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The Sound Cycle
There are many kinds of flutes, and all them as well as the various types of woodwinds have similar physics and associated mathematics to dominate their designs. These musical instruments are shaped to enclose air, some in tubular shapes, and others in tapered or horn-like shapes. They all have a mechanism to produce sound which is adjusted in pitch by the dimensions of the enclosed air column.
In the case of the flute, air is blown upon an edge so that it has energy. At the edge, the airstream is alternately routed into and out of the chamber which encloses the air. This enclosing chamber will be called the "sound chamber". The sound opening at the edge is where air initially enters and leaves the sound chamber.
When air enters the sound chamber, it is compressed until there is enough pressure for the incoming air to be blown outside. The energetic air blown by the flutist is now moving past the opening with the edge. The passing air creates a suction by pulling air molecules from the sound chamber, so that a vacuum forms within the sound chamber (this is called the Bernoulli effect). When the vacuum is strong enough, the stream of passing air is pulled back into the sound chamber and a new cycle is started. The cycles continue for as long as the flutist is blowing against the edge. The frequency of the cycle is the frequency of the sound from the instrument.
As the cycles begin, the air at the foot of the flute (the end farthest from the flutist) reacts so that, after a cycle or two, when air is pressurizing the flute, it enters from both the opening at the edge and at the foot. The same simultaneous action occurs when air leaves the flute. So the sound actually comes from two main sources: the opening at the edge and the foot.
Because of the simultaneous entering and leaving of air from both ends of the sound chamber, there is a place at the theoretical center of the sound chamber which is called the "node". At the node, there is no air motion in a lengthwise direction within the sound chamber. However, there is rotary motion of the air at the node and along the entire length of the sound chamber.
Due to inertia, air entering the flute from the foot of the sound chamber cannot turn abruptly and must adopt a whirlpool-like motion to go into the flute. After a cycle or two, this motion has affected the air along the entire length of the sound chamber.
When leaving the flute at the foot, the air is still whirling but is blown straight outward. So air entering at the foot comes from all points around it in a hemispherical shape, and air leaving at the foot is blown out lengthwise. The result is such that a piece of tissue suspended at the foot, or smoke blown past the foot, will stream outward in a direction parallel to the flute body. The air which causes this effect is the air that leaves the flute on the "exhaust" portion of each cycle. This can be illustrated by placing one's hand behind an operating electric fan. The air behind the fan is coming from the hemisphere behind the fan and is spread out over a large volume. One will not feel much of a breeze from this air because it moves into the fan slowly.
The air in front of the fan is focused in a cylindrical volume where its energy is more concentrated. Here, one can feel a distinct breeze.
The air behind the fan is analogous to the air entering the flute at the foot. The air in front of the fan is analogous to air leaving the foot of the flute. In each cycle, the air entering the flute is given a whirling motion as it arrives from a larger volume, and the air leaving the flute is given a linear motion as it exits from a smaller volume. The overall effect is as if air were being blown along the flute length to exit at the foot - but this not the case.
Some people have erroneously decided that the foregoing effect is caused by "drift air" blown through the flute. Actually, in the flute, there is no drift air, but in a horn there is a small amount of drift air which is caused by the air being blown directly into the horn from the mouth of the player.
The air in the enclosed column of a flute has inertia for the entire theoretical length of the sound chamber. This inertia is directly proportional to the theoretical length, and acts as a form of resistance in the sense that it slows the rate in which air enters and leaves the flute, and therefore lowers the frequency of the sound. So a flute with a longer sound chamber will have a lower frequency when all the playing holes are closed.
Page 2 - Flute Physics
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The cross-sectional area of the sound chamber acts as an allowance in the sense that a larger area provides greater access for air to enter and leave the sound chamber. So a sound chamber with a large cross-section will have more air entering and leaving within a particular time interval provided that the larger cross-section is extended to the foot itself.
A larger cross-section does not increase the frequency but is a means of helping to prevent an octave shift from overblowing.
