Question

An artisan designs wind instruments. She wishes to designs a pipe (one end closed, one end open) such that the fundamental frequency will correspond to middle C (261.62 Hz). The speed of sound in air is 330 m/s in the region.

(a) What should be the length of the pipe?
(b) What is the frequency of the first overtone?
(c) She now wishes to design a second pipe with fundamental frequency at treble C -- a note that is one octave higher, i.e., a note that is double the frequency of middle C. What should be the length of the second pipe?
(d) What is the frequency of the first overtone in the second pipe?

Answer 1

To produce a fundamental frequency of middle C (261.62 Hz), the length of the pipe should be approximately 0.6316 meters. The frequency of the first overtone is approximately 784.86 Hz. For a second pipe with a fundamental frequency at treble C (double the frequency of middle C), the length should be approximately 0.3158 meters. The frequency of the first overtone in the second pipe is approximately 1570.72 Hz.

Step-by-step explanation:

(a) To determine the length of the pipe required to produce a fundamental frequency of 261.62 Hz (middle C), we can use the formula:

L = (v/2f)

Where L is the length of the pipe, v is the velocity of sound in air (330 m/s), and f is the frequency.

Plugging in the known values, we get:

L = (330 m/s)/(2*261.62 Hz) = 0.6316 m

Therefore, the length of the pipe should be approximately 0.6316 meters.

(b) The frequency of the first overtone can be calculated using the formula:

f1 = 3f0

Where f1 is the frequency of the first overtone and f0 is the fundamental frequency.

Substituting the values, we get:

f1 = 3*261.62 Hz = 784.86 Hz

Therefore, the frequency of the first overtone is approximately 784.86 Hz.

(c) For a second pipe with a fundamental frequency of treble C (double the frequency of middle C), the length can be calculated using the same formula:

L = (v/2f)

Substituting the values, we get:

L = (330 m/s)/(2 * 2 * 261.62 Hz) = 0.3158 m

Therefore, the length of the second pipe should be approximately 0.3158 meters.

(d) The frequency of the first overtone in the second pipe can be calculated using the same formula as before:

f1 = 3f0

Substituting the values, we get:

f1 = 3*2*261.62 Hz = 1570.72 Hz

Therefore, the frequency of the first overtone in the second pipe is approximately 1570.72 Hz.