Applications to Wave Propagation

In Chapter 10 we have discussed various properties of the homogeneous and inhomogeneous wave equations in two and three dimensions. We derived their fundamental solutions and studied moving point, line, and surface sources. In Chapter 5 we considered various kinematic and geometrical aspects of the wave propagation in the context of surface distributions. In this chapter we consider some applications of these results and study partial differential equations whose solutions can be defined as regular singular functions. In problems of these kinds, singular surfaces can play two essentially different roles: one relates to the propagation of wave fronts, and the other to regular singular functions that arise in the study of boundary value problems. In the latter case, the surface Σ is given along with some data on it, and the solution is to be found in only some region of the space. A powerful method of attacking these problems is to embed them in the whole space. This is achieved by extending the solution to the other side of the surface in some suitable fashion, as we did in deriving the Poisson integral formula in Chapter 10. We then obtain a regular singular function that satisfies the equation in the complement of Σ.