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I have this equilateral triangle. The sun shaped object is a sound source. I know the difference of time between the arrival of the sound at the 2 vertices of the triangle. I need to find the angle alpha based on this information. I know the length of the side of the equilateral triangle.

Any help will be appreciated.

  • $\begingroup$ You know the lengths from the sound source to the two vertices, and you know the length of a side of the equilateral triangle. Try using the law of cosines to fill in more unknown angles. $\endgroup$ – user137794 Apr 15 '14 at 22:18
  • $\begingroup$ What's the length you know (assuming that's what you meant by difference of time)? $\endgroup$ – Shahar Apr 15 '14 at 22:19
  • $\begingroup$ I do not know the lengths from the source. All I know is at which vertex the sound reaches first and what is the difference in time between the arrival of sound at the 2 vertices. $\endgroup$ – user2731223 Apr 15 '14 at 22:25
  • $\begingroup$ They don't have to be similar, all they need to satisfy is that the angle alpha remains the same in all these 3 cases. $\endgroup$ – user2731223 Apr 15 '14 at 23:42
  • $\begingroup$ You haven't said this, but is the vertex of the angle $\alpha$ supposed to be the center of the equilateral triangle? $\endgroup$ – Gerry Myerson Apr 16 '14 at 7:00

You don't know the distance from the two vertices of the triangle, you only know the difference between the distances, right? In that case, you cannot deduce the location of the sound source. For example, the red line in the following illustration is the locus of all possible placements of the sound source that would lead to the same difference.

hyperbola locus

The lower branch would have the sound arriving at the lower corner first, so you are only interested in the upper branch. The red curve is a hyperbola, with the corners of your triangle as foci. That's one defining property of a hyperbola.

Different positions on the upper branch of the hyperbola would lead to different angles $\alpha$. So the angle isn't known either. In many situations, the most reasonable thing you can do is concentrate on the case where the distance between sensors and sound source is much larger than the distance between the two sensors. So you could use the direction of the asymptote as an approximation.

To do that, consider a right triangle constructed between your two sensors. The idea is that sound waves would be parallel to one of its legs, and travel in the direction of the other leg. Construct it in such a way that the length of that second leg (which is the leg incident with the sensor where the sound arrived later) matches the difference in distance that you calculated.

Construction for approximation

From that you can read off the angle of the direction where a sound source far away would be located.


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