In a tube closed at one end, what is the wavelength of the third harmonic (n=3)?

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Multiple Choice

In a tube closed at one end, what is the wavelength of the third harmonic (n=3)?

Explanation:
When a tube is closed at one end, the standing wave must have a node at the closed end and an antinode at the open end. This condition makes the length equal to an odd multiple of a quarter of the wavelength: L = n(λ/4) with n = 1, 3, 5, … Therefore the wavelength is λ = 4L/n. For the third harmonic, n = 3, giving λ = 4L/3. You can also see this by noting there are three quarter-wavelength segments inside the tube, so L = 3(λ/4) = 3λ/4, which leads to λ = 4L/3. The even harmonics aren’t allowed for a closed-end tube, so wavelengths that would imply n = 4, 2, etc., don’t occur in this setup.

When a tube is closed at one end, the standing wave must have a node at the closed end and an antinode at the open end. This condition makes the length equal to an odd multiple of a quarter of the wavelength: L = n(λ/4) with n = 1, 3, 5, … Therefore the wavelength is λ = 4L/n. For the third harmonic, n = 3, giving λ = 4L/3. You can also see this by noting there are three quarter-wavelength segments inside the tube, so L = 3(λ/4) = 3λ/4, which leads to λ = 4L/3. The even harmonics aren’t allowed for a closed-end tube, so wavelengths that would imply n = 4, 2, etc., don’t occur in this setup.

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