Page 53 - Spring 2018

Figure 1. a: Common orchestral woodwinds in descending order of pitch (from bottom): piccolo, flute, oboe, its larger version the cor anglais, clarinet, and bassoon. b: A flute head joint. A jet of air leaves the player’s lips and crosses the open embouchure hole to the edge on the far side. c: The clarinet mouthpiece is almost closed by the reed. Side (d) and end (e) views of double reeds of the oboe, cor anglais, and bassoon. Photographs courtesy of Mike Gal.
The Bore, Its Resonances,
and Its Impedance Spectrum
The ducts of the larger clarinets, saxophones, and bassoon are bent to make tone holes and keys easier for the player to reach. These bends have only a modest acoustical effect. For most flutes and the clarinet, much of the duct is approxi- mately cylindrical, whereas for the oboe, saxophone, and bassoon, the duct is nearly conical.
Let’s begin by considering a completely cylindrical bore (Figure 4), open to the air at both ends, a case approximated by the flute (Figure 1a) or the hybrid instrument in Figure 5, bottom. Inside the bore, the acoustic pressure can vary sig- nificantly, positive or negative. At the open ends, however, the acoustic pressure is small; the total pressure is close to atmospheric. So, if we look for resonant modes of oscilla- tion in the bore, the boundary condition at both ends is a node for pressure and freedom for large flows in and out. The mode diagrams in Figure 4 plot acoustic pressure (red) and acoustic flow (blue) against position along the bore. The diagram in Figure 4, top left, shows that half a sine wave fits the bore with a pressure node near each end, allowing a low- est mode whose wavelength (λ) is roughly twice the length (L) of the pipe, say λl 2L.
From its open embouchure hole to the other open end, the
not surprise us because the instrument is neither exactly cy- lindrical nor completely open at the mouthpiece end.
Considering the five mode diagrams at Figure 4, left, we see that the zero-pressure boundary conditions near the ends enclose, respectively, 1⁄2, 1, 11⁄2, 2, and 21⁄2 wavelengths. Using n for the mode number, the wavelengths and frequencies are thus approximately
(1)
Frequencies in the ratio 1:2:3, and so on, make the harmonic series. With a flute whose lowest note is C4 (262 Hz), a player can change the air-jet speed and length, thus exciting the bore to vibrate at frequencies nf1, producing the notes with f1 (C4), 2f1 (C5), 3f1 (G5), 4f1 (C6), 5f1 (E6), 6f1 (G6), and 7f1 (a note be- tween A6 and A#6), all without moving the fingers (see sound files and video at www.phys.unsw.edu.au/jw/AT).
A further important point is that a nonsinusoidal periodic sound, with period T = 1/f1, contains harmonics with fre- quencies nf1. So, for a low note on the instrument operating at f1, the resonances at 2f1, 3f1, and possibly higher multiples help radiate power at these upper harmonics of the sound and contribute to making the timbre brighter.
A clarinet is also roughly cylindrical but, unlike a flute, it is almost completely closed at the mouthpiece by the reed (Figure 1c). Figure 4, right, shows the modes; this bore can accommodate 1/4, 3/4, 5/4, and so on, wavelengths
(2)
nearly cylindrical flflute in F
Fi
gu
ig
ah
ur
re
e1
s a length of 0.63 m. ThThThe frequency , where c is the speed of sound. This is a little higher than its lowest note, B3, at 247 Hz, played with all the tone holes closed. The difference should
1a
a
ha
as
Spring 2018 | Acoustics Today | 51
=≈