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 4.5 Hz and 6.3 Hz, 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, how far is it to the end of the tunnel?

Question

Accepted Answer

Hey everyone. So this problem is dealing with standing sound waves. We're working with a pipeline which we're told is an open, close to. And that pipeline has standing wave signals that were detected at 6.7 Hertz and 7.8 Hertz with no other resonance frequencies in between. We're asked to calculate the length of the pipeline given a sound speed of 340 m per second inside the pipe. Our multiple choice answers here are a 16 m B m c 49 m or D 98 m. So the first thing we can do here is recall that for an open close tube, our frequency is given by the equation N V divided by four L. And now we can rearrange this equation for L because we're asked to calculate the length is that, so that will look like L is equal to N V divided by four. So we have our frequency at two different wave at two different um harmonics. But we don't know the N or the, the mode, the node, excuse me. Um For those frequencies we are given speed in the pipe. So really what we need to figure out is what harmonic these frequencies happen at. So we can say that uh F of N, we have a frequency of 6. Hertz. And then at that next harmonic F of N plus two, we have a frequency of 7.80 Hertz. We can recall that in and open, close to our harmonics are at odd integers. So 13579 and so on. And that's why um those harmonics must be two integers or N plus two uh away from each other. Know the length of the pipe isn't changing. So we can actually solve, we can actually set these lengths equal to each other for both of these conditions. So this looks like N V divided by four, F of N is equal to N plus two, multiplied by V divided by four, multiplied by F of N plus two. So the speeds cancel or cancel. And we're left with N divided by 6.7. Hertz is equal to N plus two divided by 7.80. Yes, we can multiply cross multiply both sides subtract to get the ends. On the same side, we're left with 1.73 Hertz multiplied by N is equal to 12.14. Divide out the hurts and, and is equal to seven. Now, we can go back up into our length equation and solve for length. So length is equal to N which we've just determined is seven multiplied by speed given to us in the problem 340 m per second, all divided by four, multiplied by that frequency, 6.7 Hertz or L you plug that in, you get 98 m. And so that aligns with answer choice D, so that's all we have for this one. We'll see you in the next video.

Verified Solution

Key Concepts

Standing Waves

Resonant Frequencies

Speed of Sound