Representations of solutions of the wave equation based on relativistic wavelets

TL;DR

This paper introduces a new integral representation of wave equation solutions using relativistic wavelets, enabling analysis of localized wave phenomena and boundary effects in two spatial dimensions.

Contribution

It develops a novel wavelet-based integral representation for the wave equation solutions utilizing affine Poincaré transformations, including Lorentz transformations.

Findings

Representation includes propagating localized solutions and boundary surface waves.

Numerical analysis of decomposition coefficients reveals dependence on source speeds.

Application to moving sources demonstrates the method's practical utility.

Abstract

A representation of solutions of the wave equation with two spatial coordinates in terms of localized elementary ones is presented. Elementary solutions are constructed from four solutions with the help of transformations of the affine Poincar\'e group, i.e., with the help of translations, dilations in space and time and Lorentz transformations. The representation can be interpreted in terms of the initial-boundary value problem for the wave equation in a half-plane. It gives the solution as an integral representation of two types of solutions: propagating localized solutions running away from the boundary under different angles and packet-like surface waves running along the boundary and exponentially decreasing away from the boundary. Properties of elementary solutions are discussed. A numerical investigation of coefficients of the decomposition is carried out. An example of the field created by sources moving along a line with different speeds is considered, and the dependence of coefficients on speeds of sources is discussed.