I have a sound source with unknown initial coordinates $\langle x,y,z\rangle$, which emits pulses which are subsequently received by two sensors with known coordinates $\langle x_1,y_1,z_1\rangle$/$\langle x_2,y_2,z_2\rangle$. The sound source is moving with an unknown velocity $\langle v_x,v_y,v_z\rangle$.
I wish to determine the magnitude of this velocity (i.e., the speed of the sound source) using only the times at which its pulses arrive at each sensor, and those sensor's coordinates. My data consists of a table containing the arrival time at each sensor for each pulse.
We may assume that the pulses travel unimpeded in a straight line from the source to the sensors.
While I'm not sure how to begin, I do know that I have insufficient information to determine the coordinates of the source at each pulse (a solution in 3D requires in principle $3$, but in practice at least $4$ receivers). Moreover, I believe that I have insufficient information to determine the full velocity (see, for example, this recent similar (but distinct) post).
Just intuitively, it seems to me that $2$ receivers should be sufficient to estimate the speed. It isn't clear to me how this can be done mathematically.
From what I understand, you have a ping table like this:
sensor1: t11, t12, t13, ... sensor2: t21, t22, t23, ...
I assume that c (speed of sound) and f0 (frequency of pings) are constant and known.
The time differences between same pings define a succession of planes where the emitter was at the time it emitted the ping. The Doppler-shift of incoming pings for each sensor give relative velocity components. So you are searching for positions within a given plane which are transfered to positions in the next plane by the given velocity components.