© 2019

Waves with Power-Law Attenuation


Table of contents

  1. Front Matter
    Pages i-xxxvii
  2. Sverre Holm
    Pages 1-21
  3. Acoustics and Linear Viscoelasticity

    1. Front Matter
      Pages 23-23
    2. Sverre Holm
      Pages 25-66
    3. Sverre Holm
      Pages 67-93
  4. Modeling of Power-Law Media

    1. Front Matter
      Pages 117-117
    2. Sverre Holm
      Pages 161-172
    3. Sverre Holm
      Pages 225-258
    4. Sverre Holm
      Pages 259-272
  5. Back Matter
    Pages 273-312

About this book


This book integrates concepts from physical acoustics with those from linear viscoelasticity and fractional linear viscoelasticity. Compressional waves and shear waves in applications such as medical ultrasound, elastography, and sediment acoustics often follow power law attenuation and dispersion laws that cannot be described with classical viscous and relaxation models. This is accompanied by temporal power laws rather than the temporal exponential responses of classical models.

The book starts by reformulating the classical models of acoustics in terms of standard models from linear elasticity. Then, non-classical loss models that follow power laws and which are expressed via convolution models and fractional derivatives are covered in depth. In addition, parallels are drawn to electromagnetic waves in complex dielectric media. The book also contains historical vignettes and important side notes about the validity of central questions. While addressed primarily to physicists and engineers working in the field of acoustics, this expert monograph will also be of interest to mathematicians, mathematical physicists, and geophysicists.

  • Couples fractional derivatives and power laws and gives their multiple relaxation process interpretation
  • Investigates causes of power law attenuation and dispersion such as interaction with hierarchical models of polymer chains and non-Newtonian viscosity
  • Shows how fractional and multiple relaxation models are inherent in the grain shearing and extended Biot descriptions of sediment acoustics
  • Contains historical vignettes and side notes about the formulation of some of the concepts discussed


Power laws acoustics Linear viscoelasticity Elastic waves acoustics Ultrasound mathematical basis Fractional linear viscoelasticity Wave propagation and attenuation Wave equations

Authors and affiliations

  1. 1.Department of InformaticsUniversity of OsloOsloNorway

About the authors

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.

Dr. Holm has authored or co-authored around 220 publications and holds 12 patents. He has spent sabbaticals at GE Global Research, NY (1998), Institut Langevin, ESPCI, Paris (2008-09), and King’s College London (2014). His research interests include medical ultrasound imaging, elastography, modeling of waves in complex media, and ultrasonic positioning.

Bibliographic information

Industry Sectors


“The book is interesting if you want to enter the field of fractional rheology. This is an actual topic of research that may be quite promising and this book is a helpful tool to take a first step in the subject. In my opinion it is well written and it is highly readable with technical details under control. I have highly appreciated the idea of integrating several ideas from physical acoustics with those from linear viscoelasticity.” (Giuseppe Saccomandi, Mathematical Reviews, November, 2019)