Justification for power laws and fractional models

Abstract

Wave equations with non-integer derivative operators describe attenuation which increases with frequency with other powers than two, unlike ordinary wave equations. It is desirable to try to understand what properties of, e.g., biological tissue that give rise to this behavior. The main attenuation mechanisms of standard acoustics are heat conduction and relaxation, as well as structural and chemical relaxation. They have fractional parallels and the first one is heat relaxation described by fractional Newton cooling due to anomalous diffusion. The most important mechanism is however the fractional parallel to structural relaxation. Instead of one there are multiple relaxation processes with a distribution of relaxation times that follows a power-law distribution, possibly indicating fractal properties. The distribution also has a relationship to the Arrhenius equation, indicating a link to chemical relaxation, albeit a quite speculative one. The multiple relaxation model may also be formulated as a hierarchical polymer model. Time-varying non-Newtonian viscosity and a medium with a fractal distribution of scatterers can also give rise to power law behavior. Existing models in sediment acoustics such as the grain shearing model and the Biot poroelastic model can also be reformulated with fractional operators. These approaches are presented in the hope of progressing towards an understanding of whether fractional wave equations give clues to some deeper reality, or if they are just a compact phenomenological description.