An old mining tunnel disappears into a hillside. You would like to know how long the tunnel is, but it's too dangerous to go inside. Recalling your recent physics class, you decide to try setting up standing-wave resonances inside the tunnel. Using your subsonic amplifier and loudspeaker, you find resonances at 5.0 and 6.4, and at no frequencies between these. It's rather chilly inside the tunnel, so you estimate the sound speed to be 340. Based on your measurements, how far is it to the end of the tunnel?

Question

Physics

Highbeam

18 August, 11:09

An old mining tunnel disappears into a hillside. You would like to know how long the tunnel is, but it's too dangerous to go inside. Recalling your recent physics class, you decide to try setting up standing-wave resonances inside the tunnel. Using your subsonic amplifier and loudspeaker, you find resonances at 5.0 and 6.4, and at no frequencies between these. It's rather chilly inside the tunnel, so you estimate the sound speed to be 340. Based on your measurements, how far is it to the end of the tunnel?

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Answers (1)

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Katie Livingston · Answer 1

121.43 m

Explanation:

Solution

Since standing waves are set up, the expression for the first mode of frequency is f₁ = nv/4L. The next mode of frequency is f₂ = (n + 2) v/4L where L is the length of the tunnel and v the speed of sound. f₁ = 5.0 Hz, f₂ = 6.4 Hz, v = 340 m/s. We now subtract f₂ - f₁ = (n + 2) v/4L - nv/4L = v/2L.

So f₂ - f₁ = v/2L. and L = v/2 (f₂ - f₁) = 340/[2 * (6.4-5.0) ] = 340/2*1.4 = 340/2.8 = 121.43 m

So, it is 121.43 m far to the end of the tunnel.