Wave Equation—Properties of Solutions—Starting Point of Hyperbolic Theory

There exists comprehensive literature on the theory of hyperbolic partial differential equations. One of the simplest hyperbolic partial differential equations is the free wave equation. First, we introduce d’Alembert’s representation in 1d and derive usual properties of solutions as finite speed of propagation of perturbations, existence of a domain of dependence, existence of forward or backward wave fronts and propagation of singularities. There a long way to get representation of solutions in higher dimensions, too. The emphasis is on two and three spatial dimensions in the form of Kirchhoff’s representation in three dimensions and by using the method of descent in two dimensions, too. Representations in higher-dimensional cases are only sketched. Some comments on hyperbolic potential theory and the theory of mixed problems complete this chapter.