IOP Physics Education Phys. Educ. 52 P A P ER Phys. Educ. 52 (2017) 025003 (9pp) iopscience.org/ped 2017 Flutist produces four resonances © 2017 IOP Publishing Ltd with a single bottle PHEDA7 Michael J Ruiz1,3 and Erika Boysen2,3 025003 1 Department of Physics, UNC Asheville, Asheville, NC, USA 2 College of Visual and Performing Arts, UNC Greensboro, Greensboro, NC, USA M J Ruiz and E Boysen E-mail: [email protected] and [email protected] Flutist produces four resonances with a single bottle Abstract In a dramatic physics demonstration, a professional flutist produces four Printed in the UK resonances with a 12 ounce Boylan soda bottle solely through her breath control. The 22 cm bottle acts like a Helmholtz resonator for the lowest PED pitch. The three higher pitches fall near the 3rd, 5th, and 7th harmonics for a 22 cm closed pipe. A video of this remarkable feat is provided (Ruiz 2016 YouTube: Four Resonances with a 12-ounce Soda Bottle (https://youtu.be/ 10.1088/1361-6552/aa532b ibtVrp2NF_k)). The video also reveals that a flutist can bend resonance pitches by as much as 10% through control of air speed. 1361-6552 The resonances and experimental data and antinodes, spaced at quarter wavelengths Published Coauthor Erika Boysen, professional flutist and apart throughout the entire length of the pipe. The Assistant Professor of Flute at the University closed-pipe standing waves occur at frequencies of North Carolina at Greensboro (NC, USA), which are odd multiples of the fundamental stand- 3 is shown in figure 1 producing the lowest reso- ing wave. After discussing the frequency data, the nance on a soda bottle. The bottle she is holding mathematics of the Helmholtz and closed-pipe 2 is made by Boylan (www.boylanbottling.com), models will be used to analyze the data. has a height of 22 cm, and holds 12 fluid ounces. A total of four resonances are produced by In a later section the dimensions of the bottle are the flutist varying her air direction, speed and used to compare experimental results with phys- embouchure (lip position) formation. Watch the ics models of resonance. video [1] of reference 1 to witness the difficult There are two applicable models for the and entertaining achievement of these resonances. bottle resonances: (1) the Helmholtz resona- The frequencies for the resonances are provided tor and (2) closed-pipe harmonics. For an ideal in figure 2 along with the closest note on the piano Helmholtz resonator, think of a large jug with keyboard for each case. a small neck. The Helmholtz resonance of the system occurs because the air in the small neck Finding the frequencies region of the bottle acts like an oscillating mass Three methods are used to measure the resonance attached to a spring produced by the large vol- frequencies: (1) matching tones with an online ume of air in the cavity below the neck. In con- keyboard [2], (2) listening to beats with an online trast to the Helmholtz resonance of neck air-mass tone generator [3], and (3) employing the free oscillation, closed-pipe resonances are standing Audacity frequency spectrum analyzer [4]. These waves with the usual alternating pressure nodes methods are demonstrated in the video abstract [1]. 3 Website: www.mjtruiz.com and www.erikaboysen.com. While the Audacity software package gives the 1361-6552/17/025003+9$33.00 1 © 2017 IOP Publishing Ltd M J Ruiz and E Boysen best results, introducing students to the other approaches allows the teacher to make a con- nection to music and demonstrate the physics of beats. Using the piano also shows students that the keyboard can be used as a frequency-measuring device. However, both of these methods have some limitations. Keyboard Matching tones with an online keyboard [2] is fun and relates the resonances to musical tones. But the keyboard method in general can only give approximate values as the actual resonance fre- quencies may not fall exactly on piano keys. Beats For the method of beats, when two frequencies Figure 1. Professional flutist and coauthor Erika f1 and f2 are near each other ( ff12≈ ), the aver- Boysen producing the lowest bottle resonance. ff age pitch f = 12+ is heard pulsating at the tone 2 technique if that is more convenient. A student in difference of the frequencies: the beat frequency the class with a sense of rhythm should be able to ffbeats2=−f1, where ff21⩾ . A clip of one of estimate the frequency pair fairly quickly without the resonances is played simultaneously with needing a clock or timer. Students can repeat the online tone generator [3] in the video of [1]. the experiment