Measure of energy loss as sound waves propagate through a medium
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]
where is the angular frequency, P the pressure, the wave propagation distance, the attenuation coefficient, and and the frequency-dependent exponent are real non-negative material parameters obtained by fitting experimental data; the value of ranges from 0 to 4. Acoustic attenuation in water is frequency-squared dependent, namely . Acoustic attenuation in many metals and crystalline materials is frequency-independent, namely .[10] In contrast, it is widely noted that the of viscoelastic materials is between 0 and 2.[7][8][11][12][13] For example, the exponent of sediment, soil, and rock is about 1, and the exponent 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:
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 varying between 0.5 and 1.5.
[25]
^
abcSzabo, 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.
PMID10830366.
^
abcdeSzabo, 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.
^{{Knopoff, L. Rev. Geophys.|title = Q|year 1964| 2, 625–660|
>
^
abcdeChen, 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.
PMID15101619.
^
abCarcione, 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.
S2CID73598914.
^
abd'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.
PMID3541334.
^Pritz, T. (2004). "Frequency power law of material damping". Applied Acoustics. 65 (11): 1027–1036.
doi:
10.1016/j.apacoust.2004.06.001.
^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.
PMID16048183.
S2CID23508424.
^
abcdNachman, 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.