The Laplace transform has been applied to solve a class of problems in wave propagation that are converse to the well-known time-dependent Boundary Value Problems (BVP). From a knowledge of time-varying conditions on a suitably restricted set of points interior to a region, it is possible to deduce time-varying boundary conditions. These standard situations in slab, cylindrical, and spherical geometry have been treated for the finite domain. The solution of an Interior Value Problem (IVP) in the semi-infinite domain explicitly demonstrates the formulation of an advanced solution in contrast to the retarded solution for a Boundary Value Problem. Corresponding problems in heat conduction will be discussed in a subsequent article.