Suppose that the range of output frequencies is from 89.2 Hz to 13.9 kHz for a pipe organ. Take 343 m/s for the speed of sound. (a) What is the length (in units of m) of the longest pipe open at both ends and producing sound at its fundamental frequency? (b) What is the length (in units of m) of the shortest pipe open at both ends and producing sound at its fundamental frequency?

a) L = 1,923 m, b) L = 0.012 m

Explanation:

This is a resonance problem, if the tube is open at both ends it has a

a maximum in each one, the waves must fulfill the relationship

λ = 2L / n

Where n is an integer and L the length of the tube

Let's use the ratio of the speed of sound

v = λ f

v = 2L f / n

L = v/2f n

The fundamental frequency corresponds to n = 1

L = v / 2f

a) for the smallest frequency

f = 89.2 Hz

L = 343 / (2 89.2)

L = 1,923 m

b) for the highest frequency

f = 13.9 kHz = 13.9 10³ Hz

L = 343 / (2 13.9 10³)

L = 12.338 10⁻³ m

L = 0.012 m