a few times and then discuss the When the frequency of the tone generator is near uncertainty in Hz for the measurement. the resonance, beats are heard. The frequency slider of the online oscillator can then be adjusted Audacity until the beats stop. At this point the reference tone matches the pitch of the unknown frequency. The best method for frequency determination A second video [5] is included where the employs the frequency spectrum analyzer in audio of each bottle resonance shown in figure 2 the free audio software Audacity [4]. As illus- is looped for one minute so that students can trated in the video abstract [1], a recorded *.wav explore measuring the frequencies using the (or *.mp3) version of the sound file is opened with online keyboard [2] and tone generator [3]. Due Audacity and a segment of the wave shown on to the looping of the sounds, it can be challeng- the Audacity display screen is selected. From the ing to zero in on the frequency using beats since menu, ‘Analyze’, is chosen, then ‘Plot Spectrum’ there are glitches from the splicing of the repeated is selected. Next, the number of frequency loops. However, one can employ a beats trick here divisions is set to 4096 and the mouse is placed in order to do the measurement over the glitches. over the tallest peak, the resonance frequency. Find the frequencies at which the beat frequency The spectrum display indicates the measured is 4 Hz on either side of the unknown frequency. value along with the closest corresponding note These rapid beats are easier to identify against on the piano. distracting background noise (glitches). Consider The experimental results using Audacity an example where the unknown pitch has a fre- are 192 Hz, 1198 Hz, 1968 Hz, and 2632 Hz, as quency of 400 Hz. Finding the pair of frequen- reported in figure 2. Comparisons of the last three cies where the beat frequency is 4 Hz will lead to values, 1198:1968:2632, are close to the ratios frequencies 396 Hz and 404 Hz on the tone gen- 3:5:7, indicating odd harmonics characteristic erator. The average of these frequencies is the fre- of a closed pipe. In the following sections physi- quency of the unknown. Instead of 4 Hz, you can cal models are used to calculate the measured use another beat frequency such as 3 Hz with this frequencies of the bottle resonances. The first March 2017 2 Phys. Educ. 52 (2017) 025003 Flutist produces four resonances with a single bottle C3 C4 A4 C5 C6 C7 C8 CDEEFFGGAABBCCDDEEFFGGAAB CCDDBBE F G A C 196 Hz keyboard pitches 1175 Hz 1976 Hz 2637 Hz G3 D6 B6 E7 192 Hz bottle resonances 1198 Hz 1968 Hz 2632 Hz Figure 2. Measured values for the four resonances, compared with piano keyboard frequencies. Middle A on the piano, to which orchestras commonly tune, is A4 = 440 Hz. consideration is the lowest resonance listed in figure 2, which has an experimental value of A = area f = 192 Hz. The bottle is taken to be a Helmholtz A resonator for this resonance. L The Helmholtz resonator The Helmholtz resonator applies to a large bottle with a small narrow neck. See figure 3. The large V0 volume of air V0 behaves as a spring driving the smaller mass of air in the neck like a harmonic oscillator. The classic Helmholtz formula for the frequency is given by vA f = , Figure 3. Helmholtz resonator. The large volume of 2π VL0 air designated by V0 acts like a spring that drives the smaller mass of air in the neck region (volume AL) as a where v is the speed of sound, A is the area at the harmonic oscillator. top of the bottle, L is the length of the neck, and V is the larger volume of air in the bottle. For a 0 The Boylan soda bottle does not have the derivation of this formula, see [6, 7]. ideal shape for a Helmholtz resonator. A big Authors today often modify the original empty apple-cider jug would be a much better Helmholtz formula to incorporate end-correction fit. Nevertheless, the Helmholtz model will be effects by taking the effective length of the neck applied for the lowest resonance. See figure 4 to be Lr+ 0.6 , where r is the inner radius at the for the dimensions of the 12 ounce Boylan soda top of the bottle [7, 8]. Replacing L with the effec- bottle. tive length, gives the frequency The soda bottle is idealized with rectangular vA and trapezoidal cross sections. A volume of 12 f = . fluid ounces is equivalent to 355 millilitres (ml). 20π VL0()+ .6r But while the model (figure 4) gives a calculated March 2017 3 Phys. Educ. 52 (2017) 025003 M J Ruiz and E Boysen Figure 4. The 12 ounce Boylan soda bottle, modeled by rectangular and trapezoidal cross sections.