Applications of Distributional Derivatives to Wave Propagation

First the concepts of the surface distributions are explained. Thereafter the first and higher-order distributional derivatives are derived for a function of several variables. These concepts are then used to study the propagation of wave fronts in continuum mechanics. Explicit formulas for the jump relations for the first and second-order partial derivatives are obtained across the wave front. A systematic method is then developed such that the algebraic work for the study of waves becomes very simple. A few illustrations in wave propagation are presented. At the end it is pointed out how this method can be effectively applied in the derivation of the jump relations for single and double-layer potentials in the theory of harmonic functions.