# Acoustic attenuation

In acoustics, acoustic attenuation is a measure of the energy loss of sound propagation through an acoustic transmission medium. 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. This effect can be quantified through the Stokes's law of sound attenuation. Sound attenuation may also be a result of heat conductivity in the media as has been shown by G. Kirchhoff in 1868.[1][2] The Stokes-Kirchhoff attenuation formula takes into account both viscosity and thermal conductivity effects.

For heterogeneous 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.[3][4][5][6]

## 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:[7][8][9]

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

where ${\displaystyle \omega }$ is the angular frequency, P the pressure, ${\displaystyle \Delta x}$ the wave propagation distance, ${\displaystyle \alpha (\omega )}$ the attenuation coefficient, and ${\displaystyle \alpha _{0}}$ and the frequency-dependent exponent ${\displaystyle \eta }$ are real non-negative material parameters obtained by fitting experimental data; the value of ${\displaystyle \eta }$ ranges from 0 to 4. Acoustic attenuation in water is frequency-squared dependent, namely ${\displaystyle \eta =2}$. Acoustic attenuation in many metals and crystalline materials is frequency-independent, namely ${\displaystyle \eta =1}$.[10] In contrast, it is widely noted that the ${\displaystyle \eta }$ of viscoelastic materials is between 0 and 2.[7][8][11][12][13] For example, the exponent ${\displaystyle \eta }$ of sediment, soil, and rock is about 1, and the exponent ${\displaystyle \eta }$ of most soft tissues is between 1 and 2.[7][8][11][12][13]

The classical dissipative acoustic wave propagation equations are confined to the frequency-independent and frequency-squared dependent attenuation, such as the damped wave equation and the approximate thermoviscous wave equation. In recent decades, increasing attention and efforts have been focused on developing accurate models to describe general power law frequency-dependent acoustic attenuation.[8][11][14][15][16][17][18] 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.[19] The multiple relaxation model considers the power law viscosity underlying different molecular relaxation processes.[17] Szabo[8] 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.[18] Chen and Holm proposed the positive fractional derivative modified Szabo's wave equation[11] and the fractional Laplacian wave equation.[11] See [20] for a paper which compares fractional wave equations with model power-law attenuation. This book on power-law attenuation also covers the topic in more detail.[21]

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:

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

Such fractional derivative models are linked to the commonly recognized hypothesis that multiple relaxation phenomena (see Nachman et al.[17]) give rise to the attenuation measured in complex media. This link is further described in[22] and in the survey paper.[23]

For frequency band-limited waves, Ref.[24] describes a model-based method to attain causal power-law attenuation using a set of discrete relaxation mechanisms within the Nachman et al. framework.[17]

In porous fluid-saturated sedimentary rocks, such as sandstone, acoustic attenuation is primarily caused by the wave-induced flow of the pore fluid relative to the solid frame, with ${\displaystyle \eta }$ varying between 0.5 and 1.5. [25]

