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The first three natural frequencies of an organ pipe are 126.0 Hz, 378.0 Hz, and 630.0 Hz. (a) Is the pipe open or closed? (b) Taking the temperature of the surrounding to be 25.0oC, find the length of the pipe.
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Added Sat, 04 Jul '15
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Answer: (a) Closed pipe (b) Length = 0.6746 metres

Explanation:
(a)
Now we have to understand that an open pipe would have natural frequencies corresponding to N =1, 2, 3, 4 ..
but a closed pipe (closed at one end ofcourse) would have natural frequencies corresponding to N =1, 3, 5, 7 ..
So to determine if the pipe is open or closed we just have to see if the second natural frequency is twice the first (depicting open pipe) or thrice the first (depicting closed pipe).
Since 378 Hz = 3 * 126 Hz and similarly 630 Hz = 5 * 126 Hz
we know that N = 1, 3, 5 and so on
Hence it is a closed pipe.

(b)
Its given that these are the first three natural frequencies. So we can use the formula
frequency in Hz = (N * velocity of sound) / 4 * Length of the pipe
where N denotes the harmonic produced only for closed pipes

So we have
126 = (1 * 340) / (4 * L)
=> L = 340 / (4 * 126) = 340 / 252 = 0.6746 metres

(Be careful as this formula is valid only for the closed pipe)
The formula for an open pipe would be
frequency in Hz = (N * velocity of sound) / 2 * Length of the pipe

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Added Sat, 04 Jul '15
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