Solving Wave Equations in the Space of Schwartz Distributions: The Beauty of Generalised functions in Physics

TL;DR

This paper develops a framework for solving wave equations with instantaneous sources in the space of Schwartz distributions, revealing how such sources relate to point-like sources and analyzing the properties of fundamental solutions across different dimensions.

Contribution

It introduces a method to represent instantaneous sources as sums of point-like sources and derives explicit solutions using convolution with fundamental solutions in various dimensions.

Findings

Wave solutions with instantaneous sources can be expressed as convolutions with fundamental solutions.

Three-dimensional solutions exhibit diffraction phenomena.

Transition from diffraction to diffusion regimes is characterized by continuation of generalized functions.

Abstract

This paper concerns the study and resolution of wave equations in the space of Schwartz distributions. Wave phenomena are widespread in many branches of physics and chemistry, such as optics, gravitation, quantum mechanics, chemical waves and often arise from instantaneous sources represented by Schwartz distributions f. Hence, there is a need to study the Cauchy problem in the space of generalised functions. Specifically, it has been proven that the instantaneous source f can always be represented as an appropriate sum of single point like sources. Under this hypothesis, each wave equation with an instantaneous source f remains associated with an equation with a point-like source represented by a Dirac delta function. The solution to the associated equation is an elementary perturbation that propagates in spacetime, defined as the fundamental solution. We proved that the solution to a wave equation with source f is given by the convolution product between one of the fundamental solutions and the generalised function f representing the instantaneous source. We investigated the physical and mathematical properties of three dimensional, two dimensional, and one dimensional fundamental solutions. Notably, we proved that the three-dimensional solution described diffraction phenomena, whereas the other two described wave diffusion phenomena. Furthermore, we demonstrated that the transition from a diffractive to a diffusive regime occurs through the continuation of an ansatz generalised function. In this paper, we discuss possible applications to solid state physics and the resolution of crystallographic structures.