Power-Law Wave Equations from Constitutive Equations

Abstract

The exponential response has been the typical medium response in the first part of this book. Examples are the creep compliances of the Kelvin–Voigt and Zener models ( 3.30) and ( 3.30) as well as the relaxation responses of the Maxwell and Zener models (( 3.37) and ( 3.45)). But in many complex media this model is too simple, and power laws are observed instead, which may both be in time-domain responses and in the frequency-domain. As the Fourier transform of a temporal power law is itself a power law, these observations are equivalent. The first section therefore starts by reviewing empirical observations of power laws in the frequency and time domains.