The Physics of Flutes
From Croom Physics Wiki
Jenna Stango Mr. Croom Physics 11 31 May 2011
The Physics of Flutes
Physics governs music in every aspect possible. It is responsible for basic sound, tone quality, and intonation. The process of making a flute demonstrated this fully. It is not only a process requiring extensive music knowledge, but also one involving math and physics. Because of the countless aspects of physics involved, it incorporated principles discussed all year. It proved fun, educational, and extremely rewarding.
Flutes are unique woodwinds in their origin of sound. Unlike clarinets and saxophones, they do not have a mouthpiece with a reed to resonate sound. Instead, they depend on a unique embrouchure style. In order to produce sound, the pressure produced by the player’s mouth must be above atmospheric pressure, which is typically 1 atmosphere. This explains why a flute does not make sound when set down on a table. Thus, in order to create sound, a player must use power (work/time [Watts]). When derived, this is equivalent to a mass times acceleration (Force) times a change in displacement over time (P= (ma∆d)/t). The acceleration of the air stream results in higher or lower pitches. The displacement refers to how far the air stream travels down the flute without leaving through any of its holes. In order to make music, a constant supply of energy is given by the player, because energy is lost to friction on the wood or metal of the flute, and some leaves through its holes. The sound technique used in the embrouchure depends on vibrations. This causes resonance, which determines the frequencies of the notes. In a vertical flute, one side of the mouthpiece is sharper than the other. The air stream strikes against it, creating vibrations. This is why horizontal flutists do not cover the complete mouthpiece with their lips. Then, the air strikes against the back of the mouth piece, travelling through the flute. In vertical flutes, like recorders, the mouthpiece is completely covered. Vibrations are made by a second hole below the mouthpiece. The bottom of this hole is filed to a 45̊ angle to create resonance (Wolfe). The velocity of air affects the sound created. For example, a quicker velocity created a high pitch. The velocity increases with an increase in pressure. Pressure equals force per given area (p=F/A). The force increases as a person blows harder, and the area decreases as a person tightens his/her lips. Thus, the pressure is increased. To calculate the vibration of the air stream, one cycle is equal to the time for one pulse to travel over twice the total length of the flute. This accounts for both directions. The speed of sound in this case is “v,” so the frequency is equal to the speed of sound divided by twice the length (v/2L) (Wolfe). In terms of harmonics, their frequencies are in increasing order in terms of “n.” For example, the second harmonic has a frequency equal to (2v/2L), the third (3v/2L), etc. (Wolfe). The picture above shows the first harmonic series on a flute tuned to C. Because its harmonics are in an even ratio (1:2:3:4:5:6:7:8:9:10), it sounds even and pleasing to the ear. This is called consonance. Dissonance is the opposite. Its ratios are not in an equal ratio. In the case of an augmented 4th (C to F sharp), the ratio of the interval is 1: √2. This is why it is the most displeasing interval in history. In fact, it was outlawed by the Catholic Church centuries ago because it was believed to be the sound of Satan (Levitin 74-80). In reality, it is simply physics. When making a copper flute, the first step was choosing a material type that would sound the best. Sound quality in a flute depends largely upon accuracy. The easier a material is to work with, the more accurate the sound quality is. Thus, metals produce a more desired effect. The resonance of the air on the metal also produces different tone colors. Therefore, copper was used instead of wood. In the hierarchy of flute quality, wood is at the bottom. Then, copper, nickel, silver, gold, and platinum follow (Woodwind Materials). For the length of the flute, a pre-purchased wooden flute in the key of C was used as a guide. It was 13 inches long, and each hole was the same distance apart. The size of the holes and distance were determined by an online calculator based on frequency. In order to get sound out, the mouthpiece was first created by sticking a piece of cork in the bore to get a small air hole. Cork is a material commonly used in musical instruments because it has good vibration quality and is durable. Then, a small round hole was cut below the mouthpiece. It was then filed at a 45̊ angle. As discussed above, in flutes, air hits the sharp edge of a mouthpiece. This splits the air column, creating a whistle. The remaining air travels down the flute, creating sound. Experimentally, when the flute was blown into without the angular component, no sound was produced. In comparison to the wooden flute, the copper flute was more in tune. The function of a chromatic tuner is to determine the sound frequencies of an instrument being played. Each note has about a 100 Hz range before it “loses” its indentity. If its frequency is below its ideal frequency in the table below, a note is considered flat. If it is above, it is considered sharp. The wooden flute, because of its inaccuracy due to wood, was about 60 Hz flat. The copper flute, because it was more accurate, was only about 10 Hz sharp.
