An air-filled tube is closed at one end by a movable piston. A tiny loudspeaker is mounted at the opposite closed end. Resonances are obtained by moving the piston. If the speed of sound in the air in the tube is 343 m/s and resonances are found at piston positions of 39.3 cm, 51.7 cm, and 64.0 cm relative to an arbitrary point, find the loudspeaker frequency.

Resonances occur when the length of the tube is an integer multiple of half the wavelength of the sound wave produced by the loudspeaker. The frequency formula f = v/λ is used to calculate the loudspeaker frequency. By determining the harmonic numbers corresponding to the given positions of the piston, the wavelength and frequency of the sound can be calculated.

Explanation:

Resonances occur when the length of the tube is an integer multiple of half the wavelength of the sound wave produced by the loudspeaker.

The formula to calculate the frequency of a sound wave is:
f = v/λ, where f is the frequency, v is the speed of sound, and λ is the wavelength.

In this case, since the tube is closed at one end, we have a standing wave with one end node and one open end antinode. The length of the tube is equal to a quarter of the wavelength of the sound wave.

We can use the formula
λ = 4L/n, where λ is the wavelength, L is the length of the tube, and n is the harmonic number.

By plugging in the given values of the lengths of the tube (39.3 cm, 51.7 cm, and 64.0 cm), we can find the corresponding harmonic numbers.

Finally, we can calculate the loudspeaker frequency using the formula
f = v/λ by plugging in the speed of sound and the calculated wavelength.

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