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Harmonics in Organ Pipes

Author: N.H. Crowhurst

Variation in overtone structure makes different stops on an organ give different tonal qualities. In a pipe organ, the wide pipes produce tones in which the fundamental predominates. Their sounds are deep and smooth. The thinner pipes of the same length produce many more harmonics than the wider pipes, and give full-bodied sounds. Still thinner pipes suppress the fundamental and give a thin or reedy tone. In an electronic organ, the harmonics put in (or left out) are controlled electronically, giving a similar variety of "tone color."

The pattern of overtones produced by a pipe also depends on whether the far end of the pipe is open or closed. Acoustic vibration is set up in the air column by a sheet of air directed toward the upper lip of the pipe. When this air passes inside the lip, it starts to compress the air inside the pipe. This inside pressure soon forces the air out again. As a result, the blown air alternately goes inside and outside the pipe, with a considerable back and forth movement of air particles occurring at the pipe mouth.

The frequency of oscillation at the mouth is determined by the pipe. The pressure wave travels up the pipe at the speed of sound. When it gets to the far end, if the end is closed, the pressure increases because it cannot be passed on. The pressure wave is therefore reflected back toward the mouth of the pipe. When it reaches the mouth, the air directed against the lip is forced to go outside the pipe. Since this represents half of a complete cycle and the pressure wave has traveled twice the length of the pipe, it is clear that a closed organ pipe is one quarter-wavelength long for its lowest frequency.

Tone generation in a closed pipe

The air at the mouth of a closed pipe can move freely, whereas the air at the far end cannot move at all. Thus the lowest (fundamental) frequency of such pipes is determined by the length of a wave that travels up to the top, produces increased pressure, and travels back down the pipe to produce outward motion of air at the mouth.

harmonics in a closed pipe
Harmonics in a closed pipe

Any wave whose length is such that the return journey gets it to the mouth when the blown air is moving in the opposite way to the initial pulse - where the pipe is *4 wavelength long, |4 wavelength long, 1J4 wavelengths long, and so on - will tend to be present in the pipe. These wavelengths correspond to the odd harmonics of the fundamental. The even harmonics (whose wavelengths are such that the pipe is iy2 wavelength long, 1 wavelength long, etc.) are absent. Notice that, at the mouth and for waves other than the fundamental, at half-wave distances along the pipe, the movement of air in the going wave adds to that in the return wave, to produce a point where air movement is a maximum. At the closed end, and for waves other than fundamental, at half-wave distances from the closed end, the air pressure in the going and returning waves adds, producing a point where air pressure fluctuation is a maximum.

If the far end of the pipe is open, the pressure will drop suddenly because the wave is no longer confined in the pipe. In this case, the pressure wave is reflected as a rarefaction. When the reflected wave reaches the mouth, the air blown against the lip is directed into the pipe again and thus completes a full cycle while the wave and its reflection have traveled twice the length of the pipe. Thus the fundamental frequency of an open pipe is such that the pipe is half a wavelength long. Any wave whose length is such that the length of the pipe is an exact multiple of a half-wavelength - 1 wavelength, iy2 wavelengths, etc. - can be sustained in an open pipe. These wavelengths correspond to all the harmonics of the fundamental frequency, and an open pipe produces a complete harmonic series.

Tone generation and harmonics in an open pipe

Just as in the closed pipe, there will be points where the pressures add up, and points where movements add up. Because of this action inside pipes, the waves do not seem to move. At the points where pressure has a maximum fluctuation, as at the end of a closed pipe, the air does not move, but changes in pressure according to the sound wave. At the points where movement is a maximum, as at the open end of an open pipe, or the mouth of either, there is little pressure fluctuation and a maximum of movement. For this reason, waves of this kind are called standing waves.

Last Update: 2011-04-01