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| Modes of Vibrations in Pipes |
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| An organ pipe is the simplest form of a wind instrument. Figure (a) shows the longitudinal section of an organ pipe whose one end is closed and figure (b) shows an organ pipe, both ends of which are open. It consists of a hollow tube BD in which air can be blown through a pipe A (also called the mouthpiece). |
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| The air moves through a narrow slit B and strikes against the sharp edge C, called the lip. This lip vibrates and sets up vibration in the air column enclosed in the pipe. These vibrations travel to the other end of the pipe and get reflected. Due to superposition of the incident wave and the reflected wave, longitudinal stationary waves are formed. The frequency of the note produced, depends mainly on the length of the pipe and the type of the pipe, i.e., whether it is closed or open. In a closed pipe, the end D is always a seat of node and in an open pipe, the end D is always a seat of antinode. In both the pipes, the end B is always the seat of an antinode. |
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In the simplest or the fundamental mode of vibration, the air column vibrates with an antinode A at the open end and a node N at the closed end as shown in figure (a). Since the distance between
a node and an antinode is  , the length of the tube l in
this case will be equal to . |
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| If v is the velocity of sound, then |
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| The frequency f1 of the fundamental note is called the |
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| 'first harmonic'. |
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| Figure (b) shows the first overtone in a closed pipe. Two nodes and two antinodes are formed. The wavelength and the frequency of the sound corresponding to this mode of vibration will be different from those corresponding to the fundamental mode. Let the wavelength be l2 and the corresponding frequency be f2. Then, as seen in the figure, |
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| Figure (c) shows the formation of the second overtone with 3 nodes and 3 antinodes. If l3 and f3 are the corresponding wavelength and frequency, then |
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| Similarly, it can be shown that |
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| Thus, in a closed pipe, only harmonics proportional to the odd natural numbers are present. Therefore, the quality of the note given out by a closed pipe lacks in fullness. |
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| In an open pipe, when a compressed wave reaches the far end, the air at that point is, for an instant, at a pressure greater than the atmospheric pressure. Being an open end, the air there can vibrate with maximum freedom and so, it suddenly expands into the surrounding air. Thus, the pressure diminishes so quickly that it becomes lesser than the pressure of the surrounding air, which causes a sudden rarefaction at the end of the pipe. This sets up a rarefied wave which passes back along the pipe. Within the tube, the reflected pulses meet the direct ones and the result is the formation of the standing waves. |
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| Figure (a) shows the fundamental mode of vibration with two
antinodes and one node. If l1 and f1
are the wavelength and frequency of the sound producing this mode of
vibration, then |
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| f1 gives the frequency of the first harmonic. |
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| The first overtone formation is shown in figure (b). Three antinodes and two nodes are formed. From the figure, |
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| Figure(c) shows the formation of the second overtone. Four antinodes and three nodes are formed. |
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| Similarly, it can be shown that |
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| The frequencies of the harmonics present in an open pipe are proportional to the natural numbers. Owing to the presence of all harmonics, the quality of the note given out by an open pipe is richer and sweeter than that given out by a closed pipe. |
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The fundamental frequency of vibration in a closed pipe is given by  and
for the same length of the tube, the fundamental frequency in an open pipe
is given by  . Therefore, the fundamental
frequency of an open pipe is said to be an octave higher than that of the closed pipe. |
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