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 ins
ide. 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?
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
