Four interpretations of temporal memory operators in the wave equation

Abstract

Attenuation and dispersion in ultrasound and elastography may be modeled with convolution memory operators in the wave equation. When the attenuation follows an arbitrary frequency power law the memory is a time domain power law. Since that is also a fractional time derivative, a large body of literature on fractional derivatives then becomes available for analysis and simulation. It can also be shown that, e.g., an elementary grain shearing process in an unconsolidated saturated sediment results in fractional wave equations for both compressional and shear waves. Second, some of these wave equations can be derived from constitutive equations with memory operators, ensuring satisfaction of causality and the Kramers-Kronig relations. The fractional Kelvin-Voigt and Zener models and their resulting attenuation and dispersion will in particular be discussed. The fractional operators can also be interpreted as the result of an infinite number of basic processes. Therefore, third, a fractional wave equation can be shown to be the result of an infinite number of elementary relaxation processes. Fourth, the fractional constitutive equation can also be expressed as an infinite sum of integer order derivatives of higher order which is also equivalent to a wave equation with higher order derivative terms.