Wave propagation with energy diffusion in a fractal solid and its fractional image

Abstract

Crucial features of seismograms and spectra with small amplitudes are explained by means of fractality and fractional calculus. Wave propagations in the elastic range of porous solids imply precursors and followers of coherent waves. They result from a non-local diffraction via force chains which is called energy diffusion. Such phenomena are captured by fractional wave equations which are deduced by means of an elastic energy and the balance of momentum for random fractal ensembles. Theoretical propagations imply precursors which were similarly observed with bender elements, and a rate of dissipation nearly proportional to the kinetic energy which suits to resonant column test results. A novel three-dimensional fractional Dirichlet-Green function implies primary and secondary wave crests with speed and alignment which do not depend on the fractal dimension. Power spectra in the dislocation-free far-field of seismogeneous chain reactions and impacts tend to a fractality-dependent power law with a peak-like cutoff, both theoretically and observed, therein a modified Huygen’s principle is employed. Limitations are discussed and possible extensions are indicated.