Can You Calculate the Pinger's Angle Using Hydrophone Data?

[Number2Pencil]

Homework Statement

This is somewhat of a design problem instead of a right/wrong answer. A pinger is placed in the ocean which produces an audible sound at random intervals from an unknown location. For simplicity say the sound travels at 1m per second. 3 hydrophones are placed on a submarine in a T shape, with a max distance of 10 m for the top of the T and 20m for the long part of the T. The only information from the hydrophones about the ping includes which hydrophone the ping hits first, and the times it takes for the ping to travel from the first hydrophone to the other two. The crewmen must use this information to calculate the angle of the pinger relative to the sub.

Homework Equations

parabola = Ax^2 + Bx + C

The Attempt at a Solution

I put hydrophone A and B on the top of the T. I tried to use the fact that at 0deg the time between hydrophone A and B would = 0, at 90deg the time between A and B would be 10s, and after acurately drawing a scaled down version on a piece of paper and properly having the pinger placed at 45 deg, when the time between A and B was 2.67s the angle was 45 deg.

I tried to create a parabola equation to make an theta(t) equation...but it didn't turn out so well.

any other ideas??

 

[Number2Pencil]

hmmm...I'm going to ask the trig forum if there is some standard way of triangulation that would help me out...

 

[Number2Pencil]

alright, well I've found no obvious answer...so I've tried taking a 2-d mathematical approach, with only this physics equation: v = d/t

I know I can place the hydrophones where i want, so I'm going to set the origin reference (0,0) right inbetween two of them so they are located at (-pos.x,0) and (pos.x,0) and the third one i will put directly below the origin at (0,pos.y)

so i know the speed of sound in water, and delta-time (i'll label it t.1 and t.2 for hydrophone 1 to 2 and 1 to 3 respectively), so i can find delta-distance (d.1 and d.2), so it's time for the distance formula on delta-d for hydrophone 1 and 2:

$$\Delta d_1 = \sqrt{y^2 + (x + pos_x)^2} - \sqrt{y^2+(x-pos_x)^2}$$

where x and y are the coordinates of my unknown pinger..

solving for this I got:

$$y = \pm \sqrt{16x^2p_x^2-4x^2\Delta d_1^2-4p_x^2\Delta d_1^2 + \Delta d_1^4}$$

which gives me a curve(s) where the relationship of delta-d.1 always holds true...
so now the distance equation for delta-d of hydrophone 1,3

$$\Delta d_2 = \sqrt{y^2+(x+p_x)^2} - \sqrt{(y+p_y)^2+x^2}$$

so now i can plug in the y= equation from above and try to solve for x...or i can try to let mathcad do it for me!

yeah...mathcad stalled out thinking for about a whole minute before it gave up and told me "no solution found". looks like I'm going to have to attempt this by hand...UNLESS somebody out there would be kind enough to chug it into their fancy math software and see if it can do it !