Wavelets in the propagation of waves in materials with microstructure

Abstract

The analysis of evolution differential equations (parabolic-hyperbolic) might be considered within the framework of the (harmonic) wavelet theory. In fact, the multiresolution analysis of wavelets seems to be a suitable scheme for the investigation of phenomena, which appears at different scales of approximation. Very often, the approximate solution is expressed in terms of functions which are significant only at a given resolution, and, some time, also the exact solution (like e.g. the D'Alembert solution of the wave equation) shows two characteristic features of wavelets: the dilation (multiscale) and the translation properties. However, from physical point of view, the wavelets still have little interpretations, especially concerning the small details expressing small deviations nearby the steady solution, since there are only a few examples of physical propagation of wavelets. In particular, the wavelet solutions of the dispersive (Klein-Gordon) wave equation show that the multiresolution approach is a kind of approximation that at each (scale) step increases the "resolution" of the solution. Thus it seems interesting to investigate this multilevel process that, at each scale (level), adds some more details to the solution. As application, the wavelet solution of the Klein Gordon equations for materials with microstructure, is defined as follows: the dispersive wave solution of the propagation equation is interpreted as a superposition of "small" waves on a basic wave. So that the wave propagation will be investigated at each given resolution, by showing that the "minor" details of the solution, neglectable at the initial time, have a significant influence on the solution on a long (time) range.