The Physics of Woodwinds Collapse
Alma C. Zook
Pomona College
All wind instruments have the same basic structure: the mouthpiece, which is the initial source of sound waves, is joined to the tube, or bore, of the instrument and with it determines the final pitch of the tone being played (Fig. 1). In addition, brass instruments and some woodwinds have a bell at the end of the bore that radiates the sound into the air.
All wind instruments except the flute use some form of reed in the mouthpiece (Fig. 2a). For brass instruments, the "reed" is the player's lips; for the woodwinds, the reed is one or more pieces of cane. The discussion that follows describes the behavior of the oboe reed, but it may be applied to any reed. The reed of an oboe is actually a double reed, as indicated in the side view in Fig. 2b. Before the instrument can be played, the reed must be thoroughly moistened to make the cane flexible. When moist, the cane behaves like a spring with a relatively low spring constant.
Sound production begins when a player places the reed in his or her mouth and pushes air through it. As a result, the speed of the air between the blades of the reed is increased from zero to some value, which in turn causes the air pressure inside the reed to decrease. The air pressure in the player's mouth, however, is equal to or slightly greater than the ambient air pressure. The difference in pressure between the inside and outside of the reed forces the blades together, stopping the flow of air through the reed. Once the air flow is cut off, the air speed inside the reed drops to zero, and the air pressure inside rises to its original value. At this point the pressure difference between the inside and outside of the reed is zero, the reed springs back to its equilibrium (open) configuration, and the cycle repeats itself.
The rapid opening and closing of the reed blades results in alternating compressions and rarefactions inside the bore of the instrument -- that is, sound waves. But the vibration of the reed is irregular and not particularly pleasant to hear, because it contains many frequencies with no particular relationship to each other. Converting the sound wave from the vibrating reed to musical sound requires that most of the sound energy be filtered out, leaving only frequencies that are integer multiples of some lowest, or fundamental, frequency. (Such frequencies are referred to as "harmonically related.") That filtering process is carried out by the bore of the instrument.
Although many frequencies are present in the sound generated by the reed, the frequencies primarily responsible for the musical sound of the instrument are those that can set up standing waves inside the bore. (Standing waves are discussed further in the next chapter.) Any other frequencies interfere destructively with themselves as they reflect off the two ends of the instrument. In this respect, the inside of a musical instrument is like a laser cavity, which also uses the occurrence of standing waves to remove all but one desired frequency with a laser, of course, the wave is light rather than sound.
The particular frequencies that can set up standing waves in a wind instrument depend on its geometry. The clarinet has a cylindrical bore for most of its length and behaves approximately like an air column that is closed at one end (the reed end) and open at the other. Its natural frequencies are therefore given by
where n is an integer, v is the speed of sound in air, and L is the distance from the reed to the first open tone hole. The fundamental frequency corresponding to n = 1 is
and all other natural frequencies are odd multiples of the fundamental frequency. We would therefore expect the sound from a clarinet to have only odd harmonics in its spectrum, and this is in fact approximately the case, at least in its lowest register Fig. 3. The absence of the even harmonics in the spectrum of the clarinet gives its sound a "hollow" quality.
The oboe, on the other hand, has a conical bore, with the reed at the narrower end (which is quite narrow ÷ about 5 mm). The conical bore causes the instrument to act acoustically like an air column that is open at both ends, with natural frequencies
where n is an integer. So the two oboe differs acoustically from the clarinet in two important respects. Although the oboe and clarinet are about the same length, the oboe's lowest possible note, corresponding to n = 1, has a fundamental frequency about twice that of the clarinet. Therefore, the clarinet can play pitches almost an octave lower than the oboe can. The oboe makes up for its more restricted pitch range by having a richer harmonic spectrum, containing all integer-multiple frequencies of the fundamental, instead of just the odd multiples Fig. 4.
Clearly, real musical instruments are not just simple cylinders and cones: the bell and tone holes introduce a complicated geometry that in turn leads to the particular balance of the harmonics we associate with a given instrument. In addition, neither the reed nor the air in the bore is a perfectly linear system, which causes the spectrum of an instrument to depend on the loudness at which it is played. Nevertheless, much of the behavior of any wind instrument can be explained using familiar physics.
Suggested Readings
Backus, J., The Acoustical Foundations of Music, New York, Norton, 1977.
Benade, A.H., "The Physics of Woodwinds," Scientific American, October 1960.
Benade, A.H., "The Physics of Brasses," Scientific American, July 1973.
Fletcher, N.H., and S. Thwaites, "The Physics of Organ Pipes," Scientific American, January 1983.
Hall, D.E., Musical Acoustics: An Introduction, Belmont, CA, Wadsworth, 1980.
Problems
where v is in m/s and T is the Celsius temperature therefore, blowing through an instrument raises the temperature of the air inside it, unless the room is very warm. For a certain fingering, a clarinet has an effective length of 24.6 cm. This fingering sounds the pitch F4 (f = 349.2 Hz) at 20¡C. During a concert, the air temperature inside the bore rises to 27¡C.
(a) What new frequency will the clarinet sound for this fingering?
(b) What is the percent change in the frequency? (Note that a 1% change in frequency would be readily noticeable to a listener.)
Figure Captions
Figure 1 Woodwind structure.
Figure 2 The oboe reed.
Figure 3 Clarinet spectrum.
Figure 4 Oboe spectrum.