Question

Problem 6: tunnel disappearing into a hillside An old mining tunnel disappears into a hillside (that means to a closed end). 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 4.5Hz and 6.3Hz, and at no frequencies between these. It's rather chilly inside the tunnel, so you estimate the sound speed to be 335 m/s. Based on your measurements you can now determine how long is the tunnel. More specifically: a) First, which harmonics are 4.5Hz and 6.3Hz? In other words are these the lowest harmonics, if not to which n do they correspond to? (Note you know that they are consecutive, since there are no other resonance frequencies in between them) b) Then, with your result from a) extract the length of the tunnel

Answer 1

(a) Given data is c = 335 m/s Fundamental frequency f - 2/2L = 335/2L High order frequency f = 335n/2L Given resonant frequency 4.5 Hz & 6.3 Hz Therefore 4.5 Hz Can only be n = 3 & 6.3 Hz must then be n = 4 (they must be consecutive) (b) length of tunnel f = 4.5 Therefore 335(n)/2L = 4.5 implies 335(3)/2L = 4.5 L = 111.667 m