This is not homework... it's way too hard! It's actually just a really intriguing problem I've wondered about for years while listening to trains echo through the mountains. Here's the simplest statement of it.


On a windless day, a train is traveling along a flat western plain on a straight track, blowing its horn once a second. Nearby is a tall mesa, making the train horn echo. You are close enough to hear the train and the echo, and you decide to record it for future study. It is very quiet otherwise. You record the train from the time it appears on the horizon to the time it disappears again. The Doppler shift is very noticeable as the train passes, and you notice the echo has a different pitch, also shifting, which must mean the mesa is echoing a different part of the Doppler shift than the sound you are hearing directly.

(If you've never heard this effect, here's an example, starting at 26 seconds: http://youtu.be/UUNnC-9XMnI?t=26s)

We know only the speed of sound (340 m/s) and the pattern of sound on the recording. Both train and echo are clearly distinguishable, except when they are the same pitch.

Can we:

  • determine the speed of the train
  • determine our distance from the track
  • determine the distance from us to the mesa
  • plot a relative map of the train track, the mesa, and ourselves

How would you approach each?


What I've worked out so far:

The speed of the train can be calculated by knowing the maximum and minimum pitch of the Doppler shift. This can only be measured directly in front or behind the train. Can you still calculate it from a distance away?

If you are able to plot the pitch over time, you can determine the relative speed to your position, which should tell you how far you are from the track.

The minimum echo time will be when the mesa is at 90 degrees from the train. The pitch from that echo will be the actual train horn pitch (I think...)

If you (a), the train (b), and the mesa (c) make a triangle, the echo length tells you (bc + ac - ab). However that doesn't tell you the length of ac.


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