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28 February, 18:49

A slide whistle is an open-closed tube with an adjustable plunger that changes the length. You are playing the slide whistle and pushing the plunger in at 8 cm/s when the tube is 15 cm long. What is the rate of change of the sound in Hz/s?

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  1. 28 February, 19:01
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    df / ft = - n 12 n = 1, 3, 5, ...

    Explanation:

    The speed of sound is

    v = λ f

    In a whistle that we approach by an open tube at one end and closed at the other, standing waves occur, which has a node in the closed part and a maximum in the open pate, whereby wavelength and the distance of the tube are related, the fundamental wave is

    λ₁ = 4L

    The harmonics are

    λ₃ = 4L / 3

    λ₅ = 4L / 5

    The general formula

    λₙ = 4L / n

    with n = 1, 3, 5, ...

    We substitute and clear in the first equation

    f = v n / 4L n = 1, 3, 5, ...

    Let's use derivatives to find the frequency change

    df / dt = v n / 4 dL⁻¹ / dt

    d / dt (1/L) = - 1 / L² dL / dt

    Where dL / dt = 8 cm / s

    We replace

    df / dt = - n v / L2 dL / dt

    Let's calculate

    df / dt = - n 340/152 8

    df / ft = - n 12 n = 1, 3, 5, ...
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