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Old Mar 31st 2015, 11:08 AM   #1
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Wave experiment - origin of a wave based on 3 points

Trying to help my kid with an experiment.

In a tank of water, if you drop a marble it creates a circular wave. If you have 3 sensors in that tank that can detect the arrival of that wave, could you mathematically calculate the position where the object was dropped into the tank based on the arrival of the wave at each sensor?

Assuming you know the locations of the sensors and how far apart they are from one another, is this possible? can anyone help with the math on the calculation?

thanks for any help!
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Old Mar 31st 2015, 01:55 PM   #2
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Yes, this should be doable. You need to know the speed of the wave on the surface of the water, which should be a constant for waves of a given height. Then by timing the difference in wave arrival time at each of the sensors it would be possible to calculate the location of the center of the disturbance.

Imagine a square tank of dimension L by L, as shown in the attached figure, with lower left hand corner at the origin of a graph - at (0,0). There are three sensors, where sensor 1 is placed at the origin (0,0), sensor 2 at the lower right corner at (L,0), and the third sensor at the upper left corner at (0,L). Let t=0 be the time when the wave hits the sensor at the origin, t2 = the time it hits sensor 2, and t3 = the time it hits the sensor 3. Note that t2 and/or t3 may be negative values, if the wave hits one or both of these sensors before hitting sensor 1. Given these times you can calculate the difference in distances of sensor 2 and 3 from the disturbance compared to sensor 1. Let unknown R = distance from sensor 1 to the disturbance, then R+d2 = distance of sensor 2 from the disturbance and R+d3 = distance of sensor 3 from the disturbance, where d2 and d3 are calculated from the time t2 and t3 divided by speed of the wave.

If we let (X,Y) = the center of the disturbance, then we have three equations :

For sensor 1: x^2 + y^2 = R^2
For sensor 2: (x-L)^2 + Y^2 = (R+d2)^2
For sensor 3: x^2 + (Y-L)^2 = (R+d3)^2

So you have three equations in three unknowns. Solving for X and Y gets a bit messy, but it's doable. Give it a shot, and let us know what you get.
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Wave experiment - origin of a wave based on 3 points-ripples.jpg  

Last edited by ChipB; Mar 31st 2015 at 02:33 PM.
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