Acoustic attenuation

Acoustic attenuation is a measure of the energy loss of sound propagation in media. Most media have viscosity, and are therefore not ideal media. When sound propagates in such media, there is always thermal consumption of energy caused by viscosity. For inhomogeneous media, besides media viscosity, acoustic scattering is another main reason for removal of acoustic energy. Acoustic attenuation in a lossy medium plays an important role in many scientific researches and engineering fields, such as medical ultrasonography, vibration and noise reduction.

Power-law frequency-dependent acoustic attenuation

Many experimental and field measurements show that the acoustic attenuation coefficient of a wide range of viscoelastic materials, such as soft tissue, polymers, soil and porous rock, can be expressed as the following power law with respect to frequency:

$P(x+\Delta x)=P(x)e^{-\alpha (\omega )\Delta x},\alpha (\omega )=\alpha _{0}\omega ^{\eta }$

where $\omega$  is the angular frequency, P the pressure, $\Delta x$  the wave propagation distance, $\alpha (\omega )$  the attenuation coefficient, $\alpha _{0}$  and frequency dependent exponent $\eta$  are real non-negative material parameters obtained by fitting experimental data and the value of $\eta$  ranges from 0 to 2. Acoustic attenuation in water, many metals and crystalline materials are frequency-squared dependent, namely $\eta =2$ . In contrast, it is widely noted that the frequency dependent exponent $\eta$  of viscoelastic materials is between 0 and 2. For example, the exponent $\eta$  of sediment, soil and rock is about 1, and the exponent $\eta$  of most soft tissues is between 1 and 2.

The classical dissipative acoustic wave propagation equations are confined to the frequency-independent and frequency-squared dependent attenuation, such as damped wave equation and approximate thermoviscous wave equation. In recent decades, increasing attention and efforts are focused on developing accurate models to describe general power law frequency-dependent acoustic attenuation. Most of these recent frequency-dependent models are established via the analysis of the complex wave number and are then extended to transient wave propagation. The multiple relaxation model considers the power law viscosity underlying different molecular relaxation processes. Szabo proposed a time convolution integral dissipative acoustic wave equation. On the other hand, acoustic wave equations based on fractional derivative viscoelastic models are applied to describe the power law frequency dependent acoustic attenuation. Chen and Holm proposed the positive fractional derivative modified Szabo's wave equation and the fractional Laplacian wave equation. See  for a paper which compares fractional wave equations which model power-law attenuation. This book on power-law attenuation also covers the topic in more detail.

The phenomenon of attenuation obeying a frequency power-law may be described using a causal wave equation, derived from a fractional constitutive equation between stress and strain. This wave equation incorporates fractional time derivatives:

${\nabla ^{2}u-{\dfrac {1}{c_{0}^{2}}}{\frac {\partial ^{2}u}{\partial t^{2}}}+\tau _{\sigma }^{\alpha }{\dfrac {\partial ^{\alpha }}{\partial t^{\alpha }}}\nabla ^{2}u-{\dfrac {\tau _{\epsilon }^{\beta }}{c_{0}^{2}}}{\dfrac {\partial ^{\beta +2}u}{\partial t^{\beta +2}}}=0.}$