Since the time of the cannon shot is unknown, only the time when it can be heard at a known location, you can't even pinpoint its unknown origin in space to a circle. Instead, the range of possibilities forms a cone in spacetime (exactly like the lightcone of electromagnetic waves, except with sound and also treating space as two-dimensional for simplicity).
The intersection of two such cones is a parabola. The intersection of the third cone with the plane containing said parabola gives another parabola that can intersect the first one in zero, one, two or an infinite number of points. In the zero-point case, you could still find the location where the two parabolas are closest.
(Or you could forget about geometry, define an objective function and do gradient descent to it.)
Good point. I think my intuition about triangulation comes from GPS, where pulses are emitted simultaneously from many points at a known time. This problem is a little trickier.
The intersection of two such cones is a parabola. The intersection of the third cone with the plane containing said parabola gives another parabola that can intersect the first one in zero, one, two or an infinite number of points. In the zero-point case, you could still find the location where the two parabolas are closest.
(Or you could forget about geometry, define an objective function and do gradient descent to it.)