Resonance and Ratios
There must be a correct range of ratios between the theoretical length of the sound chamber and its cross-sectional area. This is because the rate of pressurizing and evacuating the chamber must be such as to coincide with the speed of sound within it for resonance to occur. When there is excessive length in proportion to the cross-sectional area, no sound will come from the flute because there is too much inertia present and too little access for the air to enter and leave. When there is too much allowance and not enough inertia, there is too little containment of air for it to act as a cohesive column. In each case, the flute will simply whisper. In the second case, before approaching the whisper stage, the flute will fail to shift octaves but will have a rich sound.
Holes in the wall of the sound chamber act as separate enclosed cylinders so that they become parallel paths for air to enter and leave the chamber. When computing the effect of the holes, the equations are very similar to those used in electrical engineering for parallel paths. Like the sound chamber itself, each hole must have dimensions within a certain range for resonance to occur. Too small a hole for the wall thickness will cause the flute to whisper. and too large a hole for the wall thickness will have the same effect as shortening the flute to the point where the hole is located. The theoretical wall thickness at the hole is like the length of the sound chamber and the area of the hole is the like the cross-sectional area of the sound chamber.
The theoretical length for the sound chamber and for the wall thickness at a playing hole are not the same as the actual length and actual thickness. This is because a false length exists at each end of the sound chamber and on each side of the hole.
There is another factor that is sometimes involved when working with length, and that is the spiral motion of the air. This motion causes the length to increase because the spiral motion causes the air to travel farther than it would were it to move in a straight line.
The derivation of the math and equations used in flute design are too complex to show on a small webpage. Designing a flute by using the equations can take in excess of six weeks. Consequently, over a six-year period, a computer program was developed which allows a tailored flute design to be created in less than ten minutes. This program uses the same time-consuming equations, but solves them very quickly, and has revolutionized the designing of various flutes from the simple Middle Eastern to the more complex Native North American types.
More About Flute Theory
For those of you who are not familiar with any type except the European mechanical orchestral flute, some other more common types found about the world are the panpipes (ancient closed-end bound pipes found throughout the globe), the kena (ancient South American notched endblown), the nay (ancient Middle Eastern diagonal), the shakuhachi (ancient Japanese notched endblown), the bansuri (ancient Indian transverse), the ti (ancient Chinese transverse with a "buzzer"), the fuye (ancient Japanese transverse), the fife (ancient Keltic transverse), the ocarina (ancient South American vessel type), the tin whistle (Keltic block type), the recorder (European Renaissance block type), the flageolet (French and English 17th century block type), and the love flute (ancient Native North American block type).
Additions to Books
Since my books on flute physics were published, I ran across some old notes in which there is some information that was not published. There are times when it is advantageous to cut the foot of the flute at 45 degrees. This is especially true when working with multi-barrel flutes (flutes with more than one barrel). When the flute foot is cut at 45 degrees, one must know where the end of the flute is in theory, the end being where the flute foot would be if the flute were cut at the usual 90 degrees to the barrel length.
The inside end of the flute, when cut at 45 degrees, looks like an ellipse with a major axis (like a long diameter) and a minor axis (the actual barrel diameter). The major axis is the one measured on the 45 degree slant. The point at the end of the major axis which is nearest to the flute player (proximal end of the major axis) may be called point "A". The point at the end of the major axis which is farthest from the flute player (distal end of the major axis) may be called "B".
The theoretical end of the flute for the the 45 degree foot is approximately .4242 times the length of the major axis from point "A" toward point "B" measured on the 45 degree slant. If the flute is cut with a slant between 45 degrees and 90 degrees, the distance along the major axis increases from .4242 toward .5 times the length of the major axis. If the flute is cut with a slant between zero and 45 degrees, the distance along the major axis decreases from .4242 toward zero times the length of the major axis.
Variations in the flute barrel thickness may cause very minor changes from .4242 for the 45 degree cut.
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