Wavelet-based integral representation for solutions of the wave equation

TL;DR

This paper develops a wavelet-based integral representation for solutions of the wave equation, enabling superpositions of solutions using wavelet analysis and distribution theory, with comparisons to existing methods.

Contribution

It introduces a novel integral representation of wave equation solutions using continuous wavelet analysis, expanding the toolkit for solving wave equations.

Findings

Representation valid for a wide class of solutions

Formulas are justified via distribution theory

Comparison with G. Kaiser's results shows consistency

Abstract

An integral representation of solutions of the wave equation as a superposition of other solutions of this equation is built. The solutions from a wide class can be used as building blocks for the representation. Considerations are based on mathematical techniques of continuous wavelet analysis. The formulas obtained are justified from the point of view of distribution theory. A comparison of the results with those by G. Kaiser is carried out. Methods of obtaining physical wavelets are discussed.