--update --

Sorry, my first reasoning was far too simplified ==> removed; a retry.

You need the distance between the microphones and the two delta T arrival times of the shockwave to do the math.

If you have three points A, B and C.

arrival times of soundwave (in order)

A - T0

B - T1

C - T2

The fact that the soundwave arrived at A first defines an area(1) of all points P: d.PA < d.PB and d.PA < d.PC (d.PA = distance PA)

The delta-time AB = T1 - T0 defines a distance d1 = (T1 - T0) /so (so = speed Sound)

The delta-time AC = T2 - T0 defines a distance d2 = (T2 - T0) /so

define the curve of all points Q: d.QA - d.QB = d1 (.QA = distance QA)

define the curve of all points R: d.RA - d.RC = d2

These two curves cross each other in area (1) => the point of impact ==> Q == R.

-- update --

if you knew the time of impact the d.QA, dQB and d.QC would be known, making the math faaaaar simpler as these curves are not trivial - quadratic asymptotic beasts with sqrts in it -

Think it is easier to write an approximating algorithm that searches the point.

The fact that point A heard the soundwave first => d.QA < d.QB

-- update 2 --

from: - http://www.mathwarehouse.com/hyperbola/graph-equation-of-a-hyperbola.php

A hyperbola is a set of all points P such that the difference between the distances from P to the foci, F1 and F2, are a constant K

so my

*"quadratic asymptotic beasts with sqrts in it"* can be rewritten as hyperbola with A and B (A & C) as foci.

using the drawing of the webpage above:

Assume A = (0,-c) and B = (0,c) and the constant K = d1 = (T1 - T0) /so. The point (0,-a) where the hyperbola crosses the Y-axis is (0, -d1/2)

--> http://www.mathwarehouse.com/hyperbola/focus-of-hyperbola.php

To determine the foci one uses a^2 + b^2 = c^2 => b^2 = c^2 - a^2 = (d.AB/2)^2 - (d1/2)^2

The formula of the hyperbola becomes : y^2 / (-d1/2)^2 - x^2 / (d.AB/2)^2 - (-d1/2)^2 = 1

Same trick for the points A & C (hint: it is easier to use another reference framework to determine the formula and do a translation afterwards: X -> X-xdelta Y -> Y-ydelta)

TODO: determine intersection points of the two hyperbolas and then your close...

-- update 3 --

intersection points

-- http://www.analyzemath.com/HyperbolaProblems/hyperbola_intersection.html

Difference with the location problem is that the hyperbola defined by points AB and the one defined by points AC are 'orthogonal' - think of it as the red in the drawing rotated 90 degrees (make a drawing!!) There will be two intersection points and because the soundwave arrived first at A it becomes obvious which one to choose.

The code is left as an exercise ....