Four ways to justify temporal memory operators in the lossy wave equation

TL;DR

This paper explores four theoretical justifications for using temporal memory operators in wave equations to model complex attenuation behaviors in ultrasound and tissue shear waves, extending to nonlinear media.

Contribution

It introduces four distinct theoretical frameworks to justify temporal memory operators in wave equations, including power law kernels, relaxation processes, and higher derivatives, with an extension to nonlinear media.

Findings

Power law attenuation models with non-quadratic exponents.

Equivalence of memory operators to relaxation processes via complex compressibility.

Extension of models to nonlinear wave propagation.

Abstract

Attenuation of ultrasound often follows near power laws which cannot be modeled with conventional viscous or relaxation wave equations. The same is often the case for shear wave propagation in tissue also. More general temporal memory operators in the wave equation can describe such behavior. They can be justified in four ways: 1) Power laws for attenuation with exponents other than two correspond to the use of convolution operators with a temporal memory kernel which is a power law in time. 2) The corresponding constitutive equation is also a convolution, often with a temporal power law function. 3) It is also equivalent to an infinite set of relaxation processes which can be formulated via the complex compressibility. 4) The constitutive equation can also be expressed as an infinite sum of higher order derivatives. An extension to longitudinal waves in a nonlinear medium is also provided.