Cross-correlation of two signals
Your 2-mic array can provide an angle-of-arrival estimate with front-back ambiguity, using cross-correlation to estimate TDOA and then calculating AOA from TDOA and microphone spacing. The front-back ambiguity can be removed with sufficiently directional microphones, or just by constraining the geometry, putting the microphones against walls, for example.
You will also have a multi-path effect. If the space is cluttered, if you lose the direct path, or if the sound source is directional, this multi-path effect will be strong and the errors it produces may be overwhelming.
But even in ideal conditions, the two mics produce a single geo-observable, the TDOA, and thus can calculate only one geometric constraint. In other words, you have only one &quot;known&quot; and thus can solve only one &quot;unknown&quot;, not a full 2D position.
If the sound source and the sampling at the microphones can be synchronized, you can also measure propagation delay directly to get distance estimates from each mic - two &quot;knowns&quot;. Such absolute distance estimates are also very powerful for resolving the multi-path problem.
If the source and receivers cannot be synchronized, you will need a third microphone to produce a second independent TDOA measurement, a second &quot;known&quot;. At this point, you can use the same pseudorange/light-cone cost-minimization calculations that are used in GPS receivers, or just take the two TDOAs and try to intersect the AOA lines.
These are the general principles.
(My comments are based on personal experience with geolocation of radio emitters in several success projects, but the same principles apply to sound.)