Representations of solutions of the wave equation based on relativistic wavelets

A representation of solutions of the wave equation with two spatial coordinates in terms of localized elementary ones is presented. Ele-mentary solutions are constructed from four solutions with the help of transformations of the affine Poincare ́ 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 solu-tion as an integral representation of two types of solutions: propagat-ing localized solutions running away from the boundary under differ-ent 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. 1 a