A drainage pipe running under a freeway is 30.0 m long. Both ends of the pipe are open, and wind blowing across one end causes the air inside to vibrate.

a. If the speed of sound on a particular day is 340 m/s, what will be the fundamental frequency of air vibration in this pipe?

b. What is the frequency of the lowest harmonic that would be audible to the human ear?

c. What will happen to the frequency in the later afternoon as the air begins to cool?

Answer: fundamental frequency = 5.67Hz

Frequency at lowest harmonic relative to the audible range = 22.6Hz (fourth harmonic)

Sound frequency is independent on temperature.

Explanation:

a) pipe is an open one with length 30.0m

The relationship between the length of air and wavelength is given by the formulae below.

L = λ/2

λ = 2L = 2 * 30 = 60m

v = speed of sound = 340m/s

To get the fundamental frequency, recall that

v = fλ

f = v/λ

f = 340/60

f = 5.67Hz.

b) the audible range is 20Hz - 20,000Hz

The first harmonic produced a sound at frequency 5.67Hz.

At the second harmonic, the length of air is related to the wavelength of sound sound by the formulae

L = λ

λ = 30m

But v = fλ where v = 340m/s and λ = 30m

Hence f = 340/30

f = 11.33Hz

At the third harmonics, the length of air and wavelength is related by the formulae below

L = 3λ/2

30 = 3 * λ/2

60 = 3λ

λ = 20m.

But v = fλ where v = 340m/s and λ = 20m

f = v / λ = 340/20.

f = 17Hz

At the fourth harmonic, the length of air and wavelength are related by

L = 2λ

λ = L/2

λ = 30/2

λ = 15m.

But v = fλ where v = 340m/s and λ=15m

f = v/λ

f = 340/15

f = 22.7Hz.

You can see that the 4th harmonic is slightly above the lowest value of the audible range (20Hz).

Hence the frequency of the lowest harmonic audible to the human ear is at the fourth harmonic where f = 22.7Hz

c) the frequency of sound is independent on temperature because the energy this is because the number of particle oscillating in a tube is a constant.