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.[1][2][3][4]
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:[5][6][7]

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

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.[5][6][8][9][10] 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.[5][6][8][9][10]

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.[6][8][11][12][13][14][15] 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.[16] The multiple relaxation model considers the power law viscosity underlying different molecular relaxation processes.[14] Szabo[6] 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.[15] Chen and Holm proposed the positive fractional derivative modified Szabo's wave equation[8] and the fractional Laplacian wave equation.[8] See [17] 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.[18]

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:

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

See also[11] and the references therein.

Such fractional derivative models are linked to the commonly recognized hypothesis that multiple relaxation phenomena (see Nachman et al.[14]) give rise to the attenuation measured in complex media. This link is further described in[19] and in the survey paper.[20]

For frequency band-limited waves, Ref.[21] describes a model-based method to attain causal power-law attenuation using a set of discrete relaxation mechanisms within the Nachman et al. framework.[14]
See also

Absorption (acoustics)
Fractional calculus


Chen, Yangkang; Ma, Jitao (May–June 2014). "Random noise attenuation by f-x empirical-mode decomposition predictive filtering". Geophysics. 79 (3): V81–V91. Bibcode:2014Geop...79...81C. doi:10.1190/GEO2013-0080.1.
Chen, Yangkang; Zhou, Chao; Yuan, Jiang; Jin, Zhaoyu (2014). "Application of empirical mode decomposition in random noise attenuation of seismic data". Journal of Seismic Exploration. 23: 481–495.
Chen, Yangkang; Zhang, Guoyin; Gan, Shuwei; Zhang, Chenglin (2015). "Enhancing seismic reflections using empirical mode decomposition in the flattened domain". Journal of Applied Geophysics. 119: 99–105. Bibcode:2015JAG...119...99C. doi:10.1016/j.jappgeo.2015.05.012.
Chen, Yangkang (2016). "Dip-separated structural filtering using seislet transform and adaptive empirical mode decomposition based dip filter". Geophysical Journal International. 206 (1): 457–469. Bibcode:2016GeoJI.206..457C. doi:10.1093/gji/ggw165.
Szabo T. L., and Wu J., 2000, “A model for longitudinal and shear wave propagation in viscoelastic media,” J. Acoust. Soc. Am., 107(5), pp. 2437-2446.
Szabo T. L., 1994, “Time domain wave equations for lossy media obeying a frequency power law,” J. Acoust. Soc. Am., 96(1), pp. 491-500.
Chen W., and Holm S., 2003, “Modified Szabo’s wave equation models for lossy media obeying frequency power law,” J. Acoust. Soc. Am., 114(5), pp. 2570-2574.
Chen W., and Holm S., 2004, “Fractional Laplacian time-space models for linear and nonlinear lossy media exhibiting arbitrary frequency power-law dependency,” The Journal of the Acoustical Society of America, 115(4), pp. 1424-1430.
Carcione J. M., Cavallini F., Mainardi F., and Hanyga A., 2002, “Time-domain Modeling of Constant-Q Seismic Waves Using Fractional Derivatives,” Pure appl. geophys., 159, pp. 1719-1736.
D’astrous F. T., and Foster F. S., 1986, “Frequency dependence of ultrasound attenuation and backscatter in breast tissue,” Ultrasound Med. Biol., 12(10), pp. 795-808.
Holm S., and Näsholm S. P., 2011, "A causal and fractional all-frequency wave equation for lossy media," The Journal of the Acoustical Society of America, 130(4), pp. 2195-2201.
Pritz T., 2004, “Frequency power law of material damping,” Applied Acoustics, 65, pp. 1027-1036.
Waters K. R., Mobley J., and Miller J. G., 2005, “Causality-Imposed (Kramers-Kronig) Relationships Between Attenuation and Dispersion,” IEEE Trans. Ultra. Ferro. Freq. Contr., 52(5), pp. 822-833.
Nachman A. I., Smith J. F., and Waag R. C., 1990, “An equation for acoustic propagation in inhomogeneous media with relaxation losses,” J. Acoust. Soc. Am., 88(3), pp. 1584-1595.
Caputo M., and Mainardi F., 1971, “A new dissipation model based on memory mechanism,” Pure and Applied Geophysics, 91(1), pp. 134-147.
Thomas L. Szabo, 2004, Diagnostic ultrasound imaging, Elsevier Academic Press.
Holm S., Näsholm, S. P., "Comparison of Fractional Wave Equations for Power Law Attenuation in Ultrasound and Elastography," Ultrasound Med. Biol., 40(4), pp. 695-703, DOI: 10.1016/j.ultrasmedbio.2013.09.033 Link to e-print
Holm, S. (2019). Waves with Power-Law Attenuation. Springer and Acoustical Society of America Press.
S. P. Näsholm and S. Holm, "Linking multiple relaxation, power-law attenuation, and fractional wave equations," Journal of the Acoustical Society of America, Volume 130, Issue 5, pp. 3038-3045 (November 2011).
S. P. Näsholm and S. Holm, "On a Fractional Zener Elastic Wave Equation," Fract. Calc. Appl. Anal. Vol. 16, No 1 (2013), pp. 26-50, DOI: 10.2478/s13540-013--0003-1 Link to e-print
S. P. Näsholm: "Model-based discrete relaxation process representation of band-limited power-law attenuation." J. Acoust. Soc. Am. Vol. 133, Issue 3, pp. 1742-1750 (2013) DOI: 10.1121/1.4789001 Link to e-print

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