I'm looking into solving the wave equation for underwater sound as described in the Naval Underwater Systems Center's technical report Numerical Solutions of Underwater Acoustic Wave Propagation Problems (PDF link via dtic.mil).
The basic idea is obvious, but I'm looking for an introduction and/or tutorial that describes the matrix formulations and how they simplify the calculation.
I always recommend Finite Difference Methods for Ordinary and Partial Differential Equations: Steady-State and Time-dependent Problems by Randall LeVeque. It is a very accessible text and gives overviews and enough examples of Finite Difference Methods for you to get the hang of the method as well as covering consistency, convergence and stability questions.