Say I have some sending devices and receiving devices which I know position of ${\bf r}_k,{\bf s}_k$. Furthermore I can control what is sent on sending devices and when. In other words boundary conditions of the scalar signal $u$ at ${\bf s}_k, {\bf r}_k$. And we assume the wave equation is being followed.
$$\frac{\partial ^2 u}{\partial t^2}=c^2\nabla^2 u$$
But $c$, propagation speed is not constant, but can vary over the geometry of the solution.
How can we by varying sending $u({\bf s}_k,t)$, measuring $u({\bf r}_k,t)$ determine $u({\bf x},t)$ and $c({\bf x})$?
I am interested in practical algorithms, but any reference to the mathematics behind such algorithms are welcome as well.