An old tunnel disappears into a hillside. The ceiling bas collapsed deep inside

Question

An old tunnel disappears into a hillside. The ceiling bas collapsed deep inside the hill blocking the tunnel. The length of the surviving tunnel can be found (without walking into it) by setting up standing waves inside the tunnel using a loudspeaker (source). If you find resonances at 4.5Hz and 6.3Hz, and at NO frequencies between these, find assuming that the speed of sound inside the tunnel is 335 m/s (a) Which harmonies do these frequencies correspond to? (Do not assume they are the first and the second harmonics) (b) What is the length of the surviving tunnel? (c) What is the fundamental frequency (first harmonie) of the tunnel?

Explanation / Answer

When I am in a tunnel and I incite resonance it is the sound reflecting from the walls rather than from the ends.

So we will need to use some licence to imagine that the echoes come from the far end.

(a)The tunnel is open at one end so resonances occur when the wavelength = 4L , 4L/3, 4L/5 etc
And the corresponding frequencies are
V/4L , 3V/ 4L, 5V/4L, 7V/4L etc.

(b) So for two adjacent frequencies f2 - f1 = 2V/4L
L = V/2(f2-f1) = 335/(2* 1.8) = 93.05 m

(c) The fundamental frequency, f = v/ = v/2*L = 335/2*93.05 = 1.80 Hz

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