Note Frequency (Hz) Wavelength (cm)
C4 261.63 132. C#4/Db4 277.18 124. D4 293.66 117. D#4/Eb4 311.13 111. E4 329.63 105. F4 349.23 98.8 F#4/Gb4 369.99 93.2 G4 392.00 88.0 G#4/Ab4 415.30 83.1 A4 440.00 78.4 A#4/Bb4 466.16 74.0 B4 493.88 69.9 C5 523.25 65.9 (“Frequencies For Equal-Tempered Scale”)
The table above shows a chromatic representation of the notes of an octave on the copper flute. It was tuned to C4. Its length was 13 inches. According to the frequency equation, the frequency of C4= v/2L. The speed of sound is 350 meters per second. Therefore, 350 m/s / 2(13in) simplifies to 350/.6604 meters. This equals 530 Hz. However, that is for a full period. When this is divided by 2, it equals 265 Hz for half a cycle. According to the table, this is only 4 Hz away from the expected frequency of C4. Therefore, as stated above, the copper flute is only slightly sharp (Wolfe). In terms of waves, the overall shape of a sound wave from a perfectly enclosed pipe is a sine curve. As a copper pipe takes on more of a flute shape, it deviates more and more from the ideal sine curve. In order for the pressure outside the flute to be at equilibrium, the wave must be symmetrical on both sides. These points of atmospheric pressure are called pressure nodes. Inside the flute, the pressure is not atmospheric because of the applied force (air stream). Therefore, because of the applied force, the maximum amplitude of the applied force occurs in the middle, as shown in a standing wave. This is referred to as a pressure antinode (Wolfe). An ideal standing wave is pictured above. Inside the pipe, the net force is equal to the applied force of the air stream, the force of friction, and the parallel force. The coefficient of friction (μ) is different for every material used. To calculate this, a reference chart can be used. Wood, copper, nickel, silver, etc. all have a different μ value. The parallel force can be caluculated by the force of weight (m x g) times the sin of theta (mgsinθ). The angle that a flute is held at impacts its intonation. For a horizontal flute, if a flutist slouches, theta is affected, throwing off the parallel force. This is why good posture is stressed in music, and slouching often leads to sound falling sharp or flat. As Daniel J. Levitin said in his book This Is Your Brain On Music, “I love music and I love science- Why would I want to mix the two?” This quote applies to this final project. Music depends on science. Without physics, music simply would not exist. Music can be attributed to physics in its technicality, construction, performance, and tuning. All of these were taken into account in the construction of a copper flute. Through this process, I learned a tremendous amount about how deeply physics impacts music, which is a regular thing for me. The experiment was extremely rewarding, and I now look at music in a different way (Levitin 1).
"Does It Matter What It's Made Of?." Woodwind Materials. N.p.,2008. Web. 30 May 2011. <http://members.iinet.net.au/~mtattersall/Articles/Woodwind_Materials.htm>.
"Frequencies for Equal-Tempered Scale." Physics of Music- Notes. Web. 30 May 2011. http://www.phy.mtu.edu/~suits/notefreqs.html.
Levitin, Daniel. This Is Your Brain On Music: The Science of a Human Obsession. 1st ed. United States of America: Plume, 2006. 74-80. Print.
Levitin, Daniel. This Is Your Brain On Music: The Science of a Human Obsession. 1st ed. United States of America: Plume, 2006. 1. Print.
Wolfe, Joe. "Flute Acoustics: An Introduction." n. pag. Web. 30 May 2011. <http://www.phys.unsw.edu.au/jw/fluteacoustics.html>.
Wolfe, Joe. "Strings, Standing Waves, and Harmonics." n. pag. Web. 30 May 2011. <http://www.phys.unsw.edu.au/jw/strings.html>.