## References

1. ^ Kirchhoff, G. (1868). "Ueber den Einfluss der Wärmeleitung in einem Gase auf die Schallbewegung". Annalen der Physik und Chemie. 210 (6): 177–193. Bibcode:1868AnP...210..177K. doi:10.1002/andp.18682100602.
2. ^ Benjelloun, Saad; Ghidaglia, Jean-Michel (2020). "On the dispersion relation for compressible Navier-Stokes Equations". arXiv:2011.06394 [math.AP].
3. ^ 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.
4. ^ 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.
5. ^ 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.
6. ^ 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.
7. ^ a b c Szabo, Thomas L.; Wu, Junru (2000). "A model for longitudinal and shear wave propagation in viscoelastic media". The Journal of the Acoustical Society of America. 107 (5): 2437–2446. Bibcode:2000ASAJ..107.2437S. doi:10.1121/1.428630. PMID 10830366.
8. Szabo, Thomas L. (1994). "Time domain wave equations for lossy media obeying a frequency power law". The Journal of the Acoustical Society of America. 96 (1): 491–500. Bibcode:1994ASAJ...96..491S. doi:10.1121/1.410434.
9. ^ Chen, W.; Holm, S. (2003). "Modified Szabo's wave equation models for lossy media obeying frequency power law". The Journal of the Acoustical Society of America. 114 (5): 2570–4. arXiv:math-ph/0212076. Bibcode:2003ASAJ..114.2570C. doi:10.1121/1.1621392. PMID 14649993. S2CID 33635976.
10. ^ {{Knopoff, L. Rev. Geophys.|title = Q|year 1964| 2, 625–660| >
11. Chen, W.; 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): 1424–1430. Bibcode:2004ASAJ..115.1424C. doi:10.1121/1.1646399. PMID 15101619.
12. ^ a b Carcione, J. M.; Cavallini, F.; Mainardi, F.; Hanyga, A. (2002). "Time-domain Modeling of Constant- Q Seismic Waves Using Fractional Derivatives". Pure and Applied Geophysics. 159 (7–8): 1719–1736. Bibcode:2002PApGe.159.1719C. doi:10.1007/s00024-002-8705-z. S2CID 73598914.
13. ^ a b d'Astous, F.T.; Foster, F.S. (1986). "Frequency dependence of ultrasound attenuation and backscatter in breast tissue". Ultrasound in Medicine & Biology. 12 (10): 795–808. doi:10.1016/0301-5629(86)90077-3. PMID 3541334.
14. ^ a b Holm, Sverre; Näsholm, Sven Peter (2011). "A causal and fractional all-frequency wave equation for lossy media". The Journal of the Acoustical Society of America. 130 (4): 2195–2202. Bibcode:2011ASAJ..130.2195H. doi:10.1121/1.3631626. hdl:10852/103311. PMID 21973374.
15. ^ Pritz, T. (2004). "Frequency power law of material damping". Applied Acoustics. 65 (11): 1027–1036. doi:10.1016/j.apacoust.2004.06.001.
16. ^ Waters, K.R.; Mobley, J.; Miller, J.G. (2005). "Causality-imposed (Kramers-Kronig) relationships between attenuation and dispersion". IEEE Transactions on Ultrasonics, Ferroelectrics and Frequency Control. 52 (5): 822–823. doi:10.1109/TUFFC.2005.1503968. PMID 16048183. S2CID 23508424.
17. ^ a b c d Nachman, Adrian I.; Smith, James F.; Waag, Robert C. (1990). "An equation for acoustic propagation in inhomogeneous media with relaxation losses". The Journal of the Acoustical Society of America. 88 (3): 1584–1595. Bibcode:1990ASAJ...88.1584N. doi:10.1121/1.400317.
18. ^ a b Caputo, M.; Mainardi, F. (1971). "A new dissipation model based on memory mechanism". Pure and Applied Geophysics. 91 (1): 134–147. Bibcode:1971PApGe..91..134C. doi:10.1007/BF00879562. S2CID 121781575.
19. ^ Szabo, Thomas L. (13 November 2018). Diagnostic Ultrasound Imaging: Inside Out (Second ed.). Oxford: Academic Press. ISBN 9780123964878.
20. ^ Holm, Sverre; Näsholm, Sven Peter (2014). "Comparison of Fractional Wave Equations for Power Law Attenuation in Ultrasound and Elastography". Ultrasound in Medicine & Biology. 40 (4): 695–703. arXiv:1306.6507. doi:10.1016/j.ultrasmedbio.2013.09.033. PMID 24433745. S2CID 11983716.
21. ^ Holm, S. (2019). Waves with Power-Law Attenuation. Springer/Acoustical Society of America Press. ISBN 9783030149260.
22. ^ Näsholm, Sven Peter; Holm, Sverre (2011). "Linking multiple relaxation, power-law attenuation, and fractional wave equations". The Journal of the Acoustical Society of America. 130 (5): 3038–3045. Bibcode:2011ASAJ..130.3038N. doi:10.1121/1.3641457. hdl:10852/103312. PMID 22087931.
23. ^ Sven Peter Nasholm; Holm, Sverre (2012). "On a Fractional Zener Elastic Wave Equation". Fractional Calculus and Applied Analysis. 16: 26–50. arXiv:1212.4024. doi:10.2478/s13540-013-0003-1. S2CID 120348311.
24. ^ Näsholm, Sven Peter (2013). "Model-based discrete relaxation process representation of band-limited power-law attenuation". The Journal of the Acoustical Society of America. 133 (3): 1742–1750. arXiv:1301.5256. Bibcode:2013ASAJ..133.1742N. doi:10.1121/1.4789001. PMID 23464043. S2CID 22963787.
25. ^ Müller, Tobias M.; Gurevich, Boris; Lebedev, Maxim (September 2010). "Seismic wave attenuation and dispersion resulting from wave-induced flow in porous rocks — A review". Geophysics. 75 (5): 75A147–75A164. Bibcode:2010Geop...75A.147M. doi:10.1190/1.3463417. hdl:20.500.11937/35921.