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I quess that this is relatively easy question, but I have been struggling this for a two days now (basicly investigating different formulas) and couldn't find a solution.

So, let's think this case: Three (or more) internet nerds from same area (like in 10 kilometer radius) listens for a thunder. When each one hears that lightning strike, they tell to server that 'Wow now I heard it!' (clicks a button). Then server somehow calculates that where that lightning was. Let's assume that there is zero lag between what user hears and server got the info (network, human reflexes).

Things we know:

  • We know listeners Latitude and Longitude.
  • Also we know speed of sound in that horrible thunder.
  • Time differences between each person. (For example Listener 1 hears it first, then Listener 2 after 1.2 seconds and then Listener 3 after 2.3 seconds)

I know it can be made easily when you know how far that was (by hearing and seeing), and my awesome brain simulation tells me that it can be done with only hearing OR seeing.

2 Answers 2

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A "simple" way to do it would be solve the following equations. First, WLOG, assume that the person who heard the first sound is at the location $(0,0,0)$ and the second is $(x_{P_2},y_{P_2},z_{P_2})$ and the third is $(x_{P_3},y_{P_3},z_{P_3})$. Let $P_2$ have heard it $t_1$ seconds later, and $P_3$ have heard it $t_2$ seconds later. Then, you have the equalities

Note: speed of sound is 340 m/s and the coordinates are in meters.

$ 340t_1 + \sqrt{x^2+y^2+z^2} = \sqrt{(x-x_{P_2})^2+(y-y_{P_2})^2+(z-z_{P_2})^2} $ $ 340t_2 + \sqrt{x^2+y^2+z^2} = \sqrt{(x-x_{P_3})^2+(y-y_{P_3})^2+(z-z_{P_3})^2} $

However, note that you don't have enough info with just this.. you need another person. (because 2 equations and 3 unknowns)

EDIT: An simplifying criteria is $z=0$, which is valid if you assume that the sound originated from the ground. Note, this assumes that $z_{P_2}=z_{P_3}=0$.

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    Ahh, I get it now..2012-07-31
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If you have two listeners who see the lightning, you can calculate the distance of the strike from each one. Then just swing a circle of that radius around each one. The circles will intersect in two points (unless they are tangent) and you know the location up to that ambiguity.

If you have three listeners you have just enough data to resolve it, but I don't know an easy way. Let the listeners be at points $A,B,C$ and the time of day that they hear the thunder be $a,b,c$. Then if the strike is at $D,d$ and the speed of sound is $s$ you have $|D-A|=s(a-d)$ and two more equations like it. The three unknowns are the two coordinates of $D$ and the time $d$. Three equations in three unknowns. You generally will not have an ambiguity. If the data (positions and times) are perfect the solution will be exact.

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    If, however, you insist on using the speed of light and the ability to "see" the strike, you have an overdefined system, and you can obtain instead a least-squares solution.2012-07-31