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Submarine-related math problem.


Ensign Cthulhu

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Because it's been far too long since high school, which was the last time I was required to fit maths and physics together with any regularity.

I have set this problem for myself many times and failed to solve it many times. Perhaps someone can give me a hint.

 

"A pair of passive nondirectional sonar buoys 1km apart east-west pick up a transient noise two seconds apart. Assuming (for ease of math) that the transient is north of the buoys at the same horizontal level and assuming a 1500m/sec speed of sound in water, can the position of the transient be fixed?"

My gut feeling is that you can't; that you need at least THREE buoys and you need to switch between A and B, B and C. 

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57 minutes ago, Ensign Cthulhu said:

Because it's been far too long since high school

If your passive sonar was very accurate, and if the target was stationary, you could estimate the angle of each contact from the baseline between the buoys and then use triangulation get its approximate distance and position. However the sonar buoys would have to be directional (sound louder from a certain direction).

620px-Triangulation-boat.png

{\displaystyle d=\ell \ {\frac {\sin \alpha \sin \beta }{\sin(\alpha +\beta )}}}

Edited by Snargfargle
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3 minutes ago, Snargfargle said:

If your passive sonar was very accurate

The buoys are passive and nondirectional. The only datum they give is the time delay between discretely identifiable transients. No direct range or bearing data are available. That's the problem.

I can solve for the position with directional data available easily. Working with just a dT is a different issue.

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5 minutes ago, Ensign Cthulhu said:

The buoys are passive and nondirectional. The only datum they give is the time delay between discretely identifiable transients. No direct range or bearing data are available. That's the problem.

I can solve for the position with directional data available easily. Working with just a dT is a different issue.

Yeah, I was probably editing my post as you typed, just realizing that you mentioned that the buoys were non-directional. However, are they really? With two buoys you now have a sonar array and can determine directionality in much the same way as your "binocular" ears do.

Edited by Snargfargle
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Global Positioning Systems (GPS) perform calculations using time-differential.
But, they also communicate the positions of the satellites within the coded radio transmissions emanating from the satellites.
So, they start with more information than is available in @Ensign Cthulhu sonar-buoy word-problem.

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3 minutes ago, Wolfswetpaws said:

Global Positioning Systems (GPS) perform calculations using time-differential.

We did a training exercise in the Army (you probably did something similar in the Marines) where we had AK-47s fired over us and then determined the distance to the shooter by the time between the crack of the bullets and the sound of the shots.

When we heard the bullets crack, we counted fast "on,to,te,fo,fi,si,se" over and over until we heard the shots, counting on a finger for each seven count. Since sound travels at about 350 meters per second, each count between the crack of the bullet overhead and the sound of the gunshot was about 350/7 or 50 meters.

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36 minutes ago, Snargfargle said:

We did a training exercise in the Army (you probably did something similar in the Marines) where we had AK-47s fired over us and then determined the distance to the shooter by the time between the crack of the bullets and the sound of the shots.

When we heard the bullets crack, we counted fast "on,to,te,fo,fi,si,se" over and over until we heard the shots, counting on a finger for each seven count. Since sound travels at about 350 meters per second, each count between the crack of the bullet overhead and the sound of the gunshot was about 350/7 or 50 meters.

Nice training exercise.  🙂 
But, for whatever reason, I didn't have that kind of training.

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