The subject of this, the final chapter of this book, is perhaps the most interesting of the classical partial differential equations of mathematical physics. The theory of the wave equations, the standard example of hyperbolic equations, also builds upon the theory of Laplace’s equation, but is somewhat more complicated due to the absence of exponential decay in the time variable, as was the case with the heat equation. Our setting is again a region in En with an additional variable t for time. At various instances we shall restrict the number of dimensions for computational simplicity.