1 Introduction

1.1 Conservation laws vs constitutive equations

1.2 Conservation principles

1.3 Hookean and Newtonian medium models

1.4 Constitutive equations

1.4.1 Spring damper models

1.4.2 Exponential time responses

1.4.3 Power laws in frequency and time

1.5 Wave equations with power law solutions

1.5.1 Fractional wave equations

1.5.2 Fractal media and power law attenuation

1.5.3 Porous media

1.6 Layout

2 Classical wave equations

2.1 The lossless wave equation

2.1.1 Monochromatic plane wave

2.2 Lossless wave equations in practice

2.2.1 Acoustics

2.2.2 Elastic waves

2.2.3 Electromagnetics

2.3 Characterization of attenuation

2.3.1 Dispersion relation

2.3.2 Q, loss tangent, log decrement, and penetration depth

2.4 Viscous losses: The Kelvin-Voigt model

2.4.1 Viscous wave equation and the dispersion equation

2.4.2 Low frequency wave equation

2.5 The Zener constitutive equation

2.5.1 Wave equation

2.5.2 Dispersion relation and compressibility/compliance

2.5.3 Asymptotes

2.6 Relaxation and multiple relaxation

2.6.1 The relaxation model

2.6.2 Multiple relaxation

2.6.3 Multiple relaxation: Seawater and air

2.6.4 Higher order constitutive equations

2.6.5 Arbitrary attenuation from multiple relaxation

2.7 The Maxwell mechanical model

2.8 Losses in electromagnetics

2.8.1 A conducting medium

2.8.2 Debye dielectrics

2.8.3 Multiple Debye terms

3 Models of Linear Viscoelasticity

3.1 Constitutive equations

3.1.1 Relaxation modulus and creep compliance

3.1.2 Linear differential equation model

3.1.3 The causal fading memory model

3.1.4 Complete monotonicity

3.1.6 Spring damper model

3.2 Standard spring damper models

3.2.1 Spring and dashpot elements

3.2.2 Kelvin-Voigt model

3.2.3 Maxwell model

3.2.4 The standard linear solid

3.2.5 Higher order models

3.3 Four categories of models

3.4 Completely monotone models

3.4.1 Global vs. local passivity

3.4.2 Special role of completely monotone models

3.5 Fractional models

3.5.1 Fractional Kelvin-Voigt model

3.5.2 Fractional Zener model

3.5.3 Fractional Maxwell model

3.5.4 Fractional Newton (Scott-Blair) model

4.1 Generalization of the low-frequency wave equation

4.2 Causality

4.2.1 Impulse response and transfer function

4.2.2 Kramers-Kronig relations

4.3 Generalization of the viscous wave equation

4.3.1 Fractional temporal derivative

4.3.2 Fractional Laplacian loss term

4.3.3 Fractional biharmonic operator

4.4 Fractional diffusion-wave equation

4.5 Four term fractional wave equations

4.5.1 Fractional Zener wave equation

4.5.2 Constant power law for all frequencies

4.6 Power law solutions

5 Physically valid viscoelastic wave equations

5.1.1 Wavenumber as a function of relaxation modulus

5.1.2 Bernstein property

5.1.3 Consequences of the Bernstein property

5.1.4 Asymptotic properties

5.2 Viability of two viscous wave equations

5.3 Does the viscous model represent realistic media?

5.3.1 The Navier-Stokes equation

6 Wave equations from fractional constitutive equations

6.1 The fractional Kelvin-Voigt equation

6.1.1 Dispersion relation

6.1.
About the Author:

Sverre Holm was born in Oslo, Norway, in 1954. He received M.S. and Ph.D. degrees in electrical engineering from the Norwegian Institute of Technology (NTNU), Trondheim in 1978 and 1982, respectively.

He has academic experience from NTNU and Yarmouk University in Jordan (1984-86). Since 1995 he has been a professor of signal processing and acoustic imaging at the University of Oslo. In 2002 he was elected a member of the Norwegian Academy of Technological Sciences.

His industry experience includes GE Vingmed Ultrasound (1990-94), working on digital ultrasound imaging, and Sonitor Technologies (2000-05), where he developed ultrasonic indoor positioning. He is currently involved with several startups in the Oslo area working in the areas of acoustics and ultrasonics.