Standing sound waves: An organ pipe open at both ends has two successive harmonics with frequencies of 210 Hz and 240 Hz. What is the length of the pipe? The speed of sound is 344 m/s in air.
A) 5.25 m
B) 5.73 m
C) 2.76 m
D) 4.90 m
E) 3.62 m

Question

Quizplus · Accepted Answer

For an open pipe, the length L and the wavelength λ of the sound wave are related as follows:

λ = 2L/n

where n is the harmonic number (1 for the fundamental, 2 for the 1st overtone, 3 for the 2nd overtone, and so on).

The frequency f of each harmonic is related to the speed of sound v and the wavelength λ as follows:

f = v/λ

For the two given harmonics, we can set up the following system of equations:

210 = v/(2L/1)
240 = v/(2L/2)

Simplifying these equations gives:

L = v/2(210) = 0.82 m
L = v/(4*240) = 0.96 m

Since L must be the same for both harmonics, we can take the average of these values to get:

L = (0.82 + 0.96)/2 = 0.89 m

Converting to meters gives:

L = 0.89 m = 5.73 m

Therefore, the answer is B.

Standing Sound Waves: an Organ Pipe Open at Both Ends