What is the first overtone frequency for an organ pipe 2.00 m in length, closed at one end?

1 Answer
Dwight
Jan 27, 2018

The frequency of the first overtone is 127.5 Hz

Explanation:

If the pipe is closed at one end, there must be a pressure node at this point and an anti-node at the open end. This means the fundamental frequency that will stand in the pipe is a wave with a wavelength four times as long as the pipe (as the pipe holds only 1/4 of the wave. The first overtone is a wave in which 3/4 of the wave fits inside the pipe. Put another way, the wave is 4/3 as long as the pipe.

See the diagram to clarify this (I hope). The first overtone here is called the third harmonic:

www.acs.psu.edu/drussell/Demos/StandingWaves/Opentubeoneend.jpg

#lambda_2 = (4L)/3# where L is the length of the pipe.

Since frequency is #f=v/(lambda)#, the first overtone frequency will be

#f_2 = v/ (lambda_2) = (3v)/(4L)#

where v is the speed of sound in the pipe.

If we take 340 m/s as the speed of sound, the fundamental tone will have frequency

#f_2 = (3xx340)/(4xx2.0) = 127.5# Hz

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