Power law attenuation modeled as multiple relaxation

Abstract

Wave equations with non-integer order derivatives may model power law behavior in medical and sediment acoustics. As experiments only support a finite bandwidth, there is a limit to how much physical insight that can be gained from such models. Other ways to model a power law are with a fractional heat law, hierarchical ladder models for polymer chains, and the non-Newtonian rheology of grain shearing. Multiple relaxation processes may be motivated by a hierarchy of substructures at different scales. It is also inherent in soft glassy materials, such as cells, with disordering and metastability. Even the Biot model with contact squirt flow and shear drag (BICSQS) may be interpreted as a multiple relaxation model. A weighted sum of relaxation processes will approximate a power law over a limited band, and an even distribution of relaxation frequencies on a logarithmic frequency axis, and with equal relaxation strengths, will give a power law attenuation with unit power, y = 1. This can be generalized to other power laws if the contribution from each relaxation process varies in proportion to the relaxation frequency to the power of y - 1. This scale-invariant distribution may hint at some fractal medium properties.