hertz create standing waves between the floor and the ceiling, the lower frequency exciting the fourth harmonic and the higher one the eighth. (The sound at 990 hertz should excite the 12th harmonic, but my mouth is at the node for that resonance.) I therefore sound great in spite of the absence of resonance from my vocal tract.

Wanting to experiment with the harmonics in my shower stall but not being able to hold a note for very long and not being skilled in determining the frequency of the note, I substituted for my voice an audio oscillator and a loudspeaker. The oscillator emitted a sinusoidal wave at whatever frequency I set on the dial. The signal, strengthened by an amplifier, was directed to a speaker cone I could put anywhere in the stall. The cone had an aperture of 19 by 12 centimeters; a smaller cone would not emit the low frequencies.

If you decide to experiment in this way, bear in mind that around water an electrical appliance is dangerous. I turned off the water at the valve supplying the shower. Another danger is that the sound of a standing wave excited in the stall can be loud enough to hurt your ears, particularly if the amplifier is set at high volume. I wore a set of headphones of the type that keeps out noise. By tuning the oscillator through the range from about 200 to 1,000 hertz I could hear the resonances come and go. When the oscillator was out of resonance, the sound in the stall was relatively weak. At the resonances it increased, sometimes dramatically.

Keeping the oscillator at some resonant frequency, I could alter my perception of the sound by moving my head around in the stall. As my ears moved from an antinode to a node the level of sound dropped sharply. With a few experimental moves I could usually determine whether the standing wave was between a pair of walls (and, if it was, which pair) or between the floor and the ceiling.

The loudspeaker's efficiency at exciting a standing wave depended partly on how I pointed it. Pointed upward it could only weakly excite a standing wave between the walls even though it was set at the right frequency. I found that when I put the speaker at an angle in a corner on the floor, the sound that spread from the cone was directed more or less correctly for any possible standing wave.

To assess the resonance frequencies more objectively I installed an oscilloscope and a second speaker, which was slightly smaller than the emitting speaker. (A much smaller speaker would fail to pick up the lower frequencies.) This speaker intercepted some of the sound waves in the stall and generated a wave pattern on the screen of the oscilloscope. The vertical component of the pattern reflected the amplitude of the sound wave; the horizontal axis was the oscilloscope's time sweep.

I connected the external trigger ports of the oscilloscope to the audio oscillator so that the sweep across the screen was synchronized with the output from the oscillator. When I set the frequency from the oscillator at a harmonic for the shower stall, the pattern on the oscilloscope increased in amplitude. I measured the range of frequencies exciting a resonance by finding the two frequencies (one below the optimum frequency and one above it) at which the pattern was reduced to half the maximum amplitude.

To map out the standing waves in the stall I moved the receiving speaker either horizontally or vertically between the walls. Looking for a harmonic between the floor and the ceiling, I pointed the speaker upward and moved it vertically. For a standing wave between two walls I pointed the speaker at one wall and moved it back and forth horizontally. In this way I could ascertain whether a resonance was a standing wave between the walls or a standing wave between the floor and the ceiling. To minimize distortion caused by the movement of my body I taped the receiving speaker to a meter stick that I maneuvered in the stall. Even so, my body and the equipment distorted the standing waves enough for me to notice it.

I found and identified most of the harmonics in my table of values except for some of the lower frequencies at which the speakers were inefficient. The frequency at which I got the best pattern on the oscilloscope was not always identical with the theoretical value, but the range around that frequency usually fell across the values in the table. I should have liked to see how the harmonic frequencies shifted as I warmed the air in the stall by turning on the hot water, but for such an experiment the danger of electrocution is too great.

One's perception of the harmonics in a shower stall differs from my more objective assessment of them with electronic apparatus. The ear perceives a pure frequency as being made up of several frequencies because the perceptual - apparatus of hearing does not respond to sound waves precisely. The system is said to be nonlinear, which means that in receiving and processing a signal it distorts it.

Because of the nonlinearity of the hearing process some people can hear frequencies that are not there. Suppose two frequencies reach the ear; the listener not only hears those two frequencies but also may perceive the difference between them. For example, if the two signals are at 500 and 650 hertz, a signal of 150 hertz may also be perceived. Now let fl and f2 stand for the frequencies actually sounded; the hearer may also perceive tones at values of 2fl - f2, 3fl - 2f2 and so on. In other words, signals of low frequency may be perceived when only higher frequencies are actually reaching the ear.

These difference tones might enrich one's singing in the shower. Suppose I sing with the fundamental of my vocal cords at 289 hertz. The integral multiples of the fundamental are 578, 867 and so on. As usual, however, I fail to achieve a match between the fundamental of my vocal tract and the fundamental of the vocal cords. Instead the tract has a fundamental of 330 hertz and thus a third harmonic of 990.

The tract amplifies the sound at 289 hertz badly and at 578 worse, making the singing weak. The sound at 578 hertz, however, excites the seventh harmonic between the floor and the ceiling. A difference tone may then enhance the sound at 289 hertz. The difference between the 289-hertz signal (weakly amplified by my vocal tract) and the 578 hertz signal (amplified by the stall) is 289 hertz. Provided I can perceive a difference tone, the signal at 289 hertz sounds louder than it actually is.

I shall leave further investigation of difference tones to you. If you find interesting effects, l should like to hear from you. I should also like to be told about any flaws in my basic argument about why someone singing in the shower often sounds much better than he does when he is singing somewhere else.

Bibliography

TONE QUALITY. John Backus in The Acoustical Foundations of Music. W. W. Norton & Company, Inc., 1969.

THE PERCEPTION OF TWO PURE TONES. John S. Rigden in Physics and the Sound of Music. John Wiley & Sons, Inc., 1977.

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