Answer to Question #117780 in Physics for Susan Williams

Question

An object vibrating with a frequency 850 Hz causes resonance in a tube of air when the shortest length of air column is 10.3 cm. Calculate the second resonant length, assuming the speed of sound in air is 340 m/s.

Answer 1

The condition for a resonance in a tube of air is (see https://en.wikipedia.org/wiki/Acoustic_resonance#Closed_at_one_end):

$$f = \dfrac{nv}{4L}$$

where $n$ is positive integer (resonance mode), $v = 340 m/s$ is the speed of sound in air and $L = 0.103 m$ is the length of the tube.

For a given conditions the number of resonance mode is:

$$n = 4fL/v = 4\cdot 850\cdot 0.103/340 = 1$$

The second resonant length on the same frequency will be:

$$L = \dfrac{2v}{4f} = \dfrac{v}{2f} = \dfrac{340}{2\cdot 0.103} \approx 1650 m$$

Answer. L = 1650 m.