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Waves with Power-Law Attenuation

Sverre Holm
Publication Date: 
Number of Pages: 
[Reviewed by
Bill Satzer
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An acoustic wave propagating through a medium can incur losses due to both attenuation and dispersion. Attenuation is a frequency-dependent reduction of the amplitude of a wave whereas dispersion is a frequency-dependent change in its velocity. This monograph focuses on attenuation in its many aspects. In particular, it examines how various media affect the frequency dependence and how this effect can be modeled. In the simplest cases, the attenuation is proportional to the square of the frequency. For many materials of practical interest, the attenuation follows a power law with an exponent between 0 and 2.
One of the author’s original motivations was to understand better the attenuation of sound in human tissue both for medical ultrasound and elastography. Medical ultrasound is based on pressure or compressional waves with frequencies of 2-15 MHz with attenuation that follows a power law with an exponent from 1 to 1.5. In elastography, shear waves of frequency 30-1000 Hz probe the body and emulate the sensitivity of palpation with one’s hand. Shear waves also have power-law attenuation, and the characteristics of their attenuation can make them more suitable for the classification of tissue than medical ultrasound.
The author begins by discussing various forms of classical wave equations in the framework of linear elasticity. The two main mechanisms of attenuation are due to viscosity and relaxation of the medium. The medium – and in particular its constitutive properties – get a great deal of attention throughout the book. Those constitutive properties are of great importance to the frequency dependence of the attenuation.
A consequence of the power-law dependence of attenuation on frequency with a power that need not an integer is that – moving from the frequency domain to the time domain by Fourier transform – the time derivatives in the wave equation can be fractional derivatives. The author spends a good deal of time discussing what this means and how it is related to the constitutive properties of the medium. This leads to extended descriptions of medium models.
This is a book that is likely to be of primary interest to people working in acoustics, especially in medical applications. It provides a modest amount of introductory material and some historical background, but it would be challenging reading for those without some prior work in acoustics. For applied mathematicians interested in various forms of the wave equation, this provides insight into the complicated relationship between modeling and understanding the critical (often empirically-derived) properties of the media through which the wave travels. The book gives the very strong impression that the author wrote it because it gave him the opportunity to work through and make sense of the details and relate it to a prior body of work. The author had discovered that many of the results he knew about acoustic and elastic wave propagation were already well known in the field of linear viscoelasticity. In part, he wrote this book to make that connection clear.


Bill Satzer (, now retired from 3M Company, spent most of his career as a mathematician working in industry on a variety of applications ranging from speech recognition and network modeling to optical films and material science. He did his PhD work in dynamical systems and celestial mechanics.