A friend in another city tells you that she has two organ pipes of different len

Question

A friend in another city tells you that she has two organ pipes of different lengths, one open at both ends, the other open at one end only. In addition, she has determined that the beat frequency caused by the second-lowest frequency of each pipe is equal to the beat frequency caused by the eighth-lowest frequency of each pipe. Her challenge to you is to calculate the length of the organ pipe that is open at both ends, given that the length of the other pipe is 0.90 m. Note that there are two possible answers to this question. List them both.

Shorter pipe:

Longer pipe:

Explanation / Answer

Let f1 be the fundamental frequency of the open pipe, and f2 the fundamental frequency of the closed-end pipe. The lowest frequency is the fundamental, the next (second-lowest) is the first harmonic, or 2x the fundamental. The third lowest is 3x the fundamental. The beat frequency is the difference between the two when sounded together, so

2*f1 - 2*f2 = 3*f1 - 3*f2

since f2 = v/4L, and L is given as 0.9m f2 = v/4. Put this in the above equation:

2*f1 - v/2 = 3*f1 - 3*v/3.6.

Solve for f1

f1 = (3/4 - 1/2)*v

f1 = v/3

since f1 = v/(2L), v/3 = v/(2L)

L =3/ 2 m=1.